The time in hours which it will take them for working together for terri is about 1.71 hours.
What is the time in hours?Inverse Proportion: When two quantities are related in such a way that the product of one quantity with the reciprocal of the other quantity remains constant, it is said to be in inverse proportion.
Let's calculate their working rate:
Terri takes 3 hours to complete the painting of her living room, so she can paint her living room in [tex]\frac{1}{3} hours[/tex]
Angela takes 4 hours to complete the painting of the living room, so she can paint her living room in [tex]\frac{1}{4} hours[/tex]
If both work together, then the time taken to complete the work will be less than the time taken by each of them individually.
To find the time taken by both working together, we will add their rates.
Terri's work rate = [tex]\frac{1}{3} hours[/tex]
Angela's work rate = [tex]\frac{1}{4} hours[/tex]
Work rate when working together = Terri's rate + Angela's rate= [tex]\frac{1}{3} + \frac{1}{4} = \frac{7}{12}[/tex]
Thus, both will take [tex]\frac{12}{7} = 1.71 hours[/tex] approximately to complete the painting of the living room when they work together.
Therefore, the time in hours is about 1.71 hours.
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Please help , I’ve never been the best at math. I just need to know how to plot and connect the points and if it’s exponential or linear.
Answer: To plot and connect the points, you can use a graphing software or a graphing calculator. Once the points are plotted, you can determine if the graph is linear or exponential by looking at the slope of the line. If the slope is increasing or decreasing exponentially, the graph is exponential. If the slope is constant, the graph is linear.
Step-by-step explanation:
Jeremiah bought 9 apples and 6 apricots for $8.50 yesterday.
He bought 3 apples and 2 apricots for $7.40 today.
Enter a system of linear equations to find the cost of an apple and the cost of an apricot.
Cost of an Apple
Cost of an Apricot
The system of linear equations to find the cost of an apple and the cost of an apricot is as follows:
9x + 6y = 8.50
3x + 2y = 7.40
How to solve system of equation?Jeremiah bought 9 apples and 6 apricots for $8.50 yesterday. He bought 3 apples and 2 apricots for $7.40 today.
The system of linear equation to find the cost of an apple and the cost of an apricot can be represented as follows:
Therefore, system of equation can be solved using different method such as elimination method, substitution method and graphical method.
The linear equation is as follows:
9x + 6y = 8.50
3x + 2y = 7.40
where
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Are the following statements equivalent? a>b and there is the number c ,so that a=b+c
No, the statements are not equivalent as the statement "a > b" simply means that "a" is greater than "b" and there exists a number "c" such that when added to "b", it equals "a". This does not necessarily mean that "a" is greater than "b".
The statements "a>b" and "there is the number c, so that a=b+c" are not equivalent. The statement "a>b" simply means that "a" is greater than "b," while the statement "there is the number c, so that a=b+c" means that "a" can be expressed as the sum of "b" and another number "c."
These statements are not equivalent because even if "a" is greater than "b," it may not be possible to express "a" as the sum of "b" and another number "c."
Additionally, even if "a" can be expressed as the sum of "b" and another number "c," it may not necessarily be true that "a" is greater than "b."
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-4
-2
Intro
ty
8
4
-4
-8
N
X
Find the indicated function values.
f(-4)=
f(0) =
f(1) =
O
(>
De
Answer: 1
Step-by-step explanation:
because it not zero and not 4
A friend is building a garden with two side lengths 16 ft and exactly one right angle. What geometric figures could describe how the garden might look?
SELECT ALL THAT APPLY:
A. Kite.
B. Isosceles right triangle
C. Quadrilateral
D. Parallelogram
(Remember it is multiple choice)
Answer:
B. Isosceles right triangle
C. Quadrilateral
D. Parallelogram
Step-by-step explanation:
Answer:
The geometric figures that could describe how the garden might look are B. Isosceles right triangle and C. Quadrilateral.
