Answer: 314 square meters
Step-by-step explanation:
The formula for the area of a circle is given by A = πr^2, where r is the radius of the circle. Since the diameter of the circular patch is given as 20 meters, the radius would be half of that or 10 meters.
So, using the formula, we can calculate the area of the circular patch as follows:
A = πr^2
A = π(10)^2
A = 3.14(100)
A = 314 square meters
Therefore, the area of the circular patch is 314 square meters.
For each of the following propositions, either i. use a case-based proof to demonstrate that the proposition holds true or ii. Use a counterexample to demonstrate the proposition does not hold.
(a) Assume x is an integer that is not divisible by 3, and y is an integer that is not divisible by 3. Then the sum of x and y cannot be divisible by 3.
(b) Assume x is an integer that is not divisible by 3, and y is an integer that is divisible by 3. Then the sum of x and y cannot be divisible by 3.
In both cases, the sum of x and y is not divisible by 3, we have demonstrated that the proposition is true. and the proposition is false, and we have shown a counterexample where the sum of two integers, one of which is not divisible by 3 and the other is divisible by 3, can be divisible by 3.
(a) To prove that the sum of two integers, x and y, neither of which is divisible by 3, cannot be divisible by 3, we can use a case-based proof.
Case 1: x and y leave a remainder of 1 when divided by 3.
Let x = 3m + 1 and y = 3n + 1, where m and n are integers. Then, the sum of x and y is 3m + 3n + 2, which leaves a remainder of 2 when divided by 3. Therefore, x + y is not divisible by 3.
Case 2: x and y leave a remainder of 2 when divided by 3.
Let x = 3m + 2 and y = 3n + 2, where m and n are integers. Then, the sum of x and y is 3m + 3n + 4, which leaves a remainder of 1 when divided by 3. Therefore, x + y is not divisible by 3.
Since in both cases, the sum of x and y is not divisible by 3, we have demonstrated that the proposition is true.
(b) To prove that the sum of two integers, x and y, where x is not divisible by 3 and y is divisible by 3, cannot be divisible by 3, we can use a counterexample.
Let x = 2 and y = 6. Then, x is not divisible by 3 and y is divisible by 3. However, x + y = 8, which is not divisible by 3.
Therefore, the proposition is false, and we have shown a counterexample where the sum of two integers, one of which is not divisible by 3 and the other is divisible by 3, can be divisible by 3.
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A screen has a zoom of 140%, which means that images on the screen are 140% as long and 140% as wide as when they are printed on a sheet of paper. An image of a house is 17 cm tall when printed on a sheet of paper. How tall would the image of the house be on the screen? Give your answer in centimetres (cm).
Answer:
23.8 cm
Step-by-step explanation:
17 * 140% = 17 * 1.4 = 23.8 cm
The image of the house would be 23.8 cm tall on the screen.
To calculate the height of the image of the house on the screen, we can use the given zoom factor of 140%.
The zoom factor of 140% means that the images on the screen are 140% as long and 140% as wide compared to when they are printed on a sheet of paper.
To calculate the height of the image on the screen, we need to multiply the printed height by the zoom factor (140% or 1.4).
Height on the screen = Printed height * Zoom factor
Height on the screen = 17 cm * 1.4
Height on the screen = 23.8 cm
Therefore, the image of the house would be 23.8 cm tall on the screen.
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Cody has $7 dollars. he wants to buy at least 4 snacks. Hot dogs (x) and $2 each. Peanuts (y) are $1 each. which ordered pair is a solution
Since we can't find an ordered pair (x, y) that satisfies all the conditions, there is no solution to this problem.
What is equation?An equation is a mathematical statement that asserts the equality of two expressions. It typically contains variables, coefficients, and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. Equations can be used to model relationships between variables, solve real-world problems, and make predictions.
Here,
Let's start by defining the variables:
x: number of hot dogs
y: number of peanuts
We need to find an ordered pair (x, y) that satisfies the following conditions:
x and y are both integers
x is greater than or equal to 0
y is greater than or equal to 0
2x + y ≤ 7 (total cost of snacks can't exceed $7)
x ≥ 4 (at least 4 snacks)
We can use trial and error to find a suitable ordered pair. Let's start with x = 4 and see if we can find a corresponding y value that satisfies the conditions:
If x = 4, then the total cost of hot dogs is 4 * $2 = $8.
