The P(n) represents the maximum no of regions into which n lines divide the plain then P(k) = P(k-1) + k. (b). The formula for P(n) is [tex]1+{ }^{n+1} c_2\end{aligned}$[/tex]. (c). The 5152 regions can divide the plane for 100 lines.
When no. lines are drawn, we get one plain region.
Let the number of regions when divided by lines be represented by
P(0), P(1), P(2), P(3), P(n)
Therefore P(0) = 1
P(1) = 2
P(1) = P(0) + 1
P(2) = P(1) + 2
P(3) = P(2) + 3
P(4) = P(3) + 4………………….
P(n) = P(n-1) + n
Therefore, if P(n) represents the maximum no. of regions into which n lines divide the plain
Then P(k) = P(k-1) + k
(b).
⇒ P(0) = 1
⇒P(n) = P(n-1) + n
= P(n-2) + (n-1)+n
= P(n-3) + (n-2) +(n-1)+n……..
= P(0) + 1 + 2+ 4=3 + 4 + 5 +……+n
= P(0) + (Sum of the first n natural numbers)
[tex]$$\begin{aligned}& =P(0)+\frac{n(n+1)}{2} \\& =1+{ }^{n+1} c_2\end{aligned}$$[/tex]
Therefore, the formula for P(n) is [tex]1+{ }^{n+1} c_2\end{aligned}$[/tex].
(c)
Compute the value for n=100
[tex]$$1+{ }^{n+1} c_2=1+{ }^{101} c_2=1+\frac{101 \times 102}{2}=5152$$[/tex]
Therefore, the 5152 regions can divide the plane for 100 lines.
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A single line divides the plane into two regions. Two lines (by crossing) can divide the plane into four regions; three lines can divide it into seven regions, etc. Let P(n) be the maximum number of regions into which n lines divide a plane, where n is a positive integer.
(a) Derive a recurrence relation for P(k) in terms of P(k-1) for all integers [tex]$k \geq 2$[/tex].
(b) Use iteration to find an explicit (closed) formula for P(n).
(c) Into how many regions can 100 lines divide the plane?
Please help I don’t understand!
From the curve in the graph the function values are
h(3) = 2
h(5) = 6
h(6) = 5
What is parabolic curve?A parabola is a U-shaped plane curve in which every point is situated at an equal distance from both the focus, a fixed point, and the directrix, a fixed line.
The quadratic equation represents a parabolic curve on a graph
Examining the graph the function values can be deduced noting that the values 3 5 6 and in h(3), h(5) and h(6) are points in the x axis or the domain
The function values are points on the y axis corresponding to tracing the x values to the curve
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Prove that if parallel lines are crossed by a transversal, then the alternate interior angles are congruent. That is, use rigid transformations to map ∠CFG onto ∠BCF. Explain.
Alternate interior angles are congruent if parallel lines are crossed by a transversal.
What is congruent geometry?In congruent geometry, the shapes that are so identical. can be superimposed on themselves.
Here,
A figure has been shown below,
Where there is 2 parallel line intersected by a transversal line.
The measure of the alternate angle [besides interior and exterior angle], their measure are equal to corresponding angles as well as with alternate angles. As shown in the figure.
Thus, alternate interior angles are congruent if parallel lines are crossed by a transversal.
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need help with questions 6-15 only with all work shown.
Consequently, the square root of the [tex]\sqrt{25}[/tex] is 5 and square root of [tex]\sqrt{4}[/tex] is 2 . The square root and nth root .
what is square root ?The square root or nth root is represented by the radical sign. Square roots are a component of root expressions. There is a radical's radicand—a number or word that appears. Radicals like 7 and 2y+1 serve as examples. Additional terms that describe radicals include: Radical equations are those equations that include radicals. A "root expression" is the phrase used to describe the expression contained within the square root. To demonstrate radical representations, consider the numbers 2, 372, 2x+7, and 41p. The phrase "root expression" refers to the expression contained within the square root. Examples of radical representations are the numbers 2, 372, 2x+7, and 41p.
given
square root of (a) [tex]\sqrt{25}[/tex] because 5 is the square root, square root of
as b ) square root of [tex]\sqrt{4}[/tex] s2's square root
Consequently, the square root of the [tex]\sqrt{25}[/tex] is 5 and square root of [tex]\sqrt{4}[/tex] is 2 .
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please help me i would be happy if you helped me
The function to represent the arithmetic sequence is y = 0.17x + 0.71, and an envelop weighing 8 ounces cost $2.07
What is an equation?An equation is an expression that shows the relationship between variables and numbers.
