Gateway Arch The Gateway Arch in St. Louis is...

Question

Gateway Arch The Gateway Arch in St. Louis is 630 ft high and has a 630-ft base. Its shape can be modeled by the parabola y = 630 (1― (𝓍/315)²) . Find the average height of the arch above the ground.

Illustration of the Gateway Arch with a parabolic equation, showing its height and base dimensions.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a tunnel is shaped in a form of a parabola with a base width of 800 ft and a maximum height of 600 ft. Its cross section can be modeled by Y equal to 600, multiplied by the quantity of 1 minus the quantity of X divided by 400 in quantity squared in quantity. What is the average height of the tunnel above the ground? And we're given 4 possible choices as our answers. For choice A, we have 800 divided by 3 ft. For choice B, we have 1200 ft. For choice C, we have 400 ft, and for choice D, we have 800 ft. So we're asked to find the average height of the tunnel above ground. So what we really need to do is to find our average value of Y. So we're gonna call that average value white bar. And here we're taking the average value of a function. So recall your formula for finding the average value of a function over an interval, so that Y bar is going to be equal to 1 divided by the quantity of B minus A in quantity multiplied by the integral from A to B, of our function Y, and we're integrating that with respect to X. Now if we look at our function Y here, we see that when X is equal to 0, Y is going to be equal to 600, which is the maximum height. So that means that the apex of our parabola is located at X equal to 0. And since our parabola is symmetric and we have a base width of 800 ft, that means that our limits of integration are going to go from A being equal to -400. 2 B equal to 400. So this will give us our base width of 800 ft. So, substituting these values into our formula for Y bar, we see that Y bar is going to be equal to 1 divided by the quantity of 400 minus -400 in quantity. Multiplied by the integral from -400 to 400. Of 600. Multiplied by the quantity of 1 minus. X2 divided by 400 squared. In quantity, DX. So, to evaluate this integral, let's first move the constant out in front of the integral. So this is going to be equal to, and in front, we will have the 600, and this can be divided by 800, so we'll get 3 divided by 4. And this is going to be multiplied by the integral from -400 to 400 of the quantity of 1 minus. X 2 divided by 400 squad in quantity d x. So, if we look at our integrand here, we see that we're integrating an even function. So we can use that symmetry to change our limits of integration to 0 to 400 by multiplying the entire integral by 2. So doing that, this is going to be equal to 3 divided by 4, multiplied by 2, multiplied by the integral from 0 to 400. Of the quantity of 1 minus x 2 divided by 400 squad in quantity d x. So now we just need to integrate each of the terms individually inside the integral. And then substitute in the limits. So doing the integration, this is going to be equal to. We can simplify the coefficient out front, so that will be 3 divided by 2, and this is going to be multiplied by the quantity of, and when we integrate 1 with respect to X, we get X, then we'll have minus 1 divided by 400 squared. And that gets multiplied by the integral of X2d with respect to X, and so, recall that that is going to give you X cubed divided by 3. In quantity, and this is going to be evaluated from 0 to 400. So now we just need to substitute in our limits of integration. In doing so, this is going to be equal to. 3 divided by 2 multiplied by the quantity, and for our upper limit, we'll have 400 minus 1 divided by 400 squared. Multiplied by 400 cubed divided by 3. Then we'll have minus the quantity, and for our lower limit, we'll have 0, minus 1 divided by 400 squared. Multiplied by 0 cubed divided by 3. In quantity So now we just need to simplify this expression. So this is going to be equal to. 3 divided by 2, multiplied by the quantity of 400 minus, and here we have 400 cubed divided by 400 squares, so this will give me 400, and that's still going to be divided by 3. That we have for the main terms minus 0 + 0 in quantity. And when we evaluate this expression, we see that this is going to be equal to 400, and this result has units of feet, since both the width and the height were given in feet. So this here is the answer that we are looking for for this problem. And when we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice C. So, I hope that this explanation has been useful, and I'll see you in the next video.

Gateway Arch The Gateway Arch in St. Louis is 630 ft high and has a 630-ft base. Its shape can be modeled by the parabola y = 630 (1― (𝓍/315)²) . Find the average height of the arch above the ground.

Key Concepts

Parabolic Functions

Average Value of a Function

Integration

Watch next