We need to spend no more than $7, so we have $7 - $8 = -$1 left for peanuts.
Since we can't spend a negative amount of money, there is no solution for x = 4.
Let's try x = 5:
If x = 5, then the total cost of hot dogs is 5 * $2 = $10.
We have $7 - $10 = -$3 left for peanuts, so there is no solution for x = 5 either.
Finally, let's try x = 6:
If x = 6, then the total cost of hot dogs is 6 * $2 = $12.
We have $7 - $12 = -$5 left for peanuts, so there is no solution for x = 6 either.
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Complete question:
Cody has $7 dollars. he wants to buy at least 4 snacks. Hot dogs (x) and $2 each. Peanuts (y) are $1 each. Find the solution for this question of equation?
if cos is the 4 quadrant
The exact value of sin∅ if cos∅ =3/8 and in the fourth quadrant is 292⁰
What are the angles in the 4th quadrant?We should recall that the the angles of the 4th quadrant range from 270° to 360°. that is to say that in the fourth quadrant, 270≤x≤360
The given angle is cos∅ =3/8
cos∅ = 0.375
Find the angle we take the cos inverse
That ∅ = Cos⁻¹0.375
∅ = 68⁰
In the fourth quadrant, cos is positive that 360-68 = 292⁰
Then the exact value of sin is 292⁰
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Lori is moving and must rent a truck. There is an initial charge of $60 for the rental plus an additional fee per mile driven. Would a linear, quadratic or exponential function be the best type of equation to model this function? Exponential Quadratic Linear
Answer:
A linear function would be the best type of equation to model this situation. The total cost of renting the truck increases linearly with the number of miles driven. The initial charge of $60 can be considered as the y-intercept of the linear function, and the additional fee per mile driven can be considered as the slope of the line. Therefore, the equation that models this situation can be written in the form y = mx + b, where y is the total cost of renting the truck, x is the number of miles driven, m is the additional fee per mile driven (the slope of the line), and b is the initial charge of $60 (the y-intercept).
Answer:
A linear function would be the best type of equation to model this function.
Step-by-step explanation:
The total cost of renting the truck is composed of two parts:
Initial charge of $60.Additional fee per mile driven.The initial charge of $60 is the fixed charge, and the additional fee is the variable charge that is proportional to the number of miles driven.
Let "x" be the number of miles driven and "y" be the total cost of the rental (in dollars), then the linear equation is:
y = mx + 60
where "m" is the additional fee (in dollars) per mile driven.
Therefore, a linear function, in the form y = mx + b, where m represents the slope or rate of change, and b represents the initial fixed charge, is the most appropriate function to model this situation.
What triangles are similar to triangle ABC?
Answer:
A right angled triangle is similar to triangle ABC
Step-by-step explanation:
If you tilt the triangle and put it straight, you'll see that angle C is equal to 90°
And if a triangle has one angle of 90° then it is a right angled triangle
Hope you understand :)
Annie is concerned over a report that "a woman over age 40 has a better chance of being killed by a terrorist than of getting married." A study found that the likelihood of marriage for a never-previously-wed, 40 -year-old university-educated American woman was 2.5% . To demonstrate that this percentage is too small, Annie uses her resources at the Baltimore Sun to conduct a simple random sample of 546 never-previously-wed, university-educated, American women who were single at the beginning of their 40 s and who are now 45 . Of these women, 20 report now being married. Does this evidence support Annie’s claim, at the 0.01 level of significance, that the chances of getting married for this group is greater than 2.5% ? Step 1 of 3 : State the null and alternative hypotheses for the test. Fill in the blank below. H0Ha: p=0.025: p⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯0.025
Due to the directed nature of the alternative hypothesis, a one-tailed test is being used (greater than).
what is null hypothesis ?The null hypothesis, which is an assertion or assumption that there is no significant difference or association between two or more variables or populations, is used in statistical hypothesis testing. It is frequently indicated by the letter H0 and is typically the hypothesis that is tested against a competing hypothesis. The objective of the hypothesis test is to either reject or fail to reject the null hypothesis based on the evidence or data seen. The null hypothesis serves as the default or baseline assumption. If the alternative hypothesis is supported by evidence, the null hypothesis is likely to be rejected.
given
The test's null and alternate hypotheses are as follows:
H0: p 0.025 (The percentage of American women with university educations who had never previously been married at the start of their 40s and are now 45 and married is less than or equal to 2.5%)
Ha: p > 0.025 (More than 2.5% of American women with college degrees who were unmarried at the start of their 40s and are now 45 and married are never before married).