An arithmetic sequence is in the form:
[tex]a_n=a_1+(n-1)d\\\\where\ a_1=first\ term,d=common\ difference[/tex]
Let y represent the price to send an envelop of x ounces. The arithmetic sequence is:
y = a₁ + (x - 1)d
Given that the first ounce is 88 cents($0.88), hence a₁ = 0.88. Eadh additional ounce is 17 cents (0.17), hence d = 0.17. Therefore:
y = 0.88 + (x - 1)(0.17)
y = 0.17x + 0.71
For a cost of $2.07:
2.07 = 0.17x + 0.71
x = 8 ounces
The function to represent the arithmetic sequence is y = 0.17x + 0.71
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which of the following could be an example of a function with a domain [a, infinity) and a range (-infinity, b] where a>0 and b>0
f(x) = √x ⇒ f(x) = -√(x -a) +b . . . . . . matches choice B
What is function ?
Function is the relationship between the input or domain and the output. A relation is a function if and only if there exists one value in the range for every domain value. A function is written as f x , where x is the input value. The general form of a function is y = f x
Function transformations
To move the domain from [0, ∞) to [a, ∞) requires the function be translated 'a' units to the right. That transformation is accomplished by replacing x with (x-a) in the function definition.
To move the range from [0, ∞) to [b, ∞) requires the function be translated upward by 'b' units. This is accomplished by adding 'b' to every function value.
Therefore, the range of the function will be (-∞,b] i.e. y ≤ b.
Hence, the required function is f(x) = √x ⇒ f(x) = - √(x -a) +b . . . . . . matches choice B
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what is synthetic division method
A quantity with an initial value of 4400 grows continuously at a rate of 0.5% per year. What is the value of the quantity after 46 years, to the nearest hundredth?
Answer:
To find the value of the quantity after 46 years, we can use the formula for continuous compound interest:
A = P(1 + r)^t
Where:
A = final amount
P = initial amount (4400)
r = rate of growth (0.5%)
t = number of years (46)
We need to convert the percentage rate to decimal form, so we divide the rate by 100
0.5/100 = 0.005
Now we can substitute the values into the formula:
A = 4400(1 + 0.005)^46
A = 4400(1.005)^46
A = 4400 * 1.294624
A = 5685.89
Therefore, the value of the quantity after 46 years is 5685.89, to the nearest hundredth.
Step-by-step explanation:
Solve the following literal equation for y. -5x+8y=15
Pedro has a tax-free savings account (TFSA) with a balance of
$2350.00. His interest is earned at a rate of 2.10%, compounded monthly. If he
contributes $400.00 to the TFSA at the end of every month, how long will it take
him to save $15 000.00?
You can hold qualified investments like cash, stocks, bonds, mutual funds in a TFSA and can withdraw contributions as well as the interest, capital gains, and dividends earned in the account at any time1, without paying taxes
What is the use of TFSA?Any individual that is a non-resident of Canada who has a valid SIN and who is 18 years of age or older is also eligible to open a TFSA. However, any contributions made while a non-resident will be subject to a 1% tax for each month the contribution stays in the account.A TFSA allows you to set money aside in eligible investments and watch those savings grow tax-free throughout your lifetime. Interest, dividends, and capital gains earned in a TFSA are tax-free for life.The single biggest benefit of a TFSA is that growth of all assets within it are tax-free: this includes interest, dividends and capital gains. You won't pay a penny in income tax even when you withdraw from your account or sell the assets inside the TFSATo know more about tax visit:
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f(x) = 2x² - 2x - 12
g(x)=x+2
What is the value of f(x),
g(x)
A. x-6
B. x-3
OC. 2x-6
OD. 2x+4
Answer:
17
Step-by-step explanation:
Answer:b
Step-by-step explanation:
Sara has a 30-gallon fish tank.The tank already has 24 gallons of water in it.Sara adds more water at a rate of 0.5 gallons per minute.
Solving a linear equation we can see that the tank will be filled in 12 minutes.
How long will take Sara to fill the tank?We know that the fish tank has a volume of 30 gallons, and it already has 24 gallons of wate on it.
Sara is adding more water at a rate of 0.5 gallons per minute, so after x minutes, the volume of water on the tank is modeled by the linear equation:
V(x) = 24gal + 0.5gal*x
The tank will be full when the volume is equal to 30 gallons, then we can solve the equation:
30 gal = 24 gal + 0.5gal*x
30 gal - 24gal = 0.5gal*x
6 gal = 0.5gal*x
6gal/0.5gal = x
12 = x
The tank will be full after 12 minutes.
