On a piece of graph paper, plot the following points: A (3, 1), B (1, 5), C (9, 9), and D (11, 5). These coordinates will be the vertices of a quadrilateral. How would you use the distance formula and the slope formula to prove that this figure is actually a rectangle?
Answers
The given coordinates are actually a rectangle
How to determine the quadrilateral type?
The coordinates are given as:
A (3, 1), B (1, 5), C (9, 9), and D (11, 5).
Calculate the distance between the coordinates using:
[tex]d = \sqrt{(x_2 -x_1)^2 +(y_2 -y_1)^2[/tex]
So, we have:
[tex]AB = \sqrt{(3 -1)^2 +(1-5)^2} =\sqrt {20[/tex]
[tex]BC = \sqrt{(1 -9)^2 +(5-9)^2} =\sqrt {80[/tex]
[tex]CD = \sqrt{(9 -11)^2 +(9-5)^2} =\sqrt {20[/tex]
[tex]DA = \sqrt{(11 -3)^2 +(5-1)^2} =\sqrt {80[/tex]
The above shows that the opposite sides are congruent
Next, we calculate the slopes using:
m = (y2- y1)/(x2- x1)
So, we have:
AB = (1- 5)/(3-1) = -2
BC = (5- 9)/(1-9) = 1/2
CD = (9- 5)/(9-11) = -2
DA = (5- 1)/(11-3) = 1/2
The slopes of adjacent sides are opposite reciprocals.
This means that the sides are perpendicular
Hence, the given coordinates are actually a rectangle
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Related Questions
A farmer with 3380 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?
Answers
The largest possible total area of the four pens is 285610 ft²
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let y represent the length of the pen and x represent the width of the pen.
Hence:
Perimeter = x + y + x + y + x + x + x = 2y + 5x
2y + 5x = 3380
y = (3380 - 5x)/2 (2)
Area (A) = xy
A = x * (3380 - 5x)/2
A = (3380x - 5x²)/2
The maximum area is at A' = 0, hence:
A' = (1/2)(3380 - 10x)
0 = (1/2)(3380 - 10x)
10x = 3380
x = 338 ft.
2y + 5(338) = 3380
y = 845 ft.
Largest possible total area = 338 * 845 = 285610 ft²
The largest possible total area of the four pens is 285610 ft²
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Lines a and b are parallel and lines e and f are parallel what is the value of x
Answers
The value of x using the alternate exterior angle theorem is 82°
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
An angle is formed from the intersection of two or more lines. Types of angles are acute, obtuse and scalene.
From the diagram:
x = 82° (alternate exterior angles)
The value of x using the alternate exterior angle theorem is 82°
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Solve for x if AB = 5x and BC = 3x + 24.
Answers
The value of x according to the given task is; 12.
What is the value of x?It follows from the task content that the value of x is to be determined.
Since, lines AB and BC are both tangent to the circle and they both share a point of intersection; it follows that;
5x = 3x +24
collect like terms2x = 24
divide both sides by 2x = 24/2
x = 12.
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Solve the inequality by the method of intervals
Answers
The solution of the inequality by method of interval is x = [5,∝) and x = [-2,0]∪[8,-∝).
What is the solution of the given inequality problems ?The first inequality is (x² - 25)/(x + 10) ≥ 0
⇒ x² - 25 ≥ 0
⇒ x² ≥ 25
∴ x ≥ 5
Thus the domain of the given inequality from method of interval is x = [5,∝).
The second inequality is (x + 2)(x² - 64)/(x² + 45) ≤ 0
⇒ (x + 2)(x² - 64)≤ 0
Either (x + 2)≤ 0 or (x² - 64)≤ 0
Either x ≤ -2 or x ≤ 8
Thus , the combined domain of the given inequality from method of interval is x = [-2,0]∪[8,-∝) .
Therefore, the solution of the inequality by method of interval is x = [5,∝) and x = [-2,0]∪[8,-∝).
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simplify these fractions
xy-xz/y-z
Answers
Answer:
[tex]\frac{xy-xz}{y-z} = x[/tex]
Step-by-step explanation:
Hello!
We can simplify the fraction by factoring and canceling out common factors.
Simplify[tex]\frac{xy-xz}{y-z}[/tex][tex]\frac{x(y - z)}{y-z}[/tex][tex]\frac{x(\not y\not-\not z)}{\not y \not- \not z}[/tex][tex]x[/tex]We are left with x as the final simplified form.