Solve the system of equations:
y = 2x + 1
y=x²+2x-8
OA. (-3,-5) and (3,7)
B. (-4, 0) and (2,0)
C. (0, 1) and (2, 5)
OD. (-3, 5) and (3, 2)
The solutions of the system of equations are option (A) (3, 7) and (-3, -5)
Solving system of equations:
To solve the system of equations, use the concept of substitution, which involves solving one equation for one variable and then substituting that expression into the other equation to eliminate one variable and solve for the other variable.
In this case, solve equation (1) for y in terms of x and substitute that expression into equation (2), which allowed us to solve for x. Then we used the values of x to find the corresponding values of y.
Here we have
y = 2x + 1 --- (1)
y = x²+ 2x -8 --- (2)
From (1) and (2)
=> x²+ 2x - 8 = 2x + 1
Subtract 2x + 1 from both sides
=> x²+ 2x - 8 - 2x - 1 = 2x + 1 - 2x - 1
=> x² - 9 = 0
Now add 9 on both sides
=> x² - 9 = 0 + 9
=> x² = 9
=> x = √9
=> x = ± 3
From (1)
At x = 3
=> y = 2(3) + 1 = 7
At x = - 3
=> y = 2(-3) + 1 = - 5
Therefore,
The solutions of the system of equations are option (A) (3, 7) and (-3, -5)
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Suppose that a category of world class runners are known to run a marathon (26 miles) in an average of 146 minutes with a standard deviation of 11 minutes. Consider 49 of the races. Let x = the average of the 49 races. Part (a) two decimal places.) Give the distribution of X. (Round your standard deviation to two decimal places)Part (b) Find the probability that the average of the sample will be between 144 and 149 minutes in these 49 marathons. (Round your answer to four decimal places.) Part (c) Find the 80th percentile for the average of these 49 marathons. (Round your answer to two decimal places.) __ min Part (d) Find the median of the average running times ___ min
(a)The distribution of X (the average of the 49 races) follows a normal distribution with mean 146 minutes and standard deviation 11 minutes. (b)The probability of the average of the sample being between 144 and 149 minutes is 0.5854.(c)The 80th percentile for the average of these 49 marathons is 157.2 minutes.(d) The median of the average running times is 146 minutes.
Part(a) The distribution of X (the average of the 49 races) follows a normal distribution with mean 146 minutes and standard deviation 11 minutes. Part (b) The probability that the average of the sample will be between 144 and 149 minutes in these 49 marathons can be calculated using the z-score formula: z = (x - mean)/standard deviation
For x = 144, z = (144 - 146)/11 = -0.18
For x = 149, z = (149 - 146)/11 = 0.27,using the z-score table, the probability of the average of the sample being between 144 and 149 minutes is 0.5854 (0.4026 + 0.1828).
Part (c) The 80th percentile for the average of these 49 marathons can be calculated using the z-score formula: z = (x - mean)/standard deviation, For the 80th percentile, z = 0.84 (from z-score table). Therefore, x = 146 + (0.84 * 11) = 157.2 minutes. Part (d) The median of the average running times is 146 minutes. The median is the midpoint of the data which means half of the data is above the median and half of the data is below the median. Therefore, the median of the average running times is equal to the mean.
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The mean age of n men in a club is 50 years. Two men aged 55 and 63 left the club and mean age reduced by one year. Find the value of n.
Answer:
n=20
Sum of age of men = 50n
when 2 left,
(50n-55-63)/n-2=50-1
50n-55-63=49n-98
50n-55=49n-35
50n-49n=-35+55
n=20
The data in Exercise 1 were taken from the following functions. Compute the actual errors in Exercise 1, and find error bounds using the error formulas.
a. f ( x ) = sin x b. f (x) = ex − 2x2 + 3x – 1
The actual errors for Exercise 1 can be computed by subtracting the calculated values from the true values of the functions. For example, the actual error for sin(1.1) can be found by subtracting sin(1.1) = 0.8912 from the calculated value of 0.8890. The actual error in this case is 0.0022.
