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Chapter 5: Problem 47

Gateway Arch The Gateway Arch in St. Louis is \(630 \mathrm{ft}\) high and has a 630 -ft base. Its shape can be modeled by the parabola $$ y=630\left[1-\left(\frac{x}{315}\right)^{2}\right] $$ Find the average height of the arch above the ground.

Short Answer

Expert verified
Answer: The average height of the Gateway Arch above the ground is 420 feet.

Step by step solution

01

Set up the definite integral

To find the average height of the arch above the ground, we first need to find the total area under the curve of the given parabolic function from \(x=-315\) to \(x=315\). We can then divide this area by the total width (630 feet) to find the average height. So we set up the definite integral: $$ \int_{-315}^{315} 630\left[1-\left(\frac{x}{315}\right)^{2}\right] dx $$
02

Factor out the constant

Factor out the constant term 630 from the integral: $$ 630 \int_{-315}^{315} \left[1-\left(\frac{x}{315}\right)^{2}\right] dx $$
03

Integrate

Now, we'll integrate the function with respect to \(x\): $$ 630 \left[ \int_{-315}^{315} 1 dx - \int_{-315}^{315} \left(\frac{x}{315}\right)^{2} dx \right] $$ The first integral is: $$ \int_{-315}^{315} 1 dx = [x]_{-315}^{315} = 315 - (-315) = 630 $$ For the second integral, make a substitution: \(u = \frac{x}{315}\), which yields \(du = \frac{1}{315}dx\). Thus, the integral becomes: $$ \int_{-1}^{1} u^2 \times 315 du = 315 \int_{-1}^{1} u^2 du $$ Now, integrate \(u^2\): $$ 315 \left[ \frac{1}{3}u^3 \right]_{-1}^{1} = 315 \left[ \frac{1}{3}(1)^3 - \frac{1}{3}(-1)^3 \right] $$
04

Calculate the value of the integral and find the average height

Calculate the value of the integral: $$ 315 \left[ \frac{1}{3}(1) - \frac{1}{3}(-1) \right] = 315 \left[ \frac{1}{3}(2) \right] = 210 $$ Multiply this by the constant factor 630: $$ 630 \times (630 - 210) = 630 \times 420 $$ Now, divide this result by the total width of the base (630 feet) to find the average height: $$ \text{Average height} = \frac{630 \times 420}{630} = 420 $$ The average height of the Gateway Arch above the ground is 420 feet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
A parabola is a symmetrical, U-shaped curve that is defined by a quadratic equation. In the real world, parabolas often describe the paths of moving objects under the influence of gravity, like a basketball being thrown. The Gateway Arch can also be modeled as a parabola. This means that its shape follows the equation in the form of \[y = 630\left[1-\left(\frac{x}{315}\right)^{2}\right]\]This particular parabola opens downwards because of the negative coefficient of the squared term. The expression inside the brackets determines how wide or narrow this parabola is.
  • The vertex, which is the highest point of the Gateway Arch, is at 630 feet.
  • The arch's base is located between \(-315\) feet and 315 feet, making it symmetric around the y-axis.
  • This kind of parabolic shape is often used in architecture because it is stable and attractive.
Understanding how a parabola works is crucial in calculating the average height of a structure like the Gateway Arch.
Average Height
The average height of an object can be described as the mean elevation of its points above a fixed surface. For the Gateway Arch, we're interested in the average height above the ground surface. To compute this, it's effective to use calculus, especially when dealing with complex shapes like a parabola.The "average value" of a continuous function over a specific interval can be found by integrating the function and then dividing by the length of the interval. In the case of the Gateway Arch, this means integrating the equation for height over the span of the base:
  • First, compute the area under the curve of the parabolic function from \(x = -315\) to \(x = 315\).
  • Then, divide the result by the total width of the base, which is 630 feet, to get the average height.
These steps will give you the average height of the arch above ground as 420 feet.
Integration by Substitution
Integration by substitution is a method used to evaluate integrals more easily. This technique is useful when dealing with complex functions. For the Gateway Arch problem, we used substitution to simplify the integral of the squared term in the function describing the parabola:
  • The original term to integrate was \( \left(\frac{x}{315}\right)^{2} \).
  • To simplify, set \( u = \frac{x}{315} \), which leads to \( du = \frac{1}{315} dx \).
  • Then, change the limits of integration according to the substitution: from \(x = -315\) to \(u = -1\) and from \(x = 315\) to \(u = 1\).
  • After substitution, the integral simplifies to \( 315 \int_{-1}^{1} u^2 du \).
By integrating \(u^2\) and evaluating from \(-1\) to \(1\), the polynomial becomes much more manageable, helping us find the integral needed to determine the average height of the structure efficiently. This substitution method is crucial in many calculus problems to make integration more straightforward.

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