find a function that models the area of the parallelogram?

Question

the sum of the base and height of the parallelogram is equal to 40 inches:
A) find a function that models the area of the parallelogram:


B)Find the demensions to maximize the area (without a calculator)

Answer 1

b = 40 - h

Area = b*h, so in function language: a(x) = h(40 - h)

Part b:

a(x) = 40 h - h^2

vertex is at h = -40/-2 = 20 (using -b/2a formula)

So h = 20 in and since b = 40 - h, b = 20 inches also.

Answer 2

a) let h and b be the height and base

Then

h + b = 40

which gives (solving for h) ; b = 40 - h

The area of the parallelogram is

f(h,b) = hb

So

f(h) = h (40 - h)

b) we can rewrite this as

f(h) = h (40 - h) = 40h - h^2 = - (h - 20)^2 + 400

We note that the first part is always negative thus when it is 0 the maximum of 400 is reached.

This is maximized when h = 20 and thus b = 20

Answer 3

because the sum of the base and height are 40 inches, we could think of height being h, and base being b, and hence, b+h=40. Hence, b=40-h, and h=40-b.
The area of a parallelogram is b*h=a, so by substituting, we find that a=(40-h)(40-b), so a=160-80h-80b+hb.
To maximize this area, h and b would be equal, and hence, you would have a square.

Answer 4

Area = bh

b + h = 40
h = -b + 40

Area = (-b + 40)b
Area = -b^2 + 40b

ANS :
A(b) = -b^2 + 40b

-----------------------------------------

b = (-40)/(2(-1))
b = (20/2)
b = 10

A(10) = -(10)^2 + 40(10)
A(10) = -100 + 400
A(10) = 300

Area = 300 in^2