Chapter 12.1, Problem 48PPS

a.

To determine

Calculate the height of the second ramp to the first.

Answer to Problem 48PPS

The value of increase in second ramp is $2.7ft$ .

Explanation of Solution

Given: It is given in the question that two bicycles ramps each cover a horizontal distance of $8$ feet in which one of the ramp has a ${20}^{\circ }$ angle of elevation, and the other ramp is ${35}^{\circ }$ angle of elevation.

  

Concept Used:

In this, use the concept of trigonometric ratios and the formula will be, $\sin \theta =\frac{p}{h},\cos \theta =\frac{b}{h},\tan \theta =\frac{p}{b},\csc \theta =\frac{h}{p},\sec \theta =\frac{h}{b},\cot \theta =\frac{b}{p}$

Calculation:

Let x be the height of the first ramp and y be the height of the second ramp.

Here, use the trigonometric ratios,

For x,

  $\begin{array}{l}\tan \theta =\frac{p}{b}\\ \tan {20}^{\circ }=\frac{x}{8}\\ 0.362=\frac{x}{8}\\ x=8\cdot 0.362\\ x\approx 2.9\end{array}$

For y,

  $\begin{array}{l}\tan \theta =\frac{p}{b}\\ \tan {35}^{\circ }=\frac{y}{8}\\ 0.7=\frac{y}{8}\\ x=8\cdot 0.7\\ x\approx 5.6\end{array}$

So, the Net increase will be $5.6-2.9=2.7ft$

Conclusion:

The value of increase in second ramp is $2.7ft$ .

b.

To determine

Calculate the length of the second ramp to the first.

Answer to Problem 48PPS

The length that will be increase is $1.3ft$ .

Explanation of Solution

Given:

It is given in the question that two bicycles ramps each cover a horizontal distance of $8$ feet in which one of the ramp has a ${20}^{\circ }$ angle of elevation, and the other ramp is ${35}^{\circ }$ angle of elevation.

  

Concept Used:

In this, use the concept of trigonometric ratios and the formula will be, $\sin \theta =\frac{p}{h},\cos \theta =\frac{b}{h},\tan \theta =\frac{p}{b},\csc \theta =\frac{h}{p},\sec \theta =\frac{h}{b},\cot \theta =\frac{b}{p}$.

Calculation:

Let x be the height of the first ramp and y be the height of the second ramp.

Here, use the trigonometric ratios,

For x,

  $\begin{array}{l}\cos \theta =\frac{b}{h}\\ \cos {20}^{\circ }=\frac{8}{x}\\ 0.941=\frac{8}{x}\\ x=\frac{8}{0.941}\\ x\approx 8.5\end{array}$

For y,

  $\begin{array}{l}\cos \theta =\frac{b}{h}\\ \cos {35}^{\circ }=\frac{8}{y}\\ 0.816=\frac{8}{y}\\ x=\frac{8}{0.816}\\ x\approx 9.8\end{array}$

So, the Net increase in length will be $9.8-8.5=1.3ft$ .

Conclusion:

The increasing length will be $1.3ft$ .