Error bounds for these functions can be found using the error formulas. For the function f(x) = sin x, the error bound can be found using the formula |E| <= M|x-a|, where M is the maximum value of the first derivative of the function, and a is the value of x at which the error is computed. In this case, M = 1 and a = 1.1, so the error bound is |E| <= 1 * |1.1 - 1.1| = 0.
For the function f(x) = ex - 2x2 + 3x - 1, the error bound can be found using the formula |E| <= M|x-a|2, where M is the maximum value of the second derivative of the function, and a is the value of x at which the error is computed. In this case, M = e and a = 1.1, so the error bound is |E| <= e * |1.1 - 1.1|2 = 0.
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The data in the table below shows the number of passengers and number of suitcases on various airplanes.
Estimate to the nearest whole number the number of suitcases on a flight carrying 250 people.
[____________]
In linear equation, The number of suitcases are 503.
What in mathematics is a linear equation?
A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.
Equations with power 1 variables are known as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.
Equation
y = 1.98x + 7.97
Where,
x = Number of passengers
y = Number of suitcases
Finding the number of suitcases:
y = 1.98x + 7.97
y = 1.98(250) + 7.97
y = 495 + 7.97
y = 502.97 ≈ 503
Hence,
The number of suitcases are 503.
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a chef at a restaurant uses 12 pound of butler each day
Answer:
Every day, the chef consumes 5443.20 grams of butter, which is calculated using the conversion coefficients 16 oz/1 pound and 28.35 grams/1oz.
A restaurant chef uses 12 pounds of butter every day, as specified in the question.
We need to figure out how much butter the chef uses each day in grams.
Applying the conversion parameters provided, 16 oz/1 lb and 28.35 grams/1oz
According to the data provided, the needed solution is as follows: 12 lb 16 oz/1 lb 28.35 g/1 ounce 5443.20 grams
As a result, the chef consumes 5443.20 grams of butter every day.
Step-by-step explanation:
Brainliest pls
Model -2/3+(-1 1/6) on a number line?
Answer:
To model -2/3 + (-1 1/6) on a number line, we first need to convert the mixed number (-1 1/6) into an improper fraction.
-1 1/6 = -7/6
Now we can add -2/3 and -7/6 by finding a common denominator. The smallest common multiple of 3 and 6 is 6, so we'll convert both fractions to have a denominator of 6.
-2/3 = -4/6
-7/6 = -7/6
Now we can add them:
-4/6 + (-7/6) = -11/6
So -2/3 + (-1 1/6) = -11/6.
To model this on a number line, we would start at zero and move to the left by 1 and 2/3 units (since -2/3 is less than 1 whole unit to the left of zero). Then, we would move an additional 1 and 1/6 units to the left (since -1 1/6 is one whole unit and 1/6 of another unit to the left of zero). This would bring us to the point represented by -11/6 on the number line.
Step-by-step explanation:
Answer:
(-5/6)
Step-by-step explanation:
Start by marking the point 0 on the number line.
To represent -2/3, you'll need to move 2/3 units to the left of 0, since this value is negative. One way to do this is to divide the space between 0 and -1 into three equal parts and move two of them to the left. This brings you to the point -2/3.
To represent (-1 1/6), you'll first need to represent -1, which is one unit to the left of 0. You'll then need to add 1/6 units to the left of -1. One way to do this is to divide the space between -1 and -2 into six equal parts and move one of them to the left. This brings you to the point (-1 1/6).
To find the sum of -2/3 and (-1 1/6), you'll need to add the distances we moved in steps 2 and 3. The total distance moved to the left is 2/3 + 1/6 = 5/6. Starting from 0, we move 5/6 units to the left to arrive at the final point, which is (-5/6).
What is the ratio of the radius of circle A to the radius of circle B?
