Math  /  Geometry

QuestionQuestion A rectangular garden is to be constructed using a river as one side of the garden and wood fencing for the other three sides. Given 144 ft of wood fencing, what is the maximum area the garden can be?
Provide your answer below:

Studdy Solution

STEP 1

1. The garden is rectangular.
2. One side of the garden is bounded by a river, requiring no fencing.
3. The total length of wood fencing available is 144 144 feet.

STEP 2

1. Define the variables and set up the equation for the perimeter.
2. Express the area of the garden as a function of one variable.
3. Use calculus to find the maximum area.
4. Verify the solution.

STEP 3

Define the variables: - Let x x be the length of the side parallel to the river. - Let y y be the length of the side perpendicular to the river.
Set up the equation for the perimeter using the wood fencing:
x+2y=144 x + 2y = 144

STEP 4

Express the area A A of the garden as a function of x x :
A=x×y A = x \times y
Substitute y y from the perimeter equation:
y=144x2 y = \frac{144 - x}{2}
Thus, the area function becomes:
A(x)=x(144x2) A(x) = x \left(\frac{144 - x}{2}\right) =144xx22 = \frac{144x - x^2}{2}

STEP 5

To find the maximum area, take the derivative of A(x) A(x) and set it to zero:
A(x)=ddx(144xx22) A'(x) = \frac{d}{dx}\left(\frac{144x - x^2}{2}\right) =1442x2 = \frac{144 - 2x}{2}
Set the derivative equal to zero to find critical points:
1442x2=0 \frac{144 - 2x}{2} = 0 1442x=0 144 - 2x = 0 2x=144 2x = 144 x=72 x = 72

STEP 6

Verify the solution by checking the second derivative or using a test:
The second derivative is:
A(x)=ddx(1442x2) A''(x) = \frac{d}{dx}\left(\frac{144 - 2x}{2}\right) =1 = -1
Since A(x)<0 A''(x) < 0 , the function A(x) A(x) has a maximum at x=72 x = 72 .
Calculate y y when x=72 x = 72 :
y=144722=36 y = \frac{144 - 72}{2} = 36
Thus, the maximum area is:
A(72)=72×36=2592 square feet A(72) = 72 \times 36 = 2592 \text{ square feet}
The maximum area of the garden is:
2592 square feet \boxed{2592 \text{ square feet}}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord