Chapter 1.2, Problem 9PSA

a.

To determine

To Find: The angle between the hands of the clock at 3:00.

Answer to Problem 9PSA

The angle between the hands of the clock at 3:00 is $90°$ .

Explanation of Solution

Given:

Hands of a clock showing the time either 3:00 am or 3:00 pm.

Concept Used:

  $\begin{array}{l}12\text{hour}=360°\\ 1\text{hour}=\frac{360°}{12}=30°\\ 1\text{minute}=\frac{30°}{60}=\left(\frac{1}{2}\right)°\end{array}$

Also, the formula for exact angle between hands of a clock hh:mm

  $\theta =\left|\frac{11}{2}\times \text{minutes}-30\times \text{hour}\right|$

Calculation:

As per the given problem time is 3:00, thus the angle between hands of clock is

  $\begin{array}{l}\theta =\left|\frac{11}{2}\times \text{0}-30\times \text{3}\right|\\ \text{=}\left|\text{0}-90\right|\\ =\left|-90\right|\\ \therefore \theta =90°\end{array}$

Conclusion:

  $\theta =90°$

b.

To determine

To Find: The angle between the hands of the clock at 4:30.

Answer to Problem 9PSA

The angle between the hands of the clock at 4:30 is $45°$ .

Explanation of Solution

Given:

Hands of a clock showing the time either 4:30 am or 4:30 pm.

Concept Used:

  $\begin{array}{l}12\text{hour}=360°\\ 1\text{hour}=\frac{360°}{12}=30°\\ 1\text{minute}=\frac{30°}{60}=\left(\frac{1}{2}\right)°\end{array}$

Also, the formula for exact angle between hands of a clock hh:mm

  $\theta =\left|\frac{11}{2}\times \text{minutes}-30\times \text{hour}\right|$

Calculation:

As per the given problem time is 4:30, thus the angle between hands of clock is

  $\begin{array}{l}\theta =\left|\frac{11}{2}\times \text{30}-30\times \text{4}\right|\\ \text{=}\left|\text{165}-120\right|\\ =\left|45\right|\\ \therefore \theta =45°\end{array}$

Conclusion:

  $\theta =45°$

c.

To determine

To Find: The angle between the hands of the clock at 7:20.

Answer to Problem 9PSA

The angle between the hands of the clock at 7:20 is $100°$ .

Explanation of Solution

Given:

Hands of a clock showing the time either 7:20 am or 7:20pm.

Concept Used:

  $\begin{array}{l}12\text{hour}=360°\\ 1\text{hour}=\frac{360°}{12}=30°\\ 1\text{minute}=\frac{30°}{60}=\left(\frac{1}{2}\right)°\end{array}$

Also, the formula for exact angle between hands of a clock hh:mm

  $\theta =\left|\frac{11}{2}\times \text{minutes}-30\times \text{hour}\right|$

Calculation:

As per the given problem time is 7:20, thus the angle between hands of clock is

  $\begin{array}{l}\theta =\left|\frac{11}{2}\times \text{20}-30\times \text{7}\right|\\ \text{=}\left|\text{110}-210\right|\\ =\left|-100\right|\\ \therefore \theta =100°\end{array}$

Conclusion:

  $\theta =100°$

d.

To determine

To Find: The angle between the hands of the clock at 1:45.

Answer to Problem 9PSA

The angle between the hands of the clock at 1:45 is $217.5°$ .

Explanation of Solution

Given:

Hands of a clock showing the time either 1:45 am or 1:45 pm.

Concept Used:

  $\begin{array}{l}12\text{hour}=360°\\ 1\text{hour}=\frac{360°}{12}=30°\\ 1\text{minute}=\frac{30°}{60}=\left(\frac{1}{2}\right)°\end{array}$

Also, the formula for exact angle between hands of a clock hh:mm

  $\theta =\left|\frac{11}{2}\times \text{minutes}-30\times \text{hour}\right|$

Calculation:

As per the given problem time is 7:20, thus the angle between hands of clock is

  $\begin{array}{l}\theta =\left|\frac{11}{2}\times \text{45}-30\times \text{1}\right|\\ \text{=}\left|\text{247}\text{.5}-30\right|\\ =\left|217.5\right|\\ \therefore \theta =217.5°\end{array}$

Conclusion:

  $\theta =217.5°$