The ratios of the radius of the two circles from their circumferences is 19.09 to 25.11
How to determine the ratios of the circlesWe have the following readings from the question:
Circumference A = 19.09
Circumference B = 25.11
The formula for the circumference of a circle is:
C = 2πr
where C is the circumference and r is the radius of the circle.
We are given the circumferences of circle A and circle B. Let's use the formula to write two equations:
Circumference A = 2πrA = 19.09
Circumference B = 2πrB = 25.11
We can divide these two equations to eliminate π and get the ratio of the radii:
(2πrA) / (2πrB) = 19.09 / 25.11
Simplifying, we get:
rA / rB = 19.09 / 25.11
So, the ratio is 19.09 to 25.11
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Anna wants to make 30 mL of a 60 percent salt solution by mixing togethera 72 percent salt solution and a 54 percent salt solution. How much of each solution should dhe use
Anna should use 10 mL of the 72% salt solution and 20 mL of the 54% salt solution to make 30 mL of a 60% salt solution
Let's assume that Anna will use x mL of the 72% salt solution, and therefore she will use (30 - x) mL of the 54% salt solution (since the total volume is 30 mL).
To find out how much of each solution Anna should use, we can set up an equation based on the amount of salt in each solution.
The amount of salt in x mL of 72% salt solution is
= 0.72x
The amount of salt in (30 - x) mL of 54% salt solution is
= 0.54(30 - x)
To make a 60% salt solution, the total amount of salt in the final solution should be
0.6(30) = 18
So we can set up an equation
0.72x + 0.54(30 - x) = 1
Simplifying the equation
0.72x + 16.2 - 0.54x = 18
0.18x = 1.8
x = 10 ml
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The original price of a television is reduced by 25%.
This new price is then increased by 25%.
Calculate the price of the television now as a percentage of the original price.
The new price of the television is 15/16 of the original price
What is percentage?Percentages are essentially fractions where the denominator is 100. To show that a number is a percent, we use the percent symbol (%) beside the number.
Represent the original price by x
x × 25/100 = x/4%
The new price will be
x - x/4
= (4x- x)/4
= 3x/4
The new price is now increased by 25%
25/100 × 3x/4
= 1/4 × 3x/4
= 3x/16
the new price
3x/16 + 3x/4
= (12x + 3x)/16
= 15x/16
Therefore the new price is 15/16 of the original price.
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What is the circumference of the circle with a radius of 5.5 meters? Approximate using π = 3.14.
6.45 meters
34.54 meters
38.47 meters
199.66 meters
1. find the indefinite integral and check the result by differentiation. (use c for the constant of integration.) S (16׳ – 15x² + 6) dx 2. Find the indefinite integral and check the result by differentiation. (Use C for the constant of integration.)S (7 cos(x) +5 sin(x)) dx
For the first question, the indefinite integral of [tex]16x³ - 15x² + 6[/tex] is [tex]4x⁴ - 5x³ + 6x + C[/tex], where C is the constant of integration.
For the second question, the indefinite integral of 7cos(x) + 5sin(x) is [tex]7sin(x) + 5cos(x) + C,[/tex]where C is the constant of integration.
1)To check the result, differentiate the indefinite integral using the power rule and product rule: [tex](16x³ - 15x² + 6)' = 64x² - 30x + 6.[/tex]
2)To check the result, differentiate the indefinite integral using the sum rule and product rule: [tex](7sin(x) + 5cos(x))' = 7cos(x) - 5sin(x).[/tex]
In summary, to find the indefinite integral of a function, use the power rule and product rule to integrate the individual terms. To check the result, differentiate the indefinite integral.
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uppose m professors randomly choose from n time slots to hold their final exams. If two professors pick the same time slot, we say that they are in conflict. (If three professors all pick the same time slot, that gives three pairs of professors in conflict.) What is the expected number of pairs of professors in conflict? Your answer should depend on m and n.
The expected number of pairs of professors in conflict is given by (m choose 2) * 1/n.