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carlos borrowed 26,000 for 5 years at an APR 5.25% what will his monthly payment be?
Answer:
$568.75.
Step-by-step explanation:
Situation #6
=
It is known that ΔTOP~ΔBUG. Also, TO = 16, TP = 20, OP = 30, and BG = 105.
6. What is the perimeter of ΔBUG?
The perimeter of Δ BUG with sides BG = 105, BU = 16 and UG = 30 is 151 units.
What is perimeter?
The complete length of a shape's boundary is referred to as the perimeter in geometry. A shape's perimeter is calculated by adding the lengths of all of its sides and edges. Its dimensions are expressed in linear units like centimetres, metres, inches, and feet.
It is given that ΔTOP ~ ΔBUG.
The lengths of sides of ΔTOP is - TO = 16, TP = 20, OP = 30.
The length of one side of ΔBUG is BG = 105.
Since ΔTOP is same as ΔBUG, so sides BU and UG will same as sides TO and OP respectively.
So, BU = 16 and UG = 30.
The formula for the perimeter of a triangle is -
P = a + b + c
Where, P is the perimeter and a, b, c are the sides of the triangle.
For ΔBUG -
P = BU + BG + UG
P = 16 + 105 + 30
P = 151
Therefore, the perimeter of ΔBUG = 151 units.
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Zahara rides with her to her grandmother's house. They travel 159 miles in 3 hours. Her uncle Jorge drives there separately from his home.
a. At what speed does Zahara's dad drive in miles per hour
b. Jorge's trip takes the same time as Zahara's, but he travels a shorter distance. Does Jorge drive faster or slower than Zahara's dada
Answer: a. To find the speed at which Zahara's dad drives, we need to divide the distance traveled by the time it took to travel that distance. The formula is: speed = distance/time.
So, speed = 159 miles / 3 hours = 53 miles per hour
b. Since Zahara's trip and Jorge's trip took the same amount of time, the only way that Jorge could have traveled a shorter distance than Zahara is if he drove slower. So, it can be inferred that Jorge drives slower than Zahara's dad.
Step-by-step explanation:
Can you think of the integers for x, y, and z so that x³+y³+z³=8? Sure. One answer is x = 1, y = -1, and z = 2. But what about the integers for x, y, and z so that x³+y³+z³=42? l'll give brainiest
Answer: x3 +y3 20
Step-by-step explanation:
PLSSS HELP due at 9:00
Answer: 39.5
Step-by-step explanation: 47.5 + 93 = 140.5
180 - 140.5 = 39.5
I HOPE YOU TURN IT IN BEFORE THE DUE DATE !! GOOD LUCK!
Ms. Rios wants to store her son’s old clothes in a box. She believes the clothes will fit in a box that has a volume of 12 cubic feet³. What could be the dimensions of Ms. Rios' box?
Is it
6 ft x 4 ft x 2 ft
4 ft x 4 ft x 4 ft
4 ft x 2 ft x 2 ft
3 ft x 2 ft x 2 ft
Please Help
Answer:
the fourth one
Step-by-step explanation:
3x2x2 = 12
Answer:Last one,
Step-by-step explanation:
it equals 12, making it the only one that works.
Does each equation represent exponential decay or exponential growth?
Drag and drop the choices into the boxes to correctly complete the table.
Note: If an equation is neither exponential growth nor exponential decay, do not drag it to the table.
A = (0.97)t
y=3.7(0.2)t
y= 9/11(4)t
P=7/9(5/4)t
V=0.8(9)t
P=0.9(8.3)t
g(x) = 2.1(x)
Solving the provided question, we can say that Not exponential growth or decay (no exponent) G(x)=1.3(x)
What is exponential growth?The process of quantity growth over time is called exponential growth. occurs when the rate at which a quantity changes instantaneously with time is proportionate to the quantity itself. A data pattern that shows bigger increases with time is known as exponential growth. Compound interest yields exponential rewards in the world of finance. Savings accounts with compound interest sometimes see exponential growth. characterized by an extremely fast exponential growth rate rise (in size or extent). exponentially.
The base of the exponentiation is the strongest indicator of whether an equation exhibits an exponential growth or decline.
It will be an exponential growth if the base is greater than 1.
It will be an exponential decay if the base is less than 1.
There won't be any exponential growth or decay if the function's exponent is zero.
