Chapter 13.2, Problem 36PPS

a.

To determine

Find the time taken for the shadow to make $\frac{8\pi }{5}$ radians.

Answer to Problem 36PPS

The time taken to move is $19.2hour$ .

Explanation of Solution

Given: It is given in the question that a shadow moves around a sundial ${15}^{\circ }$ every hour.

Concept Used:

In this, use the concept of changing radian into degree or degree into radian by multiplying $\frac{180}{\pi }$ or $\frac{\pi }{180}$ and also use the unitary method.

Calculation: Since, change the radian into degree,

  $\frac{8\pi }{5}=\frac{8\pi }{5}\times \frac{180}{\pi }={288}^{\circ }$

Now ,

For every ${15}^{\circ }$ moves it takes = $1hour$

For ${288}^{\circ }$ moves it takes = $\frac{{288}^{\circ }}{{15}^{\circ }}=19.2hour$

Conclusion:

The time taken is $19.2hour$ .

b.

To determine

Find the angle of rotation in radian after $5hour$ .

Answer to Problem 36PPS

The angle of rotation will be $\frac{5\pi }{12}$ .

Explanation of Solution

Given:

It is given in the question that a shadow moves around a sundial ${15}^{\circ }$ every hour.

Concept Used:

In this, use the concept of changing radian into degree or degree into radian by multiplying $\frac{180}{\pi }$ or $\frac{\pi }{180}$ and also use the unitary method.

Calculation:

Since,

For one hour, the angle of rotation = ${15}^{\circ }$

For $5hour$ , the angle of rotation will be = ${15}^{\circ }\times 5hour={75}^{\circ }hour$

Now, change degree into radian,

  ${75}^{\circ }={75}^{\circ }\times \frac{\pi }{180}=\frac{5\pi }{12}$

Conclusion:

The angle will be $\frac{5\pi }{12}$ .

c.

To determine

Calculate the arc of the shadow formed after $14$ hours.

Answer to Problem 36PPS

The length of the arc of the shadow is $29.32inch$ .

Explanation of Solution

Given:

It is given in the question that a shadow moves around a sundial ${15}^{\circ }$ every hour and the radius will be $8$ inches.

Concept Used:

In this, use the concept of changing degree into radian by multiplying by $\frac{\pi }{180}$ and use the formula of arc length $s=r\cdot \theta$ .

Calculation:

Total degree after $14$ hours = ${15}^{\circ }\times 14hour={210}^{\circ }$

First change degree into radian,

  ${210}^{\circ }={210}^{\circ }\times \frac{\pi }{180}=\frac{7\pi }{6}$

Now, use the arc length,

  $\begin{array}{l}s=r\cdot \theta \\ s=8inch\cdot \frac{7\pi }{6}\\ s=\frac{28\pi }{3}inch\\ s=29.32inch\end{array}$

Conclusion:

The length of the arc is $29.32inch$