It can be calculated using the principle of linearity of expectation. We can first calculate the probability that any two professors pick the same time slot, which is 1/n. Then, we can count the number of pairs of professors, which is given by the binomial coefficient (m choose 2) = m(m-1)/2. Therefore, the expected number of pairs of professors in conflict is:
Expected number of pairs in conflict = (m choose 2) * 1/n
This formula holds when the selection of time slots by each professor is independent of the choices made by all other professors. Note that this formula assumes that each professor selects only one time slot, and does not consider the possibility of a professor selecting multiple time slots. If professors are allowed to select multiple time slots, then the formula would need to be modified accordingly.
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In the following code, add another array declaration thatcreates an array of 5 doubles called prices and another array of 5Strings called names and corresponding System.out.printlncommands.public class Test1{public static void main(String[] args){// Array exampleint[] highScores = new int[10];// Add an array of 5 doubles called prices.// Add an array of 5 Strings called names.System.out.println("Array highScores declared with size " +highScores.length);// Print out the length of the new arrays}
In the following code, you can add another array declaration that creates an array of 5 doubles called prices and another array of 5 Strings called names and corresponding System.out.println commands.
public class Test1 {
public static void main(String[] args) {
// Array example
int[] highScores = new int[10];
// Add an array of 5 doubles called prices.
double[] prices = new double[5];
// Add an array of 5 Strings called names.
String[] names = new String[5];
System.out.println("Array highScores declared with size " + highScores.length);
// Print out the length of the new arrays.
System.out.println("Array prices declared with size " + prices.length);
System.out.println("Array names declared with size " + names.length);
}
}
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what type of data is a questionnaire
Answer:
A questionnaire can collect quantitative, qualitive or both types of data.
Step-by-step explanation:
Answer:
Categorical data
Step-by-step explanation:
Data that relates to certain categories e.g males, females or any types of car
Aamena buys a business costing $23000 She pays part of this cost with $12000 of her own money Calculate what percentage of the $23000 this is Show your calculations
Answer: 52.173913%
Step-by-step explanation:
The answer is 52.173913%, you can round it <3
I hope this helped! <333
Answer: I got 52.17391304%
Step-by-step explanation:
If you are looking for a rate, you should use R=P/B where R is the % B is the base (23000) and P is the portion (12000) So, it would be set up like this R = 12000/23000
After I got my answer I double checked it by using the formula B = 52.17% x 12000 to see if I got 23000, but instead I got 11,999.10 because I rounded down the long percentage number from 52.17391304% to 52.17%, which would usually be ok and work just fine, but using 52.17% I got 11,999.10. Which is still extremely close to 12000, but not exact. So even though rounding down is usually a requirement in homework situations, this one seems to be an exact percentage with all the digits. Let me know if I can help further. I'm in accounting and I'm curious about this one and would like to know how it goes.
a square whose side measures 2 centimeters is dilated by a scale factor of 3 . what is the difference between the area of the dilated square and the original square?
After figuring out the given issue, we discovered that the original square's area and the dilated square's area differ by 32 square centimeters.
What is the area of square formula?Square Area = Side x Side. Hence, Side2 square units are equivalent to the area of the square. and four side units make up a square's perimeter.
The formula for calculating the area of a initial square with sides that are 2 centimeters long is:
A = s²
A = 2²
A = 4 square centimeters
The revised side length of this square after dilation by a scale factor of 3 is:
s' = 3s
s' = 3(2)
s' = 6 centimeters
Calculations for the dilated square's area are as follows:
A' = s'²
A' = 6²
A' = 36 square centimeters
The area of a dilated square differs from the size of the original square by:
A' - A = 36 - 4
A' - A = 32 square centimeters
As a result, the original square's area and the dilated square's area differ by 32 square centimeters.
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In the original plan for area codes in 1945, the first digit could be any number from 2 through 9, the second digit was either 0 or 1, and the third digit could be any number except 0. With this plan, how many different area codes are possible
Using multiple principle, The different area codes that are possible are 144
What is meant by the multiplication principle?The multiplication principle is a counting principle that states the total number of possible outcomes of a series of independent events is equal to the product of the number of outcomes for each event. It is used in probability theory and other areas of mathematics and science.