I’m confused in the exponential
Answers
Answer:
this mean in case of multiplying exponents with same base you should add the exponent and keep base same.
example:
[tex] {6}^{4} \times {6}^{5 } = {6}^{4 + 5} \\ = {6}^{9} [/tex]
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 7.9 years, and standard deviation of 0.9 years.
If you randomly purchase one item, what is the probability it will last longer than 9 years?
Answers
If you randomly purchase one item, the probability it will last longer than 9 years is; 11.12%
How to find the probability from z-score?
The formula for Z-score is;
z = (x' - µ)/σ
where;
x' = sample mean
µ = population mean
σ = standard deviation
Thus;
z = (9 - 7.9)/0.9
z = 1.22
From online p-value from z-score calculator, we have;
probability = 0.1112 = 11.12%
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The temperature at 6am is -8f , at 9am the temperature is 6f . If the temperature rises at a constant rate , what will the temperature be at 3:00pm
Answers
Answer:
24 F
Step-by-step explanation:
Temp increases by 4F per hour.
-8F - 6F = 12F / 3hours
Assume that the random variable X is normally distributed, with mean and standard deviation . Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded.
Answers
Using the normal distribution, the probability that x is of at most 40, represented by option B, is given as follows:
[tex]P(X \leq 40) = 0.1587[/tex]
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.The mean and the standard deviation of the distribution are given, respectively, by:
[tex]\mu = 51, \sigma = 11[/tex]
The probability that x is of at most 40 is the p-value of Z when X = 40, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{40 - 51}{11}[/tex]
Z = -1
Z = -1 has a p-value of 0.1587.
Hence the probability is:
[tex]P(X \leq 40) = 0.1587[/tex]
Which is the part to the left of X = 40 of the distribution, hence option B is correct.
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A Food Marketing Institute found that 29% of households spend more than $125 a week on groceries. Assume the population proportion is 0.29 and a simple random sample of 207 households is selected from the population. What is the probability that the sample proportion of households spending more than $125 a week is less than 0.31?
There is a ___ probability that the sample proportion of households spending more than $125 a week is less than 0.31. Round the answer to 4 decimal places.
Answers
Using the normal distribution, there is a 0.7357 = 73.57% probability that the sample proportion of households spending more than $125 a week is less than 0.31.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.By the Central Limit Theorem, for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1 - p)}{n}}[/tex], as long as [tex]np \geq 10[/tex] and [tex]n(1 - p) \geq 10[/tex].The estimate and the sample size are:
p = 0.29, n = 207.
Hence the mean and the standard error are given as follows:
[tex]\mu = p = 0.29[/tex].[tex]s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.29(0.71)}{207}} = 0.0315[/tex].The probability that the sample proportion of households spending more than $125 a week is less than 0.31 is the p-value of Z when X = 0.31, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.31 - 0.29}{0.0315}[/tex]
Z = 0.63
Z = 0.63 has a p-value of 0.7357.
0.7357 = 73.57% probability that the sample proportion of households spending more than $125 a week is less than 0.31.
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In a recent survey,77 % of the community favored building a health center in their neighborhood. If 14 citizens are chosen, find the probability that exactly 8 of them favor the building of the health center. Round to the nearest three decimal places.
Answers
The probability of that exactly 8 of them favor the building of the health center is 0. 399
How to determine the probabilityFrom the information given, we have that 70% of the community favors the building of the health center
Number of citizens chosen = 14
P(8) = [tex]\frac{8}{14}[/tex] × [tex]\frac{70}{100}[/tex]
P(8) = [tex]0. 57[/tex] × [tex]0. 7[/tex]
P(8) = [tex]0. 399[/tex]
Thus, the probability of that exactly 8 of them favor the building of the health center is 0. 399
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A group of people were given a personality test to determine if they were Type A or Type B. The results are shown in the table below: Male Female Type A 65 85 Type B 38 12 Compare P(Male or Type B) with P(Male | Type B). (5 points) P(Male or Type B) > P(Male | Type B) P(Male or Type B) = P(Male | Type B) P(Male or Type B) < P(Male | Type B) There is not enough informatio
Answers
The correct statement regarding the probabilities is given as follows:
P(Male or Type B) > P(Male | Type B)
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
In this problem, there is a total of 200 people, of which 103 are males and 12 are Type B females, hence:
P(Male or Type B) = 115/200 = 0.575
Of the 103 males, 38 are Type B, hence:
P(Male|Type B) = 38/103 = 0.3683.
Hence:
P(Male or Type B) > P(Male | Type B).