Therefore: Decay exponential (base less than 1:
W = 9/8(3/5)t
f(t) = (3/4)t
L = 4.2(0.6)t
Exponential Growth (base larger than 1:
L = 0.25(12)t
W = 0.5(2.1)t
f(t) = 2/3(6)t
Not exponential growth or decay (no exponent):
G(x)=1.3(x)
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The figure shows an object O in front of a plane mirror. Determine from which locations A-D the object's image is visible.
O A O B O C O D
An object O is displayed in front of a plane mirror in the figure. The image of the object is visible from locations B and C.
An object's image is created by a plane mirror by reflecting light rays that originate from the object. The object and its image appear to be equally distant from the mirror because the image created by a plane mirror is imaginary, upright, and the same size as the object.
As light from the object enters the plane mirror, it reflects each beam in accordance with the Law of Reflection. At the intersection of the reflected beams' extensions, the image is visible.
Therefore, the location must be on the reflected ray's path for the image to be visible. Here, locations B and C are on the path of the reflected ray. So the image is visible in these locations. But not in locations A and D.
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Jean joined a gym. It costs $60 per month and $2 per hour of boxing classes. Write an expression to represent the total cost of his gym, including the boxing classes for a month. Then determine how much it would cost him, if he spent 19.5 hours in gym class that month.
Answer:
y= 60+ 2x and if he spent 19.5 hours then he would pay $99 for that month
Step-by-step explanation:
$60 PER MONTH + $2 per hour of boxing classes
X represents the total hours he had the boxing classes
equation: y= 60+ 2x
if he spent 19.5 hours in boxing class
plug in 19.5 hours to X
99 = 60+ 2(19.5) = 60 + 39
Suppose that for a graduating class of seniors at a local high school, the average GPA is 3.21. If a sampling distribution of size n = 20 is created from this population of seniors, what would be the expected mean of the sampling distribution?
0.72
1.67
0.161
3.21
Suppose that for a graduating class of seniors at a local high school, the average GPA is 3.21. The expected mean of the sampling distribution would be 3.21.
What is the average GPA?Generally, This is because the sampling distribution of the mean is a probability distribution of the means of all possible samples of the same size from the same population.
The mean of the sampling distribution is equal to the population mean, in this case, 3.21.
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Listed below are the lead concentrations (in mg/g) measured in different samples of a medicine. What do the results suggest about the safety of this medicine? What do the decimal values of the listed amounts suggest about the precision of themeasurements?
O 18.5
O 14.5
O 20.5
O 7.5
O 18
O 8.5
O 8
O 9.5
O 15
The results suggest that the lead concentrations in the medicine samples are relatively low, as all of the values are below 20 mg/g. However, it would be necessary to compare these values to a standard, such as the acceptable limit of lead in medicine set by a regulatory body, in order to determine if the medicine is considered safe.
The decimal values of the listed amounts suggest that the measurements were precise, as they are all reported to one decimal place. It means that the measurement was done with a precision of 0.1. In addition to that, the measurements are consistent and close to each other, which also indicates that the measurements were precise.
It's important to keep in mind that precision and accuracy are different things. Precision refers to the degree of reproducibility of a measurement, while accuracy refers to the degree of closeness of a measurement to the true value. In this case, the measurements are precise but it's not specified if the measurements are accurate or not.
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In a basket there are 20 apples and 10 oranges. Find the percentage
Answer:
0.2% apples & 0.1 oranges resp
Haley and Emily bought a gift for their
father. Haley contributed 7 dollars for every
11 dollars the gift cost. Overall Emily
contributed 33 fewer dollars than Haley.
What was the cost of the present?
Hint: Who contributed more? Think about
what the 7 and the 11 each represent.
Haley lives with her sister Emily, and together they care for their parents' home, who have been traveling the world for the past two years.
What does Haley do as a wife in Stardew Valley?Haley moves in with you after the wedding (obviously) and spends most of her time around the house and farm. She could be. On Mondays, though, she does have unusual schedules. Due to the fact that Haley comes across as rude when first spoken to, she is one of the least liked NPCs in Stardew Valley. But as you get to know her better and she starts to show you her softer side, she'll become much more endearing. Haley's favorite gifts can be harder to find early on, making her not the most straightforward marriage candidate to woo. However, she is worth the effort because she is a kind and caring person, just like the others.To learn more about endearing refer to:
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Determine if the scenario involves mutually exclusive events.
A box of chocolates contains five milk chocolates, five dark chocolates, and three
white chocolates. You randomly select a chocolate. It is a milk chocolate or a dark chocolate.
