Answer:
[tex]P(A|B) = \frac{32}{49}[/tex]
Step-by-step explanation:
We use the conditional probability formula to solve this question. It is
[tex]P(A|B) = \frac{P(A \cap B)}{P(B)}[/tex]
In which
P(A|B) is the probability of event A happening, given that B happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(B) is the probability of B happening.
We have that:
[tex]P(A \cap B) = \frac{4}{7}, P(B) = \frac{7}{8}[/tex]
So
[tex]P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{4}{7}}{\frac{7}{8}} = \frac{4}{7}*\frac{8}{7} = \frac{32}{49}[/tex]
Then
[tex]P(A|B) = \frac{32}{49}[/tex]
The red function was transformed into the blue function. Which transformations have occurred?
Answer:
Translated 3 units down 2 units left.
Step-by-step explanation:
It looks like the functions are the same, so if you see, it was translated 3 units down and 2 units left.
Select 2 strategies that we can use to add 319 +291.
Choose 2 answers:
CORRECT (SELECTED)
Add 320 + 290.
INCORRECT (SELECTED)
Add 319 + 300 + 9.
Add 310 + 300.
Answer:
I think it is 310 plus 300 and
320 plus 390
Step-by-step explanation:
The balance on Taylor's credit card is $2000 it has an interest rate of 12.5% she wants to compare the difference between paying $75 and $100 of the monthly balance how much does she save in interest and fees if she pays $100 instead of $75?
Answer:
so she saved 82 cents in the interest the month after
Step-by-step explanation:
case 1: payment is $75
interest on 2000 = 0.125/12×2000 = $20.83
so the actual repayment on the balance = (75-20.83) = $54.17
therefore,balance =$(2000-54.7)=$1945.83
interest in the next month = $20.27
case 2: payment is $100
interest on 2000 is still $20.83
repayment = $79.73
balance = $1920.27
interest in the next month = 20.01
so she saved 82 cents in the interest the month after
Answer:
$101
Step-by-step explanation:
Check link explanation.
(-16) - (+12) + (-୨)
Answer:
-37
Step-by-step explanation:
Which of the following represents the graph of f(x) = 3x − 2?
graph of exponential rising up to the right, through the point 0, 3
graph of exponential rising up to the right, through the point 0, negative 1
graph of exponential rising up to the right, through the point 0, 1
graph of exponential rising up to the right, through the point 0, negative 2
Answer:
graph of exponential rising up to the right, through the point 0, negative 2
Step-by-step explanation:
I graphed the function on the graph below so you can see that it rises to the right and goes through the point (0,-2).
Answer:
graph of exponential rising up to the right, through the point 0, negative 2
Step-by-step explanation:
f(x)=3x-2
f(0)=3*0-2= -2
(0, -2) is the last given option
ASAP! GIVING BRAINLIEST! Please read the question THEN answer CORRECTLY! NO guessing. I say no guessing because people usually guess on my questions.
Answer:
A
Step-by-step explanation:
(f-g)(x) means that the 2 functions are being subtracted.
[tex]3^{x}[/tex] +10x -(5x-3) =[tex]3^{x}[/tex] +10x-5x+3
simplify!
[tex]3^{x}[/tex] +5x+3
the answer is A
is x^2+3x+8 a monomial, binomial, or trinomial
Answer:
A trinomial
Step-by-step explanation:
Since there are three terms, x^2, 3x, and 8, it is a trinomial
Which parent function is f(x) = x^2
O A. The linear parent function
O B. The absolute value parent function
O C. The quadratic parent function
D. An exponential parent function
Answer:
C.
Step-by-step explanation:
the standard form of a QE is ax2+bx+c. This includes x squared, and when graphed, it forms the graph of a QE, a parabola.
Hope this helps!
The parent function of f(x) = x^2 is the quadratic parent function.
We have given that,
f(x) = x^2
We have to determine the parent function of the given function.
Here the highest power of the x is 2.
We remember that the quadratic equation has the highest power is 2.
What is the formula for the quadratic equation?The standard form of a quadratic equation is ax^2+bx+c.
This includes x squared and when graphed it forms the graph of a quadratic equation is a parabola.