Due to the directed nature of the alternative hypothesis, a one-tailed test is being used (greater than).
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Consider the function h(x) = a(−2x + 1)^5 − b, where a does not=0 and b does not=0 are constants.
A. Find h′(x) and h"(x).
B. Show that h is monotonic (that is, that either h always increases or remains constant or h always decreases or remains constant).
C. Show that the x-coordinate(s) of the location(s) of the critical points are independent of a and b.
Answer:
A. To find the derivative of h(x), we can use the chain rule:
h(x) = a(-2x + 1)^5 - b
h'(x) = a * 5(-2x + 1)^4 * (-2) = -10a(-2x + 1)^4
To find the second derivative, we can again use the chain rule:
h''(x) = -10a * 4(-2x + 1)^3 * (-2) = 80a(-2x + 1)^3
B. To show that h is monotonic, we need to show that h'(x) is either always positive or always negative. Since h'(x) is a multiple of (-2x + 1)^4, which is always non-negative, h'(x) is always either positive or negative depending on the sign of a. If a > 0, then h'(x) is always negative, which means that h(x) is decreasing. If a < 0, then h'(x) is always positive, which means that h(x) is increasing.
C. To find the critical points, we need to find where h'(x) = 0:
h'(x) = -10a(-2x + 1)^4 = 0
-2x + 1 = 0
x = 1/2
Thus, the critical point is at x = 1/2. This value is independent of a and b, as neither a nor b appear in the calculation of the critical point.
5. Which of the graphs below illustrates water boiling in Denver, Colorado?
Your question is incomplete. The complete question is: Which of the graphs below illustrates water boiling in Denver, Colorado? (Altitude 1,600 meters.)
Answer:
The graphs that come with this question are in the picture attached.The answer is graph identified with the letter A.Explanation:
The normal boiling point of water is 100°C. That is the temperature at which water boils when the atmospheric pressure is 1 atm, i.e. at sea level.
The liquids boil when its vapor pressure equals the atmospheric pressure; so the higher the atmospheric pressure the higher the boiling point, and the lower the atmospheric pressure the lower the boiling point.
Since, it is stated that the altitude of Denver, Colorado is 1,600 m, the atmospheric pressure (the pressure exerted by the column of air above the place) is lower than 1 atm (atmospheric pressure at sea level).
Hence, water boiling point in Denver is lower than 100°C.
The graphs shown represent the temperature (T °C) as water is heated. Since when liquids boil their temperature remains constant during all the phase change, the flat portion of the graph represents the time during which the substance is boiling.
In the graph A, the flat portion is below 100°C; in the graph B, the flat portion is at 100 °C; in the graph C the flat part is above 100ªC, and, in graph D, there is not flat part. Then, the only graph that can illustrate water boiling in Denver, Colorado is the graph A.
Compare your answer with the average density of the giant's envelope, if it has a 0.5 solar mass and its radius is 0.6 AU . Express your answer using two significant figures.
The required average density of the giant envelope is equal to 3.30 x 10^9 kg/m^3.
Average density of the giant's envelope, required the mass and volume of the envelope.
Let us assume that the giant is a sphere with a radius of 0.6 AU,
the volume of the envelope is ,
V = (4/3)πr³
= (4/3)π(0.6 AU)³
= (4/3)(3.14)(0.6 × 149.6 ×10⁶ )³
To convert this volume to units of cubic meters,
Multiply by the conversion factor by,
1 AU = 149.6 x 10⁶m
⇒1 AU³ = (149.6 x 10⁶ km)³
⇒1 AU³ = 3.35 x 10³³ m³
Now ,calculate the density of the giant's envelope using the formula,
density = mass / volume
Giant's envelope has a mass of 0.5 solar masses,
which is equivalent to,
M = 0.5 x 1.99 x 10³⁰ kg
= 9.95 x 10²⁹ kg
Average density of the giant's envelope is equals to,
⇒ density = 9.95 x 10²⁹ kg / (4/3)(3.14)(0.6 × 149.6 ×10⁶ )³
= 3.30 x 10^9 kg/m^3
Therefore, the average density of the giant's envelope is approximately 3.30 x 10^9 kg/m^3.
