Question

The drama club is building a backdrop using arches whose shape can be represented by the function f(x)=-x^2+2x+8, where x is the length in feet. The area under each arch is to be covered with fabric.

-What is the length of the segment along the floor of each arch?
-What is the height of the arch?
-The formula A=2/3bh can be used to estimate the area A under a parabola. In this formula, b represents the length of the base and h represents the height. If there are 5 arches, calculate the total amount of fabric that is needed.

Answer 1

<<-What is the length of the segment along the floor of each arch? >>

that would be the distance between the x-intercepts
y = -x^2 + 2x + 8
= -(x^2 - 2x - 8)
= -(x-4)(x+2)

so the x-intercepts are -2 and 4, the distance between them is 6

<<-What is the height of the arch? >>

the vertex would be at x=1, (the midway between the x-intercepts)
when x=1, y = -1+2+8 = 9

according to your formula, the area under one curve is 2/3(6)(9) = 36

there are 5 of them , so the total area would be 5*36 feet^2

BTW, using Calculus the area under each curve would be exactly 36, so that approximation formula works pretty well here.