We have given function is f(x) = x^2
Therefore the value of the a,b and c are,
a=1
b=0 and c=0
Therefore, option C is correct.
The parent function of f(x) = x^2 is the quadratic parent function.
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Use a graphing calculator to sketch the graph of the quadratic equation, and then state the domain and range y=2x^2-x+3
Answer:
For the domain since we have a quadratic function then the domain would be all the real numbers:
[tex] D =[x | x \in R][/tex]
And if we want to find the range we can find the vertex:
[tex] v_x = -\frac{b}{2a}= -\frac{-1}{2*2}= \frac{1}{4}[/tex]
And now we can find th coordinate of y of the vertex like this:
[tex] f(V_x) = 2(\frac{1}{4})^2 -(1/4) +3 =2.875[/tex]
And then the range would be:
[tex] R=[x \geq 2.875][/tex]
Step-by-step explanation:
We have the following function given:
[tex] y = 2x^2 -x +3[/tex]
For this case we can plot the function with a calculator and we got the plot in the figure attached.
For the domain since we have a quadratic function then the domain would be all the real numbers:
[tex] D =[x | x \in R][/tex]
And if we want to find the range we can find the vertex:
[tex] v_x = -\frac{b}{2a}= -\frac{-1}{2*2}= \frac{1}{4}[/tex]
And now we can find th coordinate of y of the vertex like this:
[tex] f(V_x) = 2(\frac{1}{4})^2 -(1/4) +3 =2.875[/tex]
And then the range would be:
[tex] R=[x \geq 2.875][/tex]
A circle has a radius of 6 cm. Which calculation would be the correct calculation to work out the circumference?
Answer: C≈ 37.7cm
Step-by-step explanation: C=2πr=2·π·6≈37.69911cm
Hope this helps.
Answer:
C = 37.7 cm
Step-by-step explanation:
Circumference = 2πr
Where r = 6 cm, π = 3.14
C = 2(3.14)(6)
C = 37.69
C ≈ 37.7 cm
ABC represents a race path. Find the total distance of the race. Round your answer to the nearest meter.
Answer:
Race covers 1911 meters.
Step-by-step explanation:
Triangle ABC represents a race path.
Total distance covered by the race = Perimeter of the triangle ABC
We will apply Sine rule in the given triangle to find the unknown sides.
By Sine rule,
[tex]\frac{SinB}{AC}=\frac{SinA}{BC}=\frac{SinC}{AB}[/tex]
[tex]\frac{Sin35}{450}=\frac{Sin85}{AC}=\frac{Sin60}{AB}[/tex] [Since m∠A = 180° - (85 + 60)° = 35°]
[tex]\frac{Sin35}{450}=\frac{Sin85}{AC}[/tex]
AC = [tex]\frac{450\times \text{Sin85}}{\text{Sin35}}[/tex]
= 781.57 meters
[tex]\frac{Sin35}{450}=\frac{Sin60}{AB}[/tex]
AB = [tex]\frac{450\times \text{Sin60}}{\text{Sin35}}[/tex]
= 679.44 meters
Perimeter of the triangle = AB + BC + AC
= 679.44 + 450 + 781.57
= 1911.01
≈ 1911 meters
Therefore, the race covers 1911 meters.
Answer:
1911
Step-by-step explanation:
yes
What is the length of the hypotenuse of the triangle below?
45"
312
90°
312
O A. 6/2
O B. 3.2
O C.3
O D. 9.2
O E. 6
O F. 1
Answer:
Option (E)
Step-by-step explanation:
In the figure attached,
Given a isosceles right triangle with two equal legs measuring [tex]3\sqrt{2}[/tex] units
By Pythagoras theorem,
(Hypotenuse)² = (Leg 1)² + (Leg 2)²
Since, hypotenuse = h
Leg 1 = Leg 2 = 3√2
Now we substitute the values,
h² = (3√2)² + (3√2)²
h² = 18 + 18
h = √36
h = 6 units
Therefore, length of the hypotenuse is 6 units.
Option (E) will be the answer.
Using the Pythagorean Theorem, the length of the hypotenuse is: E. 6.
The Pythagorean TheoremGiven that the hypotenuse length of a right triangle is c, and the other legs are a and b, the Pythagorean Theorem states that: c = √(a² + b²).