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the answer is supposed to be 799 & 7188, but I don't know how to get there
Doing simple algebra with the two summations we can see that:
[tex]\Sigma x = 799\\\\\Sigma x^2 = 7,188[/tex]
How to find the values of the two summations?First, we know that there are 97 passengers, and we know that the summation:
[tex]\Sigma (x - 5) = 314[/tex]
Where x represents the weights.
Then we can rewrite that sum as:
[tex]\Sigma (x - 5) = \Sigma x - 97*5[/tex]
And replace that in the original equation to get:
[tex]\Sigma x - 97*5 = 314\\\Sigma x = 314 + 5*97 = 799[/tex]
So that is the first summation, now let's get the second one, we can rewrite the summation as:
[tex]\Sigma (x - 5)^2 = \Sigma x^2 - 10x +25 \\\\\Sigma x^2 - \Sigma 10x + \Sigma 25[/tex]
Where remember we have 97 terms, and the summation is equal to 1623, then:
[tex]\Sigma x^2 - \Sigma 10x + 25*97 = 1623[/tex]
Now we can replace the second term by the thing we found earlier:
[tex]\Sigma x^2 - 10\Sigma x + 25*97 = 1623\\\\\Sigma x^2 - 10*799 + 25*97 = 1623\\\\\Sigma x^2 = 1623 + 10*799 - 25*97\\\\\Sigma x^2 = 7,188[/tex]
That is the answer.
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Complete the proof of the identity by choosing the Rule that justifies each step.
To see a detailed description of a Rule, select the More Information Button to the right of the Rule.
Statement Rule 1. Algebra
cosx/sinx (sec*2x - 1) Rule ? 2. Quotient
cosx/sinx (tan*2x) Rule ? 3. Pythagorean
cosx/sinx (sin*2x/cos*2x) Rule ? 4. Odd/Even
sinx/cosx Rule ? 5. Reciprocal
tanx Rule ?
Find the rules answers towards each statement
All rules towards each statement shown in table below.
Define the term trigonometry?The relationships between the sides and angles of triangles are the subject of the mathematical discipline of trigonometry. It entails the study of trigonometric functions like sine, cosine, and tangent, which connect a triangle's angles and side lengths.
Rules towards each statement:
Statements
Rule1. Cos x/ Sin x (Sec 2x - 1) Pythagorean identity
Rule2. Cos x/Sin x (Tan 2x) Algebra
Rule3. Cos x/Sin x (Sin 2x/Cos 2x) Reciprocal
Rule4. Sin x/Cos x Odd/ Even
Rule5. Tan x Quotient
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As per trigonometric functions, rules towards each statement are:
Rules towards each statement:
Statements
Rule1. Cos x/ Sin x (Sec 2x - 1) Pythagorean identity
Rule2. Cos x/Sin x (Tan 2x) Algebra
Rule3. Cos x/Sin x (Sin 2x/Cos 2x) Reciprocal
Rule4. Sin x/Cos x Odd/ Even
Rule5. Tan x Quotient
What exactly does trigonometry mean?Trigonometry is a branch of mathematics that focuses on the relationships between triangles' sides and angles. It requires studying the trigonometric functions that link the angles and side lengths of a triangle, such as sine, cosine, and tangent.
Rules towards each statement:
Statements
Rule1. Cos x/ Sin x (Sec 2x - 1) Pythagorean identity
Rule2. Cos x/Sin x (Tan 2x) Algebra
Rule3. Cos x/Sin x (Sin 2x/Cos 2x) Reciprocal
Rule4. Sin x/Cos x Odd/ Even
Rule5. Tan x Quotient
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The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 385 grams and a standard deviation of 8 grams find the weight that corresponds to each event(use excel or appendix c to calculate the z value round your final answers to 2 decimal places
The weight that corresponds to the highest 5% is also approximately 398.12 grams.
What is Z-Score?
A score's connection to the mean within a group of scores is statistically measured by a Z-Score.
To find the weight that corresponds to each event, we need to use the standard normal distribution and convert each value to a z-score using the formula:
z = (x - μ) / σ
Here are the calculations for each event:
The weight that corresponds to the 25th percentile:
-0.68 = (x - 385) / 8
Solving for x gives:
x = 379.44 grams (rounded to two decimal places)
Therefore, the weight that corresponds to the 25th percentile is approximately 379.44 grams.
