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Chapter 7: Problem 74

Find the measure of the smaller angle between the hands of a clock when the time is 5 o'clock.

Short Answer

Expert verified
The measure of the smaller angle between the hands of a clock at 5 o'clock is 150 degrees.

Step by step solution

01

Determine the position of the hour hand

At 5 o'clock, the hour hand is at 5. Each hour on the clock represents 30 degrees, therefore the position of the hour hand in degrees is \(5 \times 30 = 150\) degrees.
02

Determine the position of the minute hand

At 5 o'clock, the minute hand is at 12. This is the starting point, which we can consider as zero degrees.
03

Calculate the measure of the smaller angle

We subtract the smaller measure from the larger measure. Since the minute hand is at zero degrees, the measure of the smaller angle is \(150 - 0 = 150\) degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle Measurement
Understanding how to measure angles on a clock is a crucial skill. The clock face is a perfect circle, and just like any circle, it can be divided into 360 degrees. Each hour on a clock represents a specific angle. Since there are 12 hours on a clock, each hour corresponds to \(\frac{360}{12} = 30\) degrees.

To find the measure of an angle formed by the clock's hands, we simply calculate the difference between the two positions. This difference gives us the angle in degrees, either clockwise or counterclockwise, depending on the position of the hour and minute hands.
  • The smaller angle is usually considered unless specifically stated otherwise.
  • This concept helps in visualizing the movement and position of the clock hands accurately.
  • Remember that each minute corresponds to 6 degrees as well (since there are 60 minutes in a clock face).
Hour Hand Position
The position of the hour hand is critical for determining the angle between it and the minute hand. Every hour, the hour hand shifts 30 degrees further along the clock face.

For example, at 5 o'clock, the hour hand points exactly at the 5, representing \(5 \times 30 = 150\) degrees. This calculation is straightforward because:
  • Every hour mark represents an increase in angle by 30 degrees.
  • Between any two consecutive numbers, like from 4 to 5, the hour hand moves 30 degrees.
  • Always ensure to refer to the correct reference point when calculating as it ensures accuracy in finding the position.
Minute Hand Position
The position of the minute hand is also essential when calculating the clock angle. Since the minute hand moves from 12 (considered as 0 degrees) across the clock face, it allows us to establish a starting point for calculations.

At 5 o'clock, the minute hand points to 12, which we take as the 0-degree mark. The critical understanding here includes:
  • The minute hand completes a full cycle, or 360 degrees, every hour.
  • Therefore, each minute marks a 6-degree change (掳).
  • This zero-degree position makes it simple to identify at which point the minute hand sits, aiding in straightforward calculations of angles with the hour hand.
  • If the time isn't an exact hour, find the minute's position by counting multiples of 6 degrees per minute past 12.

A clear understanding of both hand positions can make calculating the smaller angle easier and faster.

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Most popular questions from this chapter

You are wallpapering two walls of a room, one measuring \(9 \mathrm{ft}\) by 8 ft and the other measuring 11 ft by 8 ft. The wallpaper costs \(\$ 37\) per roll, and each roll covers \(40 \mathrm{ft}^{2}\). What is the cost to wallpaper the two walls?

Determine whether the statement is always true, sometimes true, or never true. a. The slant height of a regular pyramid is longer than the height. b. The slant height of a cone is shorter than the height. c. The four triangular faces of a regular pyramid are equilateral triangles.

Determine whether the given conditions guarantee that the triangles are congruent. One right triangle has a hypotenuse of length 10 in. and an acute angle of \(40^{\circ} .\) A second right triangle has a hypotenuse of length 10 in. and an acute angle of \(50^{\circ} .\)

A rectangular solid has a length of \(6.8 \mathrm{m},\) a width of \(2.5 \mathrm{m},\) and a height of \(2 \mathrm{m} .\) Find the volume of the solid.

Find the complement of a \(31^{\circ}\) angle.
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