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.

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The equation of the Gateway Arch is given as y = 630(1 - (x/315)^2). This represents a parabola symmetric about the y-axis, with its vertex at (0, 630) and x-intercepts at x = -315 and x = 315.

To find the average height of the arch above the ground, we calculate the average value of the function y over the interval [-315, 315]. The formula for the average value of a function f(x) over [a, b] is given by: f_avg = (1 / (b - a)) * ∫[a to b] f(x) dx.

Here, f(x) = 630(1 - (x/315)^2), a = -315, and b = 315. Substitute these values into the formula: f_avg = (1 / (315 - (-315))) * ∫[-315 to 315] 630(1 - (x/315)^2) dx.

Simplify the constant outside the integral: f_avg = (1 / 630) * ∫[-315 to 315] 630(1 - (x/315)^2) dx. The constant 630 can be factored out of the integral: f_avg = (1 / 630) * 630 * ∫[-315 to 315] (1 - (x/315)^2) dx.

Evaluate the integral ∫[-315 to 315] (1 - (x/315)^2) dx by splitting it into two parts: ∫[-315 to 315] 1 dx and -∫[-315 to 315] (x/315)^2 dx. Use symmetry properties of definite integrals to simplify the calculations, as the function (x/315)^2 is even and the integral of 1 is straightforward.

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주요 개념

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Parabolic Functions

A parabolic function is a type of quadratic function that can be represented in the form y = ax² + bx + c. In the context of the Gateway Arch, the equation y = 630(1 - (x/315)²) describes a downward-opening parabola, where the vertex represents the highest point of the arch. Understanding parabolas is essential for analyzing the shape and properties of structures modeled by such equations.

Average Value of a Function

The average value of a function over a given interval can be calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx, where f(x) is the function and [a, b] is the interval. For the Gateway Arch, this involves integrating the height function over the width of the arch to find the average height above the ground. This concept is crucial for determining how the height varies across the span of the arch.

Integration

Integration is a fundamental concept in calculus that involves finding the area under a curve represented by a function. In this case, integrating the parabolic function of the arch's height allows us to calculate the total height across its base. This process is essential for solving problems related to average height and other properties of curves in calculus.