The weight that corresponds to the 95th percentile. we find that the z-score is approximately 1.64 (rounded to two decimal places). Then we can use the formula above to solve for x:
1.64 = (x - 385) / 8
x = 398.12 grams (rounded to two decimal places)
Therefore, the weight that corresponds to the 95th percentile is approximately 398.12 grams.
The weight that corresponds to the highest 5%:
1.64 = (x - 385) / 8
x = 398.12 grams (rounded to two decimal places)
Therefore, the weight that corresponds to the highest 5% is also approximately 398.12 grams.
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I need help with this
Answer:
D
Step-by-step explanation:
when an angle is supplementary to another angle, it means that both of the angles added together equal 180 degrees.
120+60=180.
Three dice are rolled. What is the probability of getting the sum as 13?
When three dices are rolled.
Total number of outcomes = = 216
Sum of 13 can be achieved in the following ways:
From the digits 6,4,3
So, there are 3! ways = = 6
From the digits 6,2,5
So, there are 3! ways = = 6
From the digits 5,4,4
So, there are ways = 3
From the digits 6,6,1
So, there are ways = 3
From the digits 3,5,5
So, there are ways = 3
So, total numbers whose sum is 13=
So, Probability = .
Therefore, the probability of getting sum as 21 on rolling three dice = .
The roof on a house requires that every 2 yards gets covered by 3 shingles. You currently have 60 boxes that contain 120 shingles each. The roof of the house is estimated at 4500 yards that must be covered. Which sentence best describes the amount of shingles needed?
To cover the house roof, as we only need 6,750 shingles and we have 7,200 shingles available.
What are arithmetic operations ?
Arithmetic operations are basic mathematical operations used to perform calculations involving numbers. The four basic arithmetic operations are:
Addition: This operation involves combining two or more numbers to get a total or sum. The symbol used for addition is "+".Subtraction: This operation involves finding the difference between two numbers. The symbol used for subtraction is "-".Multiplication: This operation involves finding the product of two or more numbers. The symbol used for multiplication is "×" or "*".Division: This operation involves dividing a number into equal parts or finding how many times one number fits into another. The symbol used for division is "÷" or "/".According to the question:
To determine the amount of shingles needed to cover the roof of the house, we can use the fact that every 2 yards requires 3 shingles. Therefore, for 4500 yards, we need to divide by 2 and then multiply by 3 to get the total number of shingles needed.
(4500 yards) / (2 yards/2) * (3 shingles/2 yards) = 6,750 shingles
Since we have 60 boxes that contain 120 shingles each, we can calculate the total number of shingles we have:
60 boxes * 120 shingles per box = 7,200 shingles
Therefore, we have more than enough shingles to cover the roof, as we only need 6,750 shingles and we have 7,200 shingles available.
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Solve each of the following equations for the indicated variable f = (1+i)"-1 for i
Answer:
If i = -1 then the answer is f = 0
0 = ( 1 + -1 ).
f equaling 0 makes the statement true 0=0
Step-by-step explanation:
an adjusted r-squared value of 0 represents no ability of the model to explain the dependent variable.
An adjusted R-squared value of 0 indicates that the model has no ability to explain the variation in the dependent variable using the independent variables included in the model.
In other words, the model does not fit the data well and cannot make accurate predictions. An adjusted R-squared value of 1 represents a perfect fit, where the model explains all of the variation in the dependent variable using the independent variables. However, it is important to consider other factors such as the sample size, the quality of the data, and the appropriateness of the model to make valid conclusions about the model's ability to explain the dependent variable.
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The area of shape A is 3cm2 what is the area of shape B?
28.5cm^2 is the area of shape B.
What is area?A solid object's surface area is a measurement of the total area that the surface of the object takes up.
The definition polyhedra of arc length for one-dimensional curves and the definition of surface area for (i.e., objects with flat polygonal faces), where the surface area is the sum of the areas of its faces, are both much simpler mathematical concepts than the definition of surface area when there are curved surfaces.
A smooth surface's surface area is determined using its representation as a parametric surface, such as a sphere.
This definition of surface area uses partial derivatives and double much simpler mathematical concepts than the definition of surface area integration and is based on techniques used in infinitesimal calculus.sought a general definition of surface area.
