Chapter 9.1, Problem 101E

(a)

To determine

To find:Draw the sketch of the cable and label the coordinates of the known points.

Answer to Problem 101E

The sketch of the cable is shown in the figure(1) with labeled known points.

Explanation of Solution

Given information:

A cable of the Golden Gate Bridge is suspended between two towers that are $1280$ meters apart. The top of each tower is $152$ meters above the roadway. The cable touches the roadway midway between the towers.

Calculation:

The sketch of the cable is shown in the following figure:

  

Figure(1)

(b)

To determine

To calculate:The equation that models the cable.

Answer to Problem 101E

The equation that models the cable is $x^2=2694.75y$ .

Explanation of Solution

Given information:

A cable of the Golden Gate Bridge is suspended between two towers that are $1280$ meters apart. The top of each tower is $152$ meters above the roadway. The cable touches the roadway midway between the towers.

Calculation:

According to the sketch in part(a), the cable models in an upwards parabola with vertex at origin.

The general equation of the parabola is $x^2=4py$ .

As it is clear from the sketch that the parabola passes through the point $\left(640,152\right)$ , so substitute $640$ for $x$ and $152$ for $y$ in the general equation.

  $\begin{array}{c}{x}^2=4py\\ {640}^2=4p\left(152\right)\\ p=\frac{409600}{608}\\ p=673.68\end{array}$

Now substitute $673.68$ for $p$ in the equation of parabola.

  $\begin{array}{l}{x}^2=4py\\ x^2=4\left(673.68\right)y\\ x^2=2694.75y\end{array}$

Therefore, the equation of that models the cable is $x^2=2694.75y$ .

(c)

To determine

To calculate:Complete the table by finding the height $y$ of the suspension cable over the roadway at a distance of $x$ meters from the center of the bridge.

$x$	$0$	$200$	$400$	$500$	$600$
$y$

Answer to Problem 101E

The complete table is given below:

$x$	$0$	$200$	$400$	$500$	$600$
$y$	$0$	$14.84$	$59.37$	$92.77$	$133.59$

Explanation of Solution

Given information:

A cable of the Golden Gate Bridge is suspended between two towers that are $1280$ meters apart. The top of each tower is $152$ meters above the roadway. The cable touches the roadway midway between the towers.

Calculation:

As calculated in part(b) the equation that models the cable is $x^2=2694.75y$ .

Now substitute $x=0,200,400,500,600$ to find the respective $y$ values in th equation of parabola.

This gives the table:

$x$	$0$	$200$	$400$	$500$	$600$
$y$	$0$	$14.84$	$59.37$	$92.77$	$133.59$

Therefore, the complete table is shown above.