Thus:
h = √((3√2)² + (3√2)²)
h = √(18 + 18)
h = √36
h = 6
Therefore, using the Pythagorean Theorem, the length of the hypotenuse is: E. 6.
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A kite flying in the air has a 91-ft string attached to it, and the string is pulled taut. The angle of elevation of the kite is 57 degrees. Find the height of the kite. Round your answer to the nearest tenth.
Answer:
H = 49.56 m
Step-by-step explanation:
We have,
A kite is flying in the air has a 91 ft string attached to it.
The angle of elevation of the kite is 57 degrees.
It is required to find the height of the kite. If we consider a right angled triangle, 91 ft is the hypotenuse. Let H is the height of the kite.
[tex]\cos\theta=\dfrac{H}{91}\\\\H=91\times \cos(57)\\\\H=49.56\ m[/tex]
Hence, the height of the kite is 49.56 m.
A manufacturer uses 34 yard of fabric in each skirt.
How many yards of fabric will the manufacturer use in 4 skirts? in 7 skirts? in 9 skirts?
Answer:
She would use 136 yards in 4 skirts, 238 yards in 7 skirts, and 306 yards in 9 skirts. Hope this helps
Step-by-step explanation:
A scientist studying water quality measures the lead level in parts per billion (ppb) at each of 49 randomly chosen locations along a water line. Suppose that the lead levels across all the locations on this line are strongly skewed to the right with a mean of 17 ppb and a standard deviation of 14 ppb. Assume that the measurements in the sample are independent. What is the probability that the mean lead level from the sample of 49 measurements T is less than 15 ppb? Choose 1 answer: A) Plæ <15) = 0.02 B) Plū<15) – 0.16 C) Plē <15) 0.30 D) Plö < 15) – 0.44 E) We cannot calculate this probability because the sampling distribution is not normal.
Complete Question
The complete qustion is shown on the first uploaded image
Answer:
The correct option is B
Step-by-step explanation:
From the question we are told that
The sample size is [tex]n = 49[/tex]
The mean is [tex]\mu = 17ppb[/tex]
The standard deviation is [tex]\sigma = 14 ppb[/tex]
Generally the standard error of this measurement is mathematically represented as
[tex]\sigma_z = \frac{\sigma}{\sqrt{n} }[/tex]
substituting values
[tex]\sigma_{\= x} = \frac{14}{\sqrt{49} }[/tex]
[tex]\sigma_{\= x} = 2[/tex]ppb
Now the probability that the mean lead level from the sample of 49 measurements T is less than 15 ppb represented as P(X < 15 )
Next is to find the z value
[tex]z = \frac{\mu -\sigma }{\sigma_{\= x}}[/tex]
[tex]z = \frac{15-17}{2}[/tex]
[tex]z = -1[/tex]
Now checking the z-table for the z-score of -1 we have
[tex]P(X<15) = P(Z < -1 )= 0.16[/tex]
Using the normal distribution and the central limit theorem, it is found that there is a 0.16 = 16% probability that the mean lead level from the sample of 49 measurements T is less than 15 ppb, hence option B is correct.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X. By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex], considering a sample size of at least 30 for a skewed variable.In this problem:
Mean of 17 ppb, hence [tex]\mu = 17[/tex].Standard deviation of 14 ppb, hence [tex]\sigma = 14[/tex].Sample of 49, hence [tex]n = 49, s = \frac{14}{\sqrt{49}} = 2[/tex]The probability that the mean lead level from the sample of 49 measurements T is less than 15 ppb is the p-value of Z when X = 15, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{15 - 17}{2}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] has a p-value of 0.16
0.16 = 16% probability that the mean lead level from the sample of 49 measurements T is less than 15 ppb, hence option B is correct.
For more on the normal distribution and the central limit theorem, you can check https://brainly.com/question/24663213
00:00
Muriel has been a member of the Solaris Gym for 372 days,
Ben has been a member for 1 year, 2 weeks, 3 days.
Part A
Who has been a member of the Solaris Gym longer?
Use the drop-down menus to show and explain your answer.