(3×7)+(1.5×5)
21+7.5
28.5cm^2
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The local zoo buys from a supplier with an invoice amount if 17,200. The term of the sale are 5/22 n/30. What is the net amount on the order of the bill is paid by the 22nd day
The net amount on the order if the bill is paid by the 22nd day is $16,340.
What is trade credit?The terms of payment that a supplier offers to a buyer are referred to as rade credit terms. These conditions outline the deadline for payment as well as any early payment discounts. For instance, if a supplier offers "2/10 net 30" terms, the customer can choose to pay the whole amount up front or receive a 2% reduction if they pay within 10 days.
The parameters of a trade credit agreement can significantly affect a company's cash flow. If a company can benefit from an early payment discount, they can save expenses and increase cash flow.
The term of sale is given as 5/22 n/30.
Here, the discount is 5%, thus the invoice amount is:
Discount = 0.05 x $17,200 = $860
Now, the net amount is:
Net Amount = Invoice Amount - Discount
Net Amount = $17,200 - $860
Net Amount = $16,340
Hence, the net amount on the order if the bill is paid by the 22nd day is $16,340.
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A regular hexagon is inscribed into a circle. Find the length of the side of the hexagon, if the radius of the circle is 12 cm.
A. 20 cm
B. 18 cm
C. 16 cm
D. 12 cm
E. None of these
Answer:
D, 12 cm.
Step-by-step explanation:
First, draw out the circle with a hexagon in it. I did this by dividing the circle into thirds and then halving each of those because it was easier to visualize. I ended up with something like the image below. Now, we know that the sum of all of the angles that come together at the center of the circle is going to be 360. We can divide 360 by 6 to get 60. The top angle of the triangle, the angle closest to the center, is 60 degrees. We also know that the length of all of these legs that are going out will be the same, because the radius of the circle is constant. This makes each triangle an isosceles triangle. Given the isosceles base angle theorem, both base angles of the triangle must be the same. The sum of all angles in a triangle is 180 degrees. So the equation for x, one of the angles, would be 60 (angle closest to center) + 2x = 180. Let's subtract 60 from both sides to get 2x = 120. Divide both sides by two to get x = 60. Okay, so now we know that all angles in the triangle are 60 degrees. By the equilateral triangle theorem, we know these triangles must be equilateral because all of their angles are 60 degrees. Equilateral triangles also have the same side lengths. Because 12 is one side length, then all side lengths = 12. Therefore, the length of one side of the hexagon is 12.
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the annual rainfall in 2017 in opuwo was 420mm.
the annual rainfall in 2018 was 12% more than in 2017.
find the annual rainfall in 2018.
Thus, Opuwo received 470.4mm of precipitation annually in 2018.
What is the procedure for determining rainfall?Depth x Radius x Radius x 3.14 will give you the typical rainfall amount. The apex of the bucket's region can be located. To calculate the amount of rain, divide the capacity by this region.
To find the annual rainfall in 2018, we can use the fact that it was 12% more than in 2017.
Let R be the annual rainfall in 2017 (which we know to be 420mm). Then, the annual rainfall in 2018 can be expressed as:
R + 0.12R
Simplifying this expression, we get:
1.12R
Therefore, the annual rainfall in 2018 was:
1.12 x 420mm = 470.4mm
So the annual rainfall in 2018 in Opuwo was 470.4mm
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I have two fair dice each numbered 1 to 6. I am going to throw the two dice. What is
the probability that the sum of the numbers on the dice will be a square number?
The probability that the sum of the numbers on the dice will be a square number is 1/6.
What is the mathematical probability?The area of mathematics known as probability explores potential outcomes of events as well as their relative probabilities and distributions.
We can first make a list of every conceivable result when rolling two dice in order to determine the likelihood that the sum of the numbers on the two dice will be a square number:
[tex](1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6[/tex])
[tex](2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)[/tex]
[tex](3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)[/tex]
[tex](4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)[/tex]
[tex](5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)[/tex]
[tex](6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)[/tex]
The sums can then be listed, and the square numbers can be identified:
2, 3, 4, 5, 6, 7
3, 4, 5, 6, 7, 8
4, 5, 6, 7, 8, 9
5, 6, 7, 8, 9, 10
6, 7, 8, 9, 10, 11
7, 8, 9, 10, 11, 12
The likelihood that the sum of the numbers on the dice will be a square number is 6 out of a total of 36 potential outcomes.
P(square sum) = 6/36 = 1/6.
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Can you help me to solve these two questions?
The equation of the tangent line is y = (-1/49)x + 8/49. 2. The equation of the normal line is y = 49x - 342.
What do a curve's tangent and normal lines represent?A line that meets a curve at a point and has the same slope as the curve there is said to be the tangent line to that curve. A line that is perpendicular to the tangent line at a given position is the normal line to a curve at that location. In other words, the normal line's slope equals the tangent line's slope's negative reciprocal. The normal line is helpful for determining the direction of greatest change of the curve at a place whereas the tangent line gives information about the instantaneous rate of change of the curve at that point.
The given function is f(x) = 1/x.
The slope of the function is the derivative of the function thus,
f(x) = 1/x
f'(x) = -1/x²
f'(7) = -1/49
The equation of the line is given as:
y - y1 = m(x - x1)
where, m is the slope of the equation.
The equation of the tangent line is:
y - f(7) = f'(7)(x - 7)
y - 1/7 = -1/49(x - 7)
y = (-1/49)x + 8/49
b. The equation of the normal line to the graph is given as:
The slope of the normal line is negative and opposite of the tangent line.
That is,
m = 49.
y - f(7) = 49(x - 7)
y - 1/7 = 49(x - 7)
y = 49x - 342
Hence, the equation of the tangent line is y = (-1/49)x + 8/49. 2. The equation of the normal line is y = 49x - 342.
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Evaluate (can't write decimal answer).
[tex]sin(20)sin(70)-sin(14)cos(26)-cos(6)cos(84)[/tex]
The evaluation of sin(20)sin(70) - sin(14)cos(26) - cos(6)cos(84) is approximately -1.
How to solve trigonometry?Use the trigonometric identities to simplify the expression:
sin(20)sin(70) - sin(14)cos(26) - cos(6)cos(84)
= (sin(20)cos(20)) / 2 - (sin(14)sin(64)) / 2 - (cos(6)cos(6)) / 2
= [(sin(40) - sin(80)) / 2] - [(cos(50) - cos(78)) / 2] - [(1 + cos(12)) / 2]
= (sin(40) - cos(50) + cos(78) - sin(80) - 1 - cos(12)) / 2
Now, use a calculator to evaluate each trigonometric function:
sin(40) = 0.6428, cos(50) = 0.6428, cos(78) = 0.2079, sin(80) = 0.9848, cos(12) = 0.9781
Substituting these values:
= (0.6428 - 0.6428 + 0.2079 - 0.9848 - 1 - 0.9781) / 2
= -2.755 / 2
= -1.3775
Therefore, the value of the given expression is approximately -1.
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Hi, any one can solve it ?
For given function f(x)= x³ + 2x , the complete table is mentioned below. f⁻¹(3)= 1, f⁻¹(-12) = -2.
Describe function ?In mathematics, a function is a rule that assigns a unique output value for every input value in a specified set. It is a fundamental concept in algebra, calculus, and other areas of mathematics.
A function is typically denoted by a symbol, such as f(x), where x is the input variable, and f(x) is the output variable. The set of all input values for which the function is defined is called the domain, and the set of all output values is called the range.
To complete the table of values, we simply plug in the given values of x into the expression for f(x) and evaluate:
x f(x)
0 0
1 3
2 14
To find f⁻¹(3), we need to solve for x in the equation f(x) = 3:
x³ + 2x = 3
x³ + 2x - 3 = 0
We can use trial and error to find that x = 1 is a solution to this equation:
1³ + 2(1) - 3 = 0
Therefore, f⁻¹(3) = 1.
To find f⁻¹(-12), we need to solve for x in the equation f(x) = -12:
x³ + 2x = -12
x³ + 2x + 12 = 0
We can use trial and error to find that x = -2 is a solution to this equation:
(-2)³ + 2(-2) + 12 = 0
Therefore, f⁻¹(-12) = -2.
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Below are the jersey numbers of 11 players randomly selected from a football team. Find the range, variance, and standard deviation for the given sample data. What do the results tell us?