A) Not mutually exclusive
B) Mutually exclusive
The given question where we randomly select a chocolate. It is a milk chocolate or dark chocolate is a mutually exclusive chocolate
How do you calculate probability?
Simply put, probability is the likelihood that something will occur. When we don't know how an event will turn out, we might discuss the likelihood or likelihood of several outcomes. Statistics is the study of occurrences that follow a probability distribution. Probability is a measure of how likely something is to occur. The probability is computed by dividing the total number of possible outcomes by the number of possible ways the event might occur.
In the given question
we see
Selecting a milk chocolate or dark chocolate is mutually exclusive event as they are not related to each other
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ILL GIVE BRAINLIEST IF CORRECT NEED HELP ASAP PLS
Answer:
A
Step-by-step explanation:
Answer: y = (3/4)x - (7/4).
Step-by-step explanation:
Subtract 3x from both sides of the equation. Divide both sides of the equation by -4. Simplify. We get that solving 3x - 4y = 7 for y gives y = (3/4)x - (7/4).
Find the following cardinalities:
a. A when A= {5, 6, 7, 8, ..., 35}.
b. A when A= {xEZ: -2 ≤ x ≤ 90}.
c. An B| when A= {x EN: x ≤ 31} and B = {x EN: x is prime}
Therefore , the solution of the given problem of set A when A= {5, 6, 7, 8, ..., 35}.
What does the math set mean?In mathematics, a set is a logically arranged group of items that can be represented in either set-builder or roster form. Sets are typically denoted by curly brackets; for instance, A = 1, 2, 3, 4 is a set.
Here,
The set A contains integers between 5 and 35, inclusive. So, the cardinality of set A is the number of integers between 5 and 35, which is 35-5+1 = 31.
b. A when A= {xEZ: -2 ≤ x ≤ 90}.
The set A contains all integers between -2 and 90, inclusive. So, the cardinality of set A is the number of integers between -2 and 90, which is 90-(-2)+1 = 93.
c. An B| when A= {x EN: x ≤ 31} and B = {x EN: x is prime}
The set A contains all the natural numbers less than or equal to 31. The set B contains all the prime numbers less than or equal to 31. There are 15 prime numbers less than or equal to 31: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. So, the cardinality of set B is 15.
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Use perfect squares to find two consecutive integers to complete each inequality.
1) ? < √13 < ?
2) ? < √78 < ?
3) ? <√181 < ?
(Please hurry)
Following are the required perfect squares of each of the following.
A square number, sometimes known as a perfect square, is an integer that is the square of another integer, or the result of multiplying another integer by itself. For instance, 9 is a square number since it equals 32 and is represented by the symbols 3×3.
Use perfect squares to find two consecutive integers to complete each inequality -
1. ? < √13 < ?
⇒ 9 < 13 < 16
⇒ 3 < √13 < 4
2. ? < √78 < ?
⇒ 64 < 78 < 81
⇒ 8 < √78 < 9
3. ? <√181 < ?
⇒ 169 < 181 < 196
⇒ 13 < √181 < 14
Comparing values -
π/2 < √34 5/6 √22 ⇒ 29/6 > √22-√29 = -5 2/5Arranging from least to greatest -
9/2 < √10 < 22/7 < 5/3
5 2/5 < 5.3 < 3√3 < √25
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I need help ASAP
Look at the image below
Answer:
2 only
Step-by-step explanation:
Answer:2 and 4
Step-by-step explanation:
if you divide 164 by all the numbers you only get whole numbers when you divide it by 2 and 4 not 3 and 6
A car is purchased for $22,500. After each year, the resale value decreases by 30%. What will the resale value be after 4 years?
Use the calculator provided and round your answer to the nearest dollar.
On solving the provided question, exponents, resale value be $5339 after 4 years.
Exponents in math: what are they?The number of times a number has been multiplied by itself is referred to as an exponent. For instance, the expression 2 to the third (written as 23) means 2 x 2 x 2 = 8. and 2 x 3 = 6 are not equivalent. Keep in mind that any number is itself when raised to the power of 1.
y = cn^x
y is the new price, c is the starting price, n is the rate of increase (or decrease), and x is the number of years. I won't bore you with the various names this exponent use has in calculus and economics.
In our instance:
By calculating how much of the price will remain, we can determine n.
1.00 — 0.25 = 0.75
f(x) = (22500)(0.75)^x
f(5) = (22500)(0.75)^5
=5339.35
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