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One yellow daisy and two pink carnations tied with ribbon will be used for corsages at a Mother's Day banquet. Each corsage takes 6 inches of ribbon. The daises are $0.85 each, the carnations are $0.50 each, and the ribbon is $0.24 per foot. There will be 24 corsages. What is the total cost?
Answers
Answer:
$47.28
Step-by-step explanation:
The total cost is found by adding the costs of the components.
Component costThere is one daisy, at $0.85 each.
There are to carnations at $0.50 each, for a total of 2×$0.50 = $1.00.
The is 6 inches of ribbon. That amount is 1/2 foot, so its cost is ...
(1/2 ft)×$0.24/ft) = $0.12.
Total costThe total cost of the components of one corsage is ...
daisy cost + carnation cost + ribbon cost = $0.85 +1.00 +0.12 = $1.97
There are 24 corsages, so their total cost will be 24 times this amount, or ...
24 × $1.97 = $47.28
The total cost of the 24 corsages is $47.28. All the costs of the components required for making a corsage are added to get the total cost of the corsage.
How to calculate the total cost?Total cost is the sum of all the individual component costs.Consider a and b are the two components with a cost K of each. Then the total cost = aK + bKCalculation:It is given that,
The components and their costs:
The cost of each daise = $0.85
The cost of each carnation = $0.50
The cost of the ribbon per foot = $0.24 (1 foot = 12 inches)
The components for making one single corsage:
One yellow daisy + two pink carnations + 6 inches of ribbon = 1 corsage
Then, calculating the price for one single corsage,
1 × $0.85 + 2 × $0.50 + (1/2) × $0.24 = cost of 1 corsage
⇒ $0.85 + $1 + $0.12
⇒ $1.97
For 24 corsages, the total cost = 24 × $1.97 = $47.28
Therefore, the total cost of the 24 corsages is $47.28.
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Find the exact solutions of x2 − 3x − 5 = 0 using the quadratic formula. Show all work! 75 points please help!!!!!
Answers
Answer:
[tex]x=\dfrac{3+ \sqrt{29}}{2}, \quad \dfrac{3- \sqrt{29}}{2}[/tex]
Step-by-step explanation:
Quadratic Formula
[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]
Given quadratic equation:
[tex]x^2-3x-5=0[/tex]
Define the variables:
[tex]\implies a=1, \quad b=-3, \quad c=-5[/tex]
Substitute the defined variables into the quadratic formula and solve for x:
[tex]\implies x=\dfrac{-(-3) \pm \sqrt{(-3)^2-4(1)(-5)}}{2(1)}[/tex]
[tex]\implies x=\dfrac{3 \pm \sqrt{9+20}}{2}[/tex]
[tex]\implies x=\dfrac{3 \pm \sqrt{29}}{2}[/tex]
Therefore, the exact solutions to the given quadratic equation are:
[tex]x=\dfrac{3+ \sqrt{29}}{2}, \quad \dfrac{3- \sqrt{29}}{2}[/tex]
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Answer:
[tex]\boxed {\frac{3+\sqrt{29}}{2}} \boxed{\frac{3-\sqrt{29}}{2}}[/tex]
Step-by-step explanation:
Quadratic Formula :
[tex]\boxed {\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}}[/tex]
We are given the equation x² - 3x - 5 = 0.
Here,
a = 1b = -3c = -5Solving :
3 ± √3² - 4(1)(-5) / 2(1)3 ± √29 / 2Hence, the solutions are :
[tex]\boxed {\frac{3+\sqrt{29}}{2}} \boxed{\frac{3-\sqrt{29}}{2}}[/tex]
If f(x) = x and g(x) = 2, what is (fog)(x)?
Answers
Step-by-step explanation:
=fog(x)
−2x=0
x(x−2)=0
x=0,2
A certain radioactive material decays in such a way that the mass in kilograms remaining after t years is given by the function
m(t)=120e−0.018t
How much mass remains after 50 years?
Answers
The mass of radioactive material remaining after 50 years would be 48.79 kilograms
How to determine the amountIt is important to note that half - life is the time it takes for the amount of a substance to reduce by half its original size.
Given the radioactive decay formula as
m(t)=120e−0.018t
Where
t= 50 years
m(t) is the remaining amount
Substitute the value of t
[tex]m(t) = 120e^-^0^.^0^1^8^(^5^0^)[/tex]
[tex]m(t) = 120e^-^0^.^9[/tex]
Find the exponential value
m(t) = 48.788399
m(t) = 48.79 kilograms to 2 decimal places
Thus, the mass of radioactive material remaining after 50 years would be 48.79 kilograms
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1. Is the average change in population is the same now as it was 50 years ago? Explain your answer.
