Chapter 5.4, Problem 58E

Length of a Shadow On a day when the sun passes directly overhead at noon, a 6-ft-tall man casts a shadow of length

$S(t)=6\left|\cot \frac{\pi }{12}t\right|$

where S is measured in feet and t is the number of hours since 6 a.m.

1. (a) Find the length of the shadow al 8:00 a.m., noon, 2:00 p.m., and 5:45 p.m.
2. (b) Sketch a graph of the function S for 0 < t < 12.
3. (c) From the graph, determine the values of t at which the length of the shadow equals the man’s height. To what lime of day does each of these values correspond?
4. (d) Explain what happens to the shadow as the time approaches 6 p.m. (that is, as t → 12⁻).

To determine

To compute: The length of the shadow at $8.00$ A.M, 12.00 P.M, 2.00 P.M and 5.45 P.M.

Answer to Problem 58E

The length of the shadow at $8.00$ A.M is $10.393$.

The length of the shadow at $12.00$ P.M is $0$.

The length of the shadow at $2.00$ P.M is $3.4638$.

The length of the shadow at $5.45$ P.M is $91.4823$

Explanation of Solution

Given:

Shadow of length $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ and t is  the number of hours from $6.00$ A.M.

Calculation:

Compute the length of the shadow at $8.00$ A.M.

Here, number of hours from $6.00$ A.M to $8.00$ A.M is 2.

Therefore, the value of $t=2$.

Compute the value of $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ for $t=2$.

$\begin{array}{c}S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|\\ =6\left|\cot \left(\frac{2\pi }{12}\right)\right|\\ =6\left|\cot \frac{\pi }{6}\right|\\ =10.393\end{array}$

Therefore, the length of the shadow at $8.00$ A.M is $10.393$.

Compute the length of the shadow at $12.00$ P.M.

Here, number of hours from $6.00$ A.M to $12.00$ P.M is 6.

Therefore, the value of $t=6$.

Compute the value of $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ for $t=6$.

$\begin{array}{c}S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|\\ =6\left|\cot \left(\frac{6\pi }{12}\right)\right|\\ =6\left|\cot \frac{\pi }{2}\right|\\ =0\end{array}$

Therefore, the length of the shadow at $12.00$ P.M is $0$.

Compute the length of the shadow at $2.00$ P.M.

Here, number of hours from $6.00$ A.M to $2.00$ P.M is 6.

Therefore, the value of $t=8$.

Compute the value of $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ for $t=8$.

$\begin{array}{c}S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|\\ =6\left|\cot \left(\frac{8\pi }{12}\right)\right|\\ =6\left|\cot \frac{2\pi }{3}\right|\\ =3.4638\end{array}$

Therefore, the length of the shadow at $2.00$ P.M is $3.4638$.

Compute the length of the shadow at $5.45$ P.M.

Here, number of hours from $6.00$ A.M to $5.45$ P.M is 11.45.

Therefore, the value of $t=11.45$.

Compute the value of $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ for $t=11.45$.

$\begin{array}{c}S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|\\ =6\left|\cot \left(\frac{\left(11.45\right)\pi }{12}\right)\right|\\ =6\times 15.2470\\ =91.4823\end{array}$

Therefore, the length of the shadow at 11.45 P.M is $91.4823$.

To determine

To sketch: The graph of the function $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$  for $0\le t\le 12$.

Explanation of Solution

Use online graphing calculator and obtain the graph of $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ as shown below Figure 1.

From the Figure 1, observed that the time function $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ approaches to infinity as $t$ approaches to 12.

To determine

The values of $t$ at which the length of shadow equals the man’s height.

Answer to Problem 58E

The length of the shadow equals man’s height for $t=3$ and $t=9$.

The length of the shadow equals man’s height at 9 A.M and 3 P.M.

Explanation of Solution

Given:

The man’s height is 6 feet.

Observation:

From the above Figure 1, observed that the function $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ approaches to 6  feet for $t=3$ and $t=9$.

That is, the length of the shadow equal to 6 feet for $t=3$ and $t=9$.

That is, the length of the shadow equal to 6 feet at 9 A.M and 3 P.M.

Calculation:

Compute the length of the shadow at $9.00$ A.M.

Here, number of hours from $6.00$ A.M to $9.00$ A.M is 3.

Therefore, the value of $t=3$.

Compute the value of $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ for $t=3$.

$\begin{array}{c}S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|\\ =6\left|\cot \left(\frac{3\pi }{12}\right)\right|\\ =6\left|\cot \frac{\pi }{4}\right|\\ =6\end{array}$

Therefore, the length of the shadow at 9 P.M is $6$.

Compute the length of the shadow at $3.00$ P.M.

Here, number of hours from $6.00$ A.M to $3.00$ P.M is .9.

Therefore, the value of $t=9$.

Compute the value of $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ for $t=9$.

$\begin{array}{c}S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|\\ =6\left|\cot \left(\frac{9\pi }{12}\right)\right|\\ =6\left|\cot \frac{3\pi }{4}\right|\\ =6\end{array}$

Therefore, the length of the shadow at 3 P.M is $6$.

To determine

To analyze: The shadow as the time approaches to 6 P.M.

Explanation of Solution

From the above Figure 1, observed that the function $S\left(t\right)=6\left|\cot \frac{\pi }{12}t\right|$ approaches to 6 for infinity as $t$ approaches to $12$

That is, As the time reaches to 6 P.M the length of the shadow become infinity.

Observation:

Compute the length of the shadow as the time approaches to 6 P.M.

That is, compute the value of $\underset{t\to 12}{\lim }6\left|\cot \frac{\pi }{12}t\right|$.

$\begin{array}{c}\underset{t\to 12}{\lim }6\left|\cot \frac{\pi }{12}t\right|=6\left|\cot \frac{12\pi }{12}\right|\\ =6\left|\cot \pi \right|\\ =6\left|\frac{\cos \pi }{\sin \pi }\right|\\ =6\left|\frac{1}{0}\right|\\ =\infty \end{array}$

Therefore, as the time reaches to 6 P.M the length of the shadow become infinity.