Chapter 37, Problem 63PQ

(a)

To determine

The angle at which the two tangent rays diverge from the tip of the nose when distance between the gum ball and the friend’s nose is $5\text{cm}$.

Answer to Problem 63PQ

The angle at which the two tangent rays diverge from the tip of the nose is $23°$, when distance between the gum ball and the friend’s nose is $5\text{cm}$.

Explanation of Solution

Write the expression for the angle at which two tangent rays diverge.

    $\tan \theta =\frac{y}{D}$

Here, $y$ is the diameter of the gum ball and $D$ is the distance between the center of the gumball and the tip of your friend’s nose.

For very small angle $\theta$.

    $\tan \theta \approx \theta$

Substitute, $\theta$ for $\tan \theta$ in the above equation.

    $\theta =\frac{y}{D}$                                                                                                                    (I)

Conclusion:

Substitute, $2\text{cm}$ for $y$ and $5\text{cm}$ for $D$ in equation (I) to find $\theta$.

    $\begin{array}{c}\theta =\frac{2\text{cm}}{5\text{cm}}\\ =0.4\text{rad}\\ =\left(\text{0}\text{.4}\text{rad}\right)\left(\frac{180°}{\pi \text{rad}}\right)\\ =23°\end{array}$

Therefore, the angle at which the two tangent rays diverge from the tip of the nose is $23°$, when distance between the gum ball and the friend’s nose is $5\text{cm}$.

(b)

To determine

The angle at which the two tangent rays diverge from the tip of the nose when distance between the gum ball and the friend’s nose is $20\text{cm}$.

Answer to Problem 63PQ

The angle at which the two tangent rays diverge from the tip of the nose is $5.7°$, when distance between the gum ball and the friend’s nose is $20\text{cm}$.

Explanation of Solution

Rewrite equation (I).

  $\theta =\frac{y}{D}$                                                                                                                    (I)

Conclusion:

Substitute, $2\text{cm}$ for $y$ and $20\text{cm}$ for $D$ in equation (I) to find $\theta$.

    $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{2\text{cm}}{20\text{cm}}\right)\\ =\left(\text{0}\text{.1}\text{rad}\right)\left(\frac{180°}{\pi \text{rad}}\right)\\ =5.7°\end{array}$

Therefore, the angle at which the two tangent rays diverge from the tip of the nose is $5.7°$, when distance between the gum ball and the friend’s nose is $20\text{cm}$.

(c)

To determine

The angle at which the two tangent rays diverge from the tip of the nose when distance between the gum ball and the friend’s nose is $100\text{cm}$.

Answer to Problem 63PQ

The angle at which the two tangent rays diverge from the tip of the nose is $1.1°$, when distance between the gum ball and the friend’s nose is $100\text{cm}$.

Explanation of Solution

Rewrite equation (I).

  $\theta =\frac{y}{D}$                                                                                                                    (I)

Conclusion:

Substitute, $2\text{cm}$ for $y$ and $100\text{cm}$ for $D$ in equation (I) to find $\theta$.

    $\begin{array}{c}\theta =\frac{2\text{cm}}{100\text{cm}}\\ =0.4\text{rad}\\ =\text{0}\text{.02}\text{rad}\times \left(\frac{180°}{\pi \text{rad}}\right)\\ =1.1°\end{array}$

Therefore, the angle at which the two tangent rays diverge from the tip of the nose is $1.1°$, when distance between the gum ball and the friend’s nose is $100\text{cm}$.

(d)

To determine

The angle at which the two tangent rays diverge from the tip of the nose when distance between the gum ball and the friend’s nose is $100\text{km}$.

Answer to Problem 63PQ

The angle at which the two tangent rays diverge from the tip of the nose is $\left(1.1\times {10}^{-5}\right)°$, when distance between the gum ball and the friend’s nose is $100\text{km}$.

Explanation of Solution

Rewrite equation (I).

  $\theta =\frac{y}{D}$                                                                                                                    (I)

Conclusion:

Substitute, $2\text{cm}$ for $y$ and $100\text{km}$ for $D$ in equation (I) to find $\theta$.

    $\begin{array}{c}\theta =\frac{\left(2\text{cm}\right)\left(\frac{1\text{m}}{100\text{cm}}\right)}{\left(100\text{km}\right)\left(\frac{1000\text{m}}{1\text{km}}\right)}\\ =\left(\text{2}\times {\text{10}}^{-7}\text{rad}\right)\left(\frac{180°}{\pi \text{rad}}\right)\\ =\left(1.1\times {10}^{-5}\right)°\end{array}$

Therefore, the angle at which the two tangent rays diverge from the tip of the nose is $\left(1.1\times {10}^{-5}\right)°$, when distance between the gum ball and the friend’s nose is $100\text{km}$.