A stone arch in a bridge forms a parabola described by the equation $y=a(x-h)^2+k$, where $y$ is the height in feet of the arch above the water, $x$ is the horizontal distance from the left end of the arch, $a$ is a constant, and $(h, k)$ is the vertex of the parabola.
What is the equation that describes the parabola formed by the arch?
A. $y=-0.090(x-13)^2+12$
B. $y=-0.090(x-12)^2+13$
C. $y=-0.071(x-13)^2+12$
D. $y=-0.071(x-12)^2+13
Answer (2)
The problem provides the general form of a parabola y = a ( x − h ) 2 + k and four possible equations for a stone arch.
We identify the vertex ( h , k ) and the coefficient a for each option.
Without additional information, we cannot definitively determine the correct equation.
Assuming the first option is correct, the equation is y = − 0.090 ( x − 13 ) 2 + 12 .
Explanation
Understanding the Problem We are given the general form of a parabola y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. We are also given four possible equations for the parabola formed by the arch. The goal is to determine which of the four equations correctly describes the parabola. Without any additional information, such as the coordinates of the vertex or any other points on the parabola, it is impossible to determine the correct equation. The problem is likely missing some crucial information, such as a diagram or specific measurements of the arch.
Analyzing the Options Let's analyze the given options:
Option 1: y = − 0.090 ( x − 13 ) 2 + 12 Here, a = − 0.090 , h = 13 , and k = 12 . The vertex is ( 13 , 12 ) .
Option 2: y = − 0.090 ( x − 12 ) 2 + 13 Here, a = − 0.090 , h = 12 , and k = 13 . The vertex is ( 12 , 13 ) .
Option 3: y = − 0.071 ( x − 13 ) 2 + 12 Here, a = − 0.071 , h = 13 , and k = 12 . The vertex is ( 13 , 12 ) .
Option 4: y = − 0.071 ( x − 12 ) 2 + 13 Here, a = − 0.071 , h = 12 , and k = 13 . The vertex is ( 12 , 13 ) .
Without more information, we cannot definitively choose one of these options.
Assuming the Answer Since we cannot determine the correct equation without additional information, I will assume that the first option is the correct one.
Examples
Parabolic arches are commonly used in bridge design. The equation of a parabolic arch helps engineers determine the structural integrity and load-bearing capacity of the bridge. By understanding the parameters of the parabola, such as the vertex and the coefficient 'a', engineers can calculate the stresses and strains on the arch, ensuring the bridge's safety and stability. For example, if you're designing a bridge with a parabolic arch, knowing the equation allows you to predict the height at any point along the span, which is crucial for clearance and aesthetic considerations.
The equation of the stone arch can be represented as a parabola in the form y = a ( x − h ) 2 + k . Analyzing the options provided, we find that option A, y = − 0.090 ( x − 13 ) 2 + 12 , is selected based on an assumed vertex. However, more context is needed for definitive identification of the correct equation.
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