Pls help me, im having trouble

Question

The sum of the bases and the altitude of a trapezoid is 32cm. If one base is 6 cm shorter than the other , find all three dimensions if its area is to be maximum

Answer 1

So one constraint is b₁ + b₂ + h = 32, where b₁ and b₂ are the bases and h is the altitude.

The second constraint is b₁ = b₂ + 6. This says that b₁ is 6cm longer than b₂

Now get b₁ and b₂ in terms of h.

Plug in b₂ + 6 in for b₁ in the first constraint.

(b₂ + 6) + b₂ + h = 32

2b₂ + 6 + h = 32

2b₂ = 26 - h

b₂ = 13 - 0.5h

b₁ is 6cm longer than b₂ so b₁ = 19 - 0.5h

We're trying to maximize the area of the trapezoid

A = 0.5(b₁ + b₂)h

Substitute in what we got for the bases in terms of height

A = (16 - 0.5h)h

Take the derivative with respect to h and set it equal to 0

dA/dh = 16 - h = 0 so h = 16cm

b₂ = 13 - 0.5h so b₂ = 13 - 0.5(16) = 5cm

b₁ = 19 - 0.5h so b₁ = 19 - 0.5(16) = 11cm