HURRY PLEASE NEED IT URGENTLY!!!!!!!!!!!!
Answers
No, the average change in population is not the same as it was 50 years ago.
Reason in Support of the Answer:
In many important ways, the demographic future of the United States and the rest of the world is substantially different from the recent past. The world's average population approximately tripled between 1950 and 2010, and the U.S. population nearly doubled.
However, it is anticipated that between 2010 and 2050, both globally and in the United States, average population growth will be substantially slower and will disproportionately favor the oldest age groups. Hence it is seen that the average change in population is never constant. It depends on the demographic trends and conditions, whether the average change in population will be larger or comparatively trivial in the future. And, similarly, it can be said that the average change in population is not the same as it was 50 years ago.
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Raymond’s gas tank is 1/3 full. After he buys 11 gallons of gas, it is 5/6 full. How many gallons can Raymond’s tank hold?
Answers
A = {1, 2, 6}
B = {x|x is an odd whole number less than 8}.
Find An B.
Answers
Answer:
A∩B = { 1 }
Step-by-step explanation:
Consider the sets :
A = {1, 2, 6}
B = {x ;where x is an odd whole number less than 8}
Suppose the numbers of B are natural numbers (not integers).
Then
B = {1 , 3 , 5 , 7}
We obtain :
A = {1, 2, 6} and B = {1 , 3 , 5 , 7}
The intersection of A and B or the set A∩B :
is the set of the common numbers of A and B.
Then
A∩B = { 1 }
The set A intersection B(or, A∩B) for the given set A and set B exists {1}.
What is A intersection B, (A∩B)?"The set A intersection B(or, A∩B) exists described as the set composed of all elements that belong to both A and B".
The intersection of a set A with a B exists the set of elements that exist in both sets A and B. The intersection exists represented as A∩B.
The intersection of two mathematical objects exists where they overlap. For geometric objects, the intersection exists as a point or group of points where the objects cross.
For numerical or non-numerical sets, the intersection exists every number or object the two sets contain in common.
Given: A = {1, 2, 6}
B = {x | x is an odd whole number less than 8}.
From the above information, we get
⇒ B = {1, 3, 5, 7}
If A = {1, 2, 6} and B = {1, 3, 5, 7}
then A∩B = {1, 2, 6} ∩ {1, 3, 5, 7}
A∩B = {1}
Therefore, the correct answer A∩B exists {1}.
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Follow the instructions below to show two different ways of filling a square that has sided of length a + b with triangles and squared without gaps or overlaps.
Answers
The proofing for the triangle is illustrated below.
How to illustrate the information?The proofing for a² + b² = c² in the triangle is illustrated.
It should be noted that the longest side in the triangle is the hypothenuse. The hypothenuse is gotten by finding the square root of the opposite and the adjacent.
Therefore, a² + b² = c²
Let's say a = 3 and b = 4, the value of c will be:
c² = a² + b²
c² = 3² + 4²
c² = 9 + 16
c² = 25
c = ✓25
c = 5
Also, the square and triangle is illustrated and attached.
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will give brainliest please help
Answers
Answer:
A. 7x + 3x = 10x, B. 6s - 3s + 4s = 7s, C. 4T - 6T = -2T, D. 4xy + 3xy + x = 7xy + x, E. (6x) ^2 = 36x^2, F. 6x/2x = 3.
Step-by-step explanation:
A. explanation: Add 7x and 3x. B. explanation: Simplify the expression. C. explanation: Subtract 6T from 4T. D. explanation: Add 4xy and 3xy. E. explanation: Simplify the explanation. F. explanation: Simplify the expression.
Answer: give her brainliest
Step-by-step explanation:
hey mackenzie :)
How many more people attended from the 40-60 class than the 60-80
Answers
Answer:7 people
Step-by-step explanation:
Answer:
7
Step-by-step explanation:
Claire has 18 nail polish bottles while Becky has 2/3 of that amount. How many bottles of nail polish does Becky have?
Answers
Answer: 12
Step-by-step explanation:
Your car broke down and it's time to get a new one. You've found just the right one to fit your needs but you want to be sure you are making a good financial decision,
so you ask the lender for an amortization table. What key pieces of information will you be able to see in the amortization table to help you make your decision?
