Chapter 2.2, Problem 75E

To determine

To draw: The curve of the arch and find the domain and the range of the function, and their representation.

Answer to Problem 75E

The domain of the graph represents the distance covered by the arch and the domain is $\left(126,0\right)$ .

The range represents the distance covered vertically by the arch and the range is, $\left(0,60\right)$ .

The equation of the graph is $y=-0.01259{\left(x-63\right)}^2+50$ .

Explanation of Solution

Given Information: The arch is modeled by the parabola and can reach a maximum height of 50 meters at a point approximately 63 meters across the river.

Calculation:

The domain of the graph represents the distance covered by the arch. This can be calculated by calculating the x -coordinates of the graph. Here, the arch reaches maximum height at approximately 63 meters. The total distance covered by the arch can be calculated by multiplying 63 by 2,

  $\begin{array}{c}x=63\times 2\\ =126\end{array}$

Hence, the domain is $\left(126,0\right)$ .

The range represents the distance covered vertically by the arch.

Since the maximum height that could be reached is 50, the range is, $\left(0,60\right)$ .

To graph the arch, which is in the shape of parabola, first write the equation.

Equation of the parabola in standard form is,

  $y=a{\left(x-h\right)}^2+k$ where $\left(h,k\right)$ is the vertex.

Since the vertex of the parabola made by arch is $\left(63,50\right)$ , the value of a at the starting point that $\left(0,0\right)$ is,

  $\begin{array}{c}0=a{\left(0-63\right)}^2+50\\ 0=3969a+50\\ a=-\frac{50}{3969}\\ =-0.01259\end{array}$

Hence the equation of the graph is,

  $y=-0.01259{\left(x-63\right)}^2+50$

The graph is,