According to the question
Under the original plan for area codes in 1945, the first digit could be any number from 2 through 9, the second digit was either 0 or 1, and the third digit could be any number except 0.
Therefore, the number of possible area codes can be calculated as follows:
For the first digit, there are 8 possibilities (2, 3, 4, 5, 6, 7, 8, or 9). Here 9 is inclusive.
For the second digit, there are 2 possibilities (0 or 1).
For the third digit, there are 9 possibilities (any digit except 0).
Using the multiplication principle, the total number of possible area codes can be determined by multiplying the number of choices for each digit:
8 × 2 × 9 = 144
So there are 144 different area codes possible according to this plan.
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find an equation of the line throught the point (3,5) that cuts iff the least area from the first quadrant
The equation of the line y = (5/3)x + (10/3) represents the line passing through the point (3, 5) that cuts off the smallest area from the first quadrant.
To find the equation of the line that passes through the point (3, 5) and cuts off the least area from the first quadrant, we need to consider the slope of the line.
Any line passing through the point (3, 5) can be written in a point-slope form as:
y - 5 = m(x - 3)
where m is the slope of the line. We want to find the slope that minimizes the area cut off by the line.
Consider a line passing through the origin with slope m. The area cut off by this line in the first quadrant is given by:
A(m) = (1/2)(3)(m*3) = (9/2)m
Note- that the area cut off by the line passing through (3, 5) with slope m is equal to the area cut off by the line passing through the origin with slope m plus the area of the triangle formed by the point (3, 5), the origin, and the point where the line intersects the y-axis. The y-intercept of the line passing through (3, 5) with slope m is given by:
y - 5 = m(x - 3)
y = mx - 3m + 5
Setting x = 0, we get:
y = -3m + 5
The coordinates of the point where the line intersects the y-axis are (0, -3m + 5), and the area of the triangle is:
(1/2)(3)(|-3m + 5 - 0|) ⇒ (3/2)|-3m + 5|
Therefore, the total area cut off by the line passing through (3, 5) with slope m is:
A(m) = (9/2)m + (3/2)|-3m + 5|
To find the slope that minimizes this expression, we need to consider two cases:
Case 1: -3m + 5 ≥ 0, i.e., m ≤ 5/3
In this case, the expression simplifies to:
A(m) = (9/2)m + (9/2)m - (3/2)(5)
= (9m - (15/2)
This expression is minimized when m = 5/3, which is within the range of possible slopes.
Case 2: -3m + 5 < 0, i.e., m > 5/3
In this case, the expression simplifies to:
A(m) = (9/2)m - (9/2)m + (3/2)(5)
= (15/2)
This expression is minimized when m = 5/3, which is again within the range of possible slopes.
Therefore, the line passing through (3, 5) with slope m = 5/3 cuts off the least area from the first quadrant. The equation of the line is:
y - 5 = (5/3)(x - 3)
Simplifying, we get:
y = (5/3)x + (10/3)
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1) y = |x+2| how do i make a table and graph this solution! someone help!!
Answer:
(-1,1)
(0,2
(1,3)
(2,4)
please answer im on number 10 easy question
One paperclip has the mass of 1 gram. 1,000 paperclips have a mass of 1 kilogram. How many kilograms are 5,600 paperclips?
560 kilograms
56 kilograms
5.6 kilograms
0.56 kilograms
Answer:
Since 1 000 paperclips = 1 kilogram
5,600 paperclips x 0.001 kilograms per paperclip = 5.6 kilograms
Answer: 5.6 kilograms
Step-by-step explanation:
5,600 grams = 5.6 kilograms
Use the coordinate plane to answer the question. Information A coordinate plane is shown. No points are plotted. Question What is the perimeter of a quadrilateral with vertices at the point negative 5 comma negative 2, the point negative 4 comma negative 2, the point negative 5 comma 1, and the point negative 4 comma 1? Enter the answer in the box. Response area with 1 text input box units
The perimeter of the quadrilateral is 8 units. Therefore, the answer to the question is 8 units.