Answers
Answer:
what is the monthly payment or ask for the mileage use for the car
What is the solution of p - 12 > -18
p> -30
p<-6
p> -6
p < -30
Answers
Answer:
p > -6
Step-by-step explanation:
1) Add 12 to both sides.
p > -18 + 12
2) Simplify -18 + 12 to -6.
p > -6
Which inequality is represented by the given graph?
C) y ≤ -3x+6
D) y ≥ 3x+6
Answers
Answer: c, y ≤ -3x+6
Step-by-step explanation:
for example we could use a point on the graph such as (-2, 0)
now we would insert the numbers into the inequality:
y ≤ -3x+6
= 0 ≤ -3(-2) +6
= 0 ≤ 6+6
= 0 ≤ 12
which is true! therefore it is y ≤ -3x+6
) Consider the following probability density function of the random variable X. f(x) = k(2x + 3x2 ), 0 ≤ x ≤ 2. (i) Determine the value of the constant k.
Answers
I assume [tex]f(x)=0[/tex] otherwise. If [tex]f(x)[/tex] is indeed a proper PDF, then its integral over the support of [tex]X[/tex] is 1.
[tex]\displaystyle \int_{-\infty}^\infty f(x) \, dx = k \int_0^2 (2x + 3x^2) \, dx = 1[/tex]
Compute the integral.
[tex]\displaystyle \int_0^2 (2x + 3x^2) \, dx = (x^2 + x^3)\bigg|_{x=0}^{x=2} = (2^2 + 2^3) - (0^2 + 0^3) = 12[/tex]
Then
[tex]12k = 1 \implies \boxed{k=\dfrac1{12}}[/tex]
Select all the terms that are like terms
Answers
Answer:
-8 x^2
1.25 x^2
1/2 x^2
Step-by-step explanation:
They are like terms because they have the same variable x^2, yx^2 is not the same because it has the variable y
Answer:
the 2nd, 3rd and 4th options (all with x²)
Step-by-step explanation:
"like terms" means the exactly same variable(s) with exactly the same exponents of these variables.
constants in the terms can be different and even have different signs.
5yx² is not "like", because of the additional y variable.
f is a trigonometric function of the form f(x)=asin(bx+c)+d. The function intersects its midline at (π,−4) and has a minimum point at ( π/4 , − 5.5 ). Find a formula for f(x). Give an exact expression.
Answers
The given point on the midline of (π, -4), and the minimum point of (π/4, -5.5), give the exact expression for f(x) as; f(x) = 1.5•sin((2/3)•(x - π)) - 4.
Which method can be used to find the trigonometric function?The amplitude, a = The difference between the y-values at the minimum point and the midline
Therefore;
a = -4 - (-5.5) = 1.5
d = The difference between the y-values of the midline and the x-axis
Therefore;
d = -4 - 0 = -4
The given function is presented as follows;
f(x) = a•sin(b•x + c) + d
Which gives;
-4 = 1.5•sin(b•π + c) - 4
sin(b•π + c) = 0
(b•π + c) = 0
-c/b = π
c = -b•π
Therefore;
-5.5 = 1.5•sin(b•π/4 + c) - 4
-1.5 = 1.5•sin(b•π/4 + c)
-1 = sin(b•π/4 - b•π) = sin(-3•b•π/4)
-3•b•π/4 = arcsine (-1) = -π/2
-3•b•π/4 = -π/2
3•b/4 = 1/2
b = 2/3
c = -(2/3)•π
Therefore, f(x) = a•sin(b•x + c) + d, gives;
f(x) = 1.5•sin((2/3)•x - (2/3)•π) - 4
By simplification, the exact expression for f(x) is therefore;
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PLEASE HELP!!!!
Cooling towers are used to remove or expel heat from a process. A cooling tower's walls are modeled by x squared over 400 minus quantity y minus 110 end quantity squared over 2304 equals 1 comma where the measurements are in meters. What is the width of the cooling tower at the base of the structure? Round your answer to the nearest whole number.
100 meters
50 meters
48 meters
40 meters
Answers
Answer:
(a) 100 meters
Step-by-step explanation:
You want the width at the base of a cooling tower whose walls are modeled by the equation (x/20)² -((y-110)/48)² = 1.
BaseThe width at the base will be the difference in the values of x when y=0.
(x/20)² -((0 -110)/48)² = 1
(x/20)² = 1 +3025/576
x = ±20/24√3601 ≈ ±50.01
The width will be 50.01 -(-50.01) ≈ 100 meters.
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