Chapter 10.2, Problem 72E

(a)

To determine

To Find: The sketch for the bridge on the rectangular coordinate system with the cables touching the roadway at the origin and label the coordinates of the known pair.

Answer to Problem 72E

The required sketch is shown in Figure 1

Explanation of Solution

Given:

The two towers are 1280 meters apart.

The top of each tower is 152 meter above the roadway.

Calculation:

Consider the origin is at the midpoint such that top of towers coordinates are $\left(-640,152\right),\left(640,152\right)$ . This indicates the horizontal distance $640-\left(-640\right)=1280\text{m}$ and the height is $152\text{m}$ .

The required graph is shown in Figure 1

  

Figure 1

(b)

To determine

To Find: The equation that models the cables.

Answer to Problem 72E

The required model is $y=\frac{19}{51200}{x}^2$ .

Explanation of Solution

Considered the standard equation when the parabola opens upward is,

  ${\left(x-h\right)}^2=4p\left(y-k\right)$

Then,

  $\begin{array}{l}{\left(640-0\right)}^2=4p\left(152-0\right)\\ 409600=608p\\ p=\frac{12800}{19}\end{array}$

Then, the equation is,

  $\begin{array}{l}{\left(x-0\right)}^2=4\left(\frac{12800}{19}\right)\left(y-0\right)\\ y=\frac{19}{51200}{x}^2\end{array}$

(c)

To determine

To Find: The complete form of the given table over the roadway at the distance of $x$ meters from the point where the cables touch the roadway.

Answer to Problem 72E

The completed table is shown in Table 2

Explanation of Solution

Given:

The given table is shown in Table 1

Table 1

Distance, $x$	Height $y$
0
100
250
400
500

Calculation:

Consider the required model is,

  $y=\frac{19}{51200}{x}^2$

From, the above equation the complete form of table is shown in Table 2

Table 2

Distance, $x$	Height $y$
0	0
100	3.71
250	23.19
400	59.38
500	92.77