Choose...
has been a member longer because 1 year, 2 weeks, 3 days is
Choose...
than 372 days
00:00
Part B
How much longer? Assume that it is not a leap year. Enter your answer in the box.
days
Answer:
A: Ben has been a member longer because 1 year, 2 weeks, 3 days is longer than 372 days.
B: 10 days longer
Step-by-step explanation:
1 year, 2 weeks, 3 days is 382 days -> 365 + 14 + 3 = 382
A car travels at an average speed of 48 miles per hour. How long it take to travel 156 miles
Answer:
3.25 h = 3 h 15 m
Step-by-step explanation:
156 mi * 1h/48 mi = 3.25 h = 3 h 15 m
Answer: 195 min or 3 hr and 15 min
Step-by-step explanation:
We can set up a proportion to solve this problem.
[tex]\frac{48mi}{60 min} =\frac{156 mi}{x}[/tex]
[tex]48x=156*60[/tex]
[tex]48x=9360[/tex]
[tex]x=195 min[/tex]
We can also write this in terms of hours and minutes.
[tex]\frac{60 min}{1 hr} =\frac{195 min}{x}[/tex]
[tex]60x=195[/tex]
[tex]x=3.25[/tex]
3 hr and 15 min
Eighty percent of all California drivers wear seat belts. If three drivers are pulled over,
what is the probability that all would be wearing their seat belts? Write as a percent to the nearest tenth. Thanks!
Answer:
The probability that the three drivers would wear seat belts is 0.5
Step-by-step explanation:
Given
Percentage of drivers using seat belt = 80%
Number of drivers pulled over = 3
Required
Probability that all three drivers wore seat belt
First, the probability that a driver would wear seat belt has to be calculated.
Let's represent that with P(D)
P(D) is equivalent to the percentage of drivers using seat belt
[tex]P(D) = 80%[/tex]%
[tex]P(D) = \frac{80}{100}[/tex]
[tex]P(D) = 0.8[/tex]
Let the probability that the three drivers would wear seat belts be represented as P(All).
P(All) is calculated as thus;
(Probability that the first driver would wear seat belt) and (Probability that the second driver would wear seat belt) and (Probability that the first driver would wear seat belt).
Mathematically, this means
[tex]P(All) = P(D) * P(D) * P(D)[/tex]
Substitute [tex]P(D) = 0.8[/tex]
[tex]P(All) = 0.8 * 0.8 * 0.8[/tex]
[tex]P(All) = 0.512[/tex]
[tex]P(All) = 0.5[/tex] --- Approximated
Hence, the probability that the three drivers would wear seat belts is 0.5
A stock's price fluctuations are approximately normally distributed with a mean of $104.50 and a standard deviation of $23.62. You decide to purchase whenever the price reaches its lowest 10% of values. What is the most you would be willing to pay for the stock?
a) $80.88
b) $74.23
c) $84.62
d) $134.77
Answer:
[tex]P(z<\frac{a-\mu}{\sigma})=0.10[/tex]
and we can set up the following equation
tex]z=-1.282<\frac{a-104.5}{23.62}[/tex]
And if we solve for a we got
[tex]a=104.5 -1.282*23.62=74.22[/tex]
And the best answer for this case would be:
b) $74.23
Step-by-step explanation:
Let X the random variable that represent the stocks price of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(104.5,23.62)[/tex]
Where [tex]\mu=104.5[/tex] and [tex]\sigma=23.62[/tex]
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.90[/tex] (a)
[tex]P(X<a)=0.10[/tex] (b)
As we can see on the figure attached the z value that satisfy the condition with 0.10 of the area on the left and 0.90 of the area on the right it's z=-1.282. On this case P(Z<-1.282)=0.10 and P(z>-1.282)=0.90
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.10[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.10[/tex]
and we can set up the following equation
tex]z=-1.282<\frac{a-104.5}{23.62}[/tex]
And if we solve for a we got
[tex]a=104.5 -1.282*23.62=74.22[/tex]
And the best answer for this case would be:
b) $74.23
My rule is: y= 1/3 x+ 11/15 Find x, if y=1.
