Question

You are building a right-angled triangular flower garden along a stream as shown in the figure.

The fencing of the left border costs $8 per foot, while the fencing of the lower border costs $2 per foot. (No fencing is required along the river.) You want to spend $320 and enclose as much area as possible. What are the dimensions of your garden, and what area does it enclose? [The area of a right-triangle is given by
A = xy/2.]

left border = ?? bottom border =?? garden area = ??

Answer 1

This is the figure

Answer 2

Left border cost = 8y
Bottom border cost = 2x
Total cost = 2x + 8y = 320 ----- (1)

Area A = 1/2xy --------------- (2)

Find x from (1) and substitute in (2)

To maximize the area, find the derivative dA/dy, set it to 0 and solve for y.
Then find x and the area A.

Answer 3

i got x=160-4y

Is that right?

Answer 4

yes.

Answer 5

So when I plug it into the area, do I have to distribute the y?

Answer 6

yes.

Answer 7

so i get 1/(2)(160y-4y^2)

Then I distribute the 2?

Answer 8

distribute the half (1/2)

Answer 9

I got 80y-2y^2

Answer 10

Yes, A = 80y-2y^2
Find dA/dy and set it to 0 and solve for y.

Answer 11

So find the derivative of 80y-2y^2 ?

Answer 12

Yes. To find the minimum/maximum we have to find the critical point. That is done by taking the derivative and setting it to 0.

Answer 13

Ok, so y = 20

Answer 14

Yes. Find x and A.

Answer 15

x= 80

Answer 16

yes. A = ?

Answer 17

A = 800 ?

Answer 18

That is it.

Answer 19

Yay! Thank you :)

Answer 20

you are welcome.