What is perimeter?The perimeter of a quadrilateral is the sum of the lengths of its sides. To calculate the perimeter, we need to calculate the distances between the points of the quadrilateral.
To draw a quadrilateral with the given points on the coordinate plane, we can draw four lines connecting the given points. The first line will connect the points (–5, –2) and (–4, –2), which is a horizontal line with a length of 1 unit. The second line will connect the points (–4, –2) and (–4, 1), which is a vertical line with a length of 3 units. The third line will connect the points (–4, 1) and (–5, 1), which is a horizontal line with a length of 1 unit. The fourth line will connect the points (–5, 1) and (–5, –2), which is a vertical line with a length of 3 units.
The total length of the sides of the quadrilateral is calculated by adding the lengths of the four lines. Thus, the perimeter of the quadrilateral is 8 units.
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Write the prime factorization of 84.
Answer:
2 × 2 × 3 × 7=84
Step-by-step explanation:
7. Complete a dilation with scale factor of ½ around the origin and then
reflect over the y-axis. What are the new ordered pair of A'?
The new ordered pairs of points A', B', and C' after the reflection over the y-axis are (1,-2.5), (0,1.5), and (-3,-1.5), respectively.
WHAT IS AXIS ?
An axis is a straight line around which an object rotates or is symmetrical. In geometry, it is a reference line that is used to measure distances, angles, and other geometric properties of objects. In a two-dimensional plane, there are two axes: the x-axis and the y-axis. The x-axis is the horizontal line that runs from left to right, while the y-axis is the vertical line that runs from bottom to top. Together, the x-axis and y-axis form a coordinate system that is used to plot points and graph functions. In three-dimensional space, there is also a z-axis that runs perpendicular to the x-axis and y-axis
To perform a dilation with a scale factor of 1/2 around the origin, we need to multiply the coordinates of each point by 1/2. This gives us:
A' = (1/2)A = (1/2)(-2,-5) = (-1,-2.5)
B' = (1/2)B = (1/2)(0,3) = (0,1.5)
C' = (1/2)C = (1/2)(6,-3) = (3,-1.5)
The new coordinates of points A, B, and C after the dilation are (-1,-2.5), (0,1.5), and (3,-1.5), respectively.
To reflect each point over the y-axis, we need to change the sign of its x-coordinate, while leaving the y-coordinate unchanged. This gives us:
A" = (-x,y) = (-(-1),-2.5) = (1,-2.5)
B" = (-x,y) = (-0,1.5) = (0,1.5)
C" = (-x,y) = (-3,-1.5) = (-3,-1.5)
The new ordered pairs of points A', B', and C' after the reflection over the y-axis are (1,-2.5), (0,1.5), and (-3,-1.5), respectively.
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Find the derivative of the function f(x), below. It may be to your advantage to simplify first. f(x)=x⋅5x
f′(x)=
The derivative of f(x) = x⋅5x is f'(x) = 10x, which means that the rate of change of the function at any point x is 10 times the value of x at that point.
Using the product rule of differentiation, we can find the derivative of the function f(x) = x⋅5x as follows:
f'(x) = (x)'(5x) + x(5x)'
where (x)' and (5x)' are the derivatives of x and 5x with respect to x, respectively.
(x)' = 1 (the derivative of x with respect to x is 1)
(5x)' = 5 (the derivative of 5x with respect to x is 5)
Substituting these values, we get:
f'(x) = 1⋅5x + x⋅5
Simplifying further, we get:
f'(x) = 5x + 5x
Therefore, f'(x) = 10x.
In conclusion, the derivative of f(x) = x⋅5x is f'(x) = 10x, which means that the rate of change of the function at any point x is 10 times the value of x at that point.
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