Average Value of a Function: The Mean Value Theorem for Integrals gives us an expression that can be used to calculate the average value of a function. In order to find this average value, we need to be able to evaluate an integral where our function is the integrand. If this is possible, then this average value is found as follows.
Answer to question 1
Since this function gives us the height of this building, the average height can be found by finding the average value of this function. We are able to integrate this function, so this average value can be found by evaluating the definite integral below. Reversing the Power Rule will allow us to construct the required antiderivative for this computation.
We do, however, need to know the bounds of integration, which we aren’t given. If we assume that the left wall is at x=0x=0, let’s integrate until the height function equals zero. We have to look to a computer to determine when this would occur, as setting this function equal to zero does not give us an equation that we can solve algebraically. The one real solution to this equation is approximately x=421.72x=421.72, so let’s use this as our right endpoint.
The average height of this building is 22.5 feet.
Answer to question 2
We would find the average height from x = 0 up to the point it touches the ground, or H(x) = 0. We first find x when H(x) = 0.
Now, we would be applying the formula for the average value of a function,
Unfortunately, the correct answer is not within the choices.
Answer to question 3