92 19 41 24 75 53 70 3 67 64 9
Step-by-step explanation:
To find the range, we need to subtract the smallest value from the largest value in the dataset:
Range = Largest value - Smallest value
Range = 92 - 3
Range = 89
To find the variance and standard deviation, we need to calculate the mean first:
Mean = (Sum of all values) / (Number of values)
Mean = (92+19+41+24+75+53+70+3+67+64+9) / 11
Mean = 45.09 (rounded to two decimal places)
Next, we need to calculate the variance:
Variance = (Sum of squared differences from the mean) / (Number of values - 1)
Variance = [(92-45.09)^2 + (19-45.09)^2 + (41-45.09)^2 + (24-45.09)^2 + (75-45.09)^2 + (53-45.09)^2 + (70-45.09)^2 + (3-45.09)^2 + (67-45.09)^2 + (64-45.09)^2 + (9-45.09)^2] / (11-1)
Variance = 1071.45 (rounded to two decimal places)
Finally, we can calculate the standard deviation by taking the square root of the variance:
Standard deviation = Square root of variance
Standard deviation = Square root of 1071.45
Standard deviation = 32.74 (rounded to two decimal places)
The range tells us the difference between the highest and lowest values in the dataset, which in this case is 89. The variance and standard deviation tell us how spread out the data is from the mean. The higher the variance and standard deviation, the more spread out the data is. In this case, the variance and standard deviation are both relatively high, indicating that the data is fairly spread out.
Given f(x) = x³ + kx + 9, and the remainder when f(x) is divided by x − 2 is 7,
then what is the value of k?
Answer:
k = -5
Step-by-step explanation:
According to the Remainder Theorem, when we divide a polynomial f(x) by (x − c), the remainder is f(c).
Therefore, if we divide polynomial f(x) = x³ + kx + 9 by (x - 2) and the remainder is 7 then:
f(2) = 7To find the value of k, simply substitute x = 2 into the function, equate it to 7 and solve for k.
[tex]\begin{aligned}f(2)=(2)^3 + k(2) + 9 &= 7\\8+2k+9&=7\\2k+17&=7\\2k&=-10\\k&=-5\end{aligned}[/tex]
Therefore, the value of k is -5.
in a batch of 10,000 clock radios 500 are defective. A sample of 10 clock radios is randomly selected without replacement from the 10,000 and tested. The entire batch will be rejected if at least one of those tested is defective. what is the probability that the entire batch will be rejected?
Answer:
Step-by-step explanation:
This is an example of a hypergeometric distribution problem, where we have a population of 10,000 clock radios with 500 defective ones, and we want to calculate the probability of getting at least one defective radio in a random sample of 10 without replacement.
The probability of getting no defective radios in the sample is:
(9500/10000) * (9499/9999) * (9498/9998) * ... * (9491/9992)
This is because, for the first radio, there are 9500 good radios out of 10,000, and for the second radio, there are 9499 good radios out of 9,999, and so on.
The probability of getting at least one defective radio in the sample is then:
1 - (9500/10000) * (9499/9999) * (9498/9998) * ... * (9491/9992)
which is approximately equal to 0.401.
Therefore, the probability that the entire batch will be rejected is 0.401.
The pens in a box are repackaged equally into 9 packs. Each pack has more than 15 pens.
1. Find an inequality to represent n, the possible number of pens in the box.
2. Explain why you chose this inequality.
Therefore, the possible number of pens in the box is p, where p is greater than 135.
What is inequality?Inequality refers to a situation in which there is a difference or disparity between two or more things, usually in terms of value, opportunity, or outcome. Inequality can take many forms, including social, economic, and political inequality.
Inequalities are mathematical expressions that compare two values using the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). To solve an inequality, you need to isolate the variable (the unknown quantity) on one side of the inequality symbol and determine the range of values for which the inequality holds true.
Here are some general steps to solve an inequality:
Simplify both sides of the inequality as much as possible. This may involve combining like terms, distributing terms, or factoring.Get all the variable terms on one side of the inequality symbol and all the constant terms on the other side. Remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.Solve for the variable by isolating it on one side of the inequality symbol. If the variable has a coefficient, divide both sides of the inequality by that coefficient.Write down the solution as an inequality. If you have solved for x, the solution will be in the form of x < a or x > b, where a and b are numbers.Check your solution by testing a value in the original inequality that is within the range of the solution. If the inequality holds true for that value, then the solution is correct. If not, then you may need to recheck your work or adjust your solutionby the question.
Let's say there are 'p' pens in the box. Each pack has more than 15 pens, so we can write the inequality:
p/9 > 15
Multiplying both sides by 9, we get:
p > 135
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