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.
<<-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.
To answer the given questions, let's break them down step by step:
1. What is the length of the segment along the floor of each arch?
To find the length of the segment along the floor, we need to determine the x-values where the graph of the arch intersects the x-axis. These points are where f(x) equals zero since the x-axis is represented by y=0.
To find these points, set f(x) equal to zero and solve for x:
0 = -x^2 + 2x + 8
This is a quadratic equation, so we can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:
0 = (x + 2)(-x + 4)
Setting each factor equal to zero gives us:
x + 2 = 0 or -x + 4 = 0
x = -2 or x = 4
So the length of the segment along the floor of each arch is from x = -2 to x = 4.
2. What is the height of the arch?
To find the height of the arch, we need to determine the maximum point of the graph, which corresponds to the vertex of the parabola f(x).
The x-coordinate of the vertex can be found using the formula -b/2a for a quadratic equation in the form ax^2 + bx + c. In this case, the equation is -x^2 + 2x + 8, so a = -1, b = 2, and c = 8.
x = -b/2a = -2/(2*(-1)) = 1
Now, substitute x = 1 into the equation f(x) = -x^2 + 2x + 8 to find the corresponding y-value:
f(1) = -(1)^2 + 2(1) + 8 = -1 + 2 + 8 = 9
Therefore, the height of the arch is 9 feet.
3. Calculate the total amount of fabric needed for 5 arches using the formula A = (2/3)bh.
To use the formula, we need to find the length of the base and the height for each arch.
The length of the base is the same as the length of the segment along the floor of each arch, which we found to be 4 - (-2) = 6 feet.
The height of each arch is 9 feet (as calculated in the previous question).
Now, substitute these values into the formula A = (2/3)bh for each arch:
A = (2/3)(6 ft)(9 ft) = 36 ft²
Since there are 5 arches, the total amount of fabric needed is:
Total fabric = 5 arches * 36 ft²/arch = 180 ft².
Therefore, the drama club needs 180 square feet of fabric to cover all 5 arches.