Chapter 13.1, Problem 49PPS

a.

To determine

Find the approximate distance z between the falcon and mouse B.

Answer to Problem 49PPS

The distance between the two falcon and mouse is $647$ feet.

Explanation of Solution

Given: It is given in the question that a falcon at a height of $200$ feet sees two mice A and B, as shown in the diagram below:

  

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 z be the distance between the falcon and mouse B.

Here, use the trigonometric ratios,

  $\begin{array}{l}\cos \theta =\frac{b}{h}\\ \cos {72}^{\circ }=\frac{200}{z}\\ 0.309=\frac{200}{z}\\ z=\frac{200}{0.309}\\ z\approx 647\end{array}$

Conclusion:

The distance between the two is $647$ feet.

b.

To determine

Find the difference between the two mouse A and B.

Answer to Problem 49PPS

The difference between the two is $239ft$ .

Explanation of Solution

Given:

It is given in the question that a falcon at a height of $200$ feet sees two mice A and B, as shown in the diagram below:

  

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 distance between the tree and mouse B.

Here, use the trigonometric ratios,

For x,

  $\begin{array}{l}\tan \theta =\frac{p}{b}\\ \tan {72}^{\circ }=\frac{x}{200}\\ 3.075=\frac{x}{200}\\ x=200\cdot 3.075\\ x\approx 615\end{array}$

Let y be the distance between the tree and mouse A.

For y,

  $\begin{array}{l}\tan \theta =\frac{p}{b}\\ \tan {62}^{\circ }=\frac{y}{200}\\ 1.88=\frac{y}{200}\\ y=200\cdot 1.88\\ y\approx 376\end{array}$

So, the Net increase in length will be $615-376=239ft$ .

Conclusion:

The increasing length will be $239ft$ .