Answer:
4/5
Step-by-step explanation:
[tex]y=\dfrac{1}{3}x+\dfrac{11}{15} \\\\1=\dfrac{1}{3}x+\dfrac{11}{15}\\\\\dfrac{4}{15}=\dfrac{1}{3}x\\\\\dfrac{4}{5}=x[/tex]
Hope this helps!
Answer:
x=4/5
Step-by-step explanation:
y=1/3x+11/15
1=1/3x+11/15
4/15=1/3x
multiply both sides by 3
4/5=x
The length of time a person takes to decide which shoes to purchase is normally distributed with a mean of 8.21 minutes and a standard deviation of 1.90. Find the probability that a randomly selected individual will take less than 5 minutes to select a shoe purchase. Is this outcome unusual?
Answer:
4.55% probability that a randomly selected individual will take less than 5 minutes to select a shoe purchase.
Since Z > -2 and Z < 2, this outcome is not considered unusual.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
If [tex]Z \leq 2[/tex] or [tex]Z \geq 2[/tex], the outcome X is considered to be unusual.
In this question:
[tex]\mu = 8.21, \sigma = 1.9[/tex]
Find the probability that a randomly selected individual will take less than 5 minutes to select a shoe purchase.
This is the pvalue of Z when X = 5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{5 - 8.21}{1.9}[/tex]
[tex]Z = -1.69[/tex]
[tex]Z = -1.69[/tex] has a pvalue of 0.0455.
4.55% probability that a randomly selected individual will take less than 5 minutes to select a shoe purchase.
Since Z > -2 and Z < 2, this outcome is not considered unusual.
According to Net Market Share, Microsoft's Internet Explorer browser has 53.4% of the global market. A random sample of 70 users was selected. What is the probability that 32 or more from this sample used Internet Explorer as their browser?
Answer:
Probability that 32 or more from this sample used Internet Explorer as their browser is 0.9015.
Step-by-step explanation:
We are given that according to Net Market Share, Microsoft's Internet Explorer browser has 53.4% of the global market.
A random sample of 70 users was selected.
Let [tex]\hat p[/tex] = sample proportion of users who used Internet Explorer as their browser.
The z score probability distribution for sample proportion is given by;
Z = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ~ N(0,1)
where, p = population proportion of users who use internet explorer = 53.4%
[tex]\hat p[/tex] = sample proportion = [tex]\frac{32}{70}[/tex] = 0.457
n = sample of users = 70
Now, probability that 32 or more from this sample used Internet Explorer as their browser is given by = P( [tex]\hat p[/tex] [tex]\geq[/tex] 0.457)
P( [tex]\hat p[/tex] [tex]\geq[/tex] 0.457) = P( [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] [tex]\geq[/tex] [tex]\frac{0.457-0.534}{\sqrt{\frac{0.457(1-0.457)}{70} } }[/tex] ) = P(Z [tex]\geq[/tex] -1.29)
= P(Z [tex]\leq[/tex] 1.29) = 0.9015
The above probability is calculated by looking at the value of x = 1.29 in the z table which has an area of 0.9015.
A well is 7 meters deep, and a snail climbs up from the bottom of the well.It climbs 3meters during the day and falls 2 meters at night.How many days can the snail crawl out of the well?
Answer:
1
Step-by-step explanation:
300 students in a high school freshman class are surveyed about what kinds of pets they have. Of the 300
students, 200 have a dog, 180 have a cat, and 150 have a cat and a dog. Using this information, answer
• each of the following questions.
Let D be the event that a randomly selected student has a dog and C be the event that a randomly
selected student has a cat.
What is P(D), the probability that a student in the class has a dog?
What is P(C), the probability that a student in the class has a cat?
What is P(D and C), the probability that a student in the class has a dog and a cat?
What is P(D or C), the probability that a student in the class has a dog or a cat?
Answer:
Step-by-step explanation:
What is P(D), the probability that a student in the class has a dog?
D=66%
What is P(C), the probability that a student in the class has a cat?
C=60%
What is P(D and C), the probability that a student in the class has a dog and a cat?
DC=50%
A self storage center is a storage room that is 8 feet long, 6 feet wide, and 10 feet high. What is the volume of the room?
