ALGEBRA!! HELP ME PLEASE!!!!

8. You have 180 feet of fence to make a regular pen. What are the dimensions of the rectangle with the maximum area?
The answer's supposed to be 45ft by 45ft but idk y

9. Of all the pairs of real numbers that have sum 90, which pair has the greatest product?
do i write an equation?

10. You have 180 feet of fence to make a rectangle pen. One side of the pen will be against a 200 ft wall, so it requires no fence. What are the dimensions of the rectangle with the maximum.

8) the 180 feet of fence to make a regular pen in shape of rectangle

ok you want to maximize the area function

let x = one side dimension

let y = other side dimension

now we know that we have only 180 feet so

2x + 2y = 180

that is 2 y sides and 2 x sides make a rectangle

notice we didnt make the assumption that the sides are the same

now for area

A = xy

this is complicated because it has y in it

if we use the other formula for length solve for y we can substitute that in for the y in the area and have an equation in x only

2x + 2y = 180
2y = 180 - 2x
y = 90 - x

putting in the area equation
Area = xy = x(90 -x)
area = 90x - x^2
we want to maximize the area do the derivative

d(area)/dx = 90 -2x


this is max or min when it equals 0

0 = 90 -2x
2x = 90
x = 45


d^2(area)/dx^2 = -2
since second derivative is negative slope is decreasing and so its a hump and therefor a maximum

so when x = 45 Area is maximum that makes

y = 90 -x = 90 -45 = 45

9)

same thing with 9 make the equations

a pair of numbers is x and y

so x + y = 90 means there sum is 90

x * y = product

use the first one to solve for y again .. you could solve for x if you wanted but I picked y

y = 90 - x

x(90 -x) =product

product = 90x - x^2