Answer:
480 cubic feet
Step-by-step explanation:
The volume of any rectangular prism can be found by multiplying together the length, width and height. In this case, 8*6*10=48*10=480 cubic feet. Hope this helps!
Answer:
[tex]480 {ft}^{3} [/tex]
Step-by-step explanation:
[tex]area \\ = l \times b \times h \\ = 8 \times 6 \times 10 \\ = 48 \times 10 \\ = 480 {ft}^{3} [/tex]
hope this helps
brainliest appreciated
good luck! have a nice day!
Using either the critical value rule or the p-value rule, if a one-sided null hypothesis for a single mean cannot be rejected at a given significance level, then the corresponding two-sided null hypothesis (i.e., the same sample size, the same standard deviation, and the same mean) will ______________ be rejected at the same significance level.
Answer:
Step-by-step explanation:
Using either the critical value rule or the p-value rule, a conclusion can be drawn at a level of significance (alpha)
The null hypothesis: u = hypothesized mean
Alternative hypothesis: u > u0 or u < u0 for a one tailed test
Alternative hypothesis for a two tailed test: u =/ u0
To draw a conclusion by failing to reject the null hypothesis as stated then: using critical value
Observed z score > critical z score for both the one and two tailed test.
Or using p value:
P-value > alpha for a one tailed test
P-value > alpha/2 for a two tailed test
Thus, if a one-sided null hypothesis for a single mean cannot be rejected at a given significance level, then the corresponding two-sided null hypothesis will also not be rejected at the same significance level.
Reduce the fraction to lowest terms. Do not use spaces in your answer.
Answer:
-2x/yz
Step-by-step explanation:
You can cancel out terms using division and properties of exponents. x^a/x^b = x^a-b
How many solutions does this linear system have?
y = 2x-5
-8x - 4y = -20
one solution: (-2.5, 0)
O one solution: (2.5, 0)
O no solution
O infinite number of solutions
Answer:
B (2.5, 0)
Step-by-step explanation:
2x - y = 5 (multiply all by 4)
8x- 4y = 20
-8x - 4y = - 20
eliminate the 4y
8x- 4y = 20
-8x - 4y = - 20
-------------------- –
16x = 40
x = 40/16 = 2.5
now we substitute x with 2.5
2x - y = 5
2(2.5) - y = 5
y = 0
Answer:
its B
Step-by-step explanation:
took the test
If eggs cost $3 per dozen, how much would 8 eggs cost?
Answer:
$2
Step-by-step explanation:
Because 3/12 is 0.25 and then you multiply it by 8 to get 2.
A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Six hundredSix hundred and sixtyand sixty feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?
Answer:
Dimensions: 165 feet by 110 feet.
Maximum Area =18,150 Square feet
Step-by-step explanation:
Let the dimension of the playground be x and y.
The rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground.
Let the side parallel to one side of the playground =x
Therefore, the total length of fencing =2x+x+2y
P=3x+2y
Six hundred and sixty feet of fencing is used.
We have: 3x+2y=660
3x=660-2y
[tex]x=\dfrac{660-2y}{3}[/tex]
Area of the Playground A=xy
We write the area in terms of y by substitution of x derived above.
[tex]A(y)=y\left(\dfrac{660-2y}{3}\right )\\A(y)=\dfrac{660y-2y^2}{3}[/tex]
We want to maximize the total enclosed area.
To do this, we first find the derivative of A(y).
[tex]A'(y)=\dfrac{660-4y}{3}[/tex]
Next, we solve A'(y) for its critical point.
[tex]A'(y)=\dfrac{660-4y}{3}=0\\660-4y=0\\660=4y\\y=660 \div 4\\y=165$ feet\\[/tex]
Recall that: [tex]x=\dfrac{660-2y}{3}[/tex]
Therefore:
[tex]x=\dfrac{660-2(165)}{3}=\dfrac{660-330}{3}=\dfrac{330}{3}\\x=110$ feet[/tex]
Therefore, the dimensions of the playground that maximize the total enclosed area is 165 feet by 110 feet.
Maximum Area =165 X 110
=18,150 Square feet
Answer:
Dimensions: 165 feet by 110 feet.
Maximum Area =18,150 Square feet
Maximum Area =165 X 110
=18,150 Square feet
