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Chapter 8: Problem 92

The second hand on a clock sweeps out a complete circle in 1 min. A circle forms a \(360^{\circ}\) arc. Use this information. (IMAGE CAN'T COPY) How many degrees does a second hand on a clock move in 45 sec?

Short Answer

Expert verified
270 degrees

Step by step solution

01

Understand the Problem

The second hand of a clock completes a full circle, which is a 360-degree rotation, in 1 minute (60 seconds). We need to find out how many degrees it moves in 45 seconds.
02

Set Up the Proportion

Since the second hand moves 360 degrees in 60 seconds, we can set up a proportion to find the degrees it moves in 45 seconds. Let \(\theta\) be the number of degrees:\[ \frac{360 \, \text{degrees}}{60 \, \text{seconds}} = \frac{\theta \, \text{degrees}}{45 \, \text{seconds}} \]
03

Solve the Proportion

Cross-multiply to solve for \(\theta\):\[ 360 \, \text{degrees} \times 45 \, \text{seconds} = 60 \, \text{seconds} \times \theta \, \text{degrees} \]Simplify:\[ 16200 = 60\theta \]Divide both sides by 60:\[ \theta = \frac{16200}{60} = 270 \]
04

Conclusion

The second hand on a clock moves 270 degrees in 45 seconds.

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Key Concepts

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

Proportions
In math, proportions are used to show that two ratios are equal. They allow us to solve problems involving comparisons between different quantities. Think of a proportion as a way to set up a relationship between two fractions. They are extremely useful in calculating unknown quantities.
Here's a basic idea:
  • If you know that one part of a relationship, say A, is to B as C is to D, then you can set it up as a proportion: \(\frac{A}{B} = \frac{C}{D}\)
  • This means that if you know three parts of the relationship, you can solve for the fourth part by cross-multiplying.
In the context of the clock problem, we use a proportion to compare the seconds and degrees:
\[ \frac{360 \text{ degrees}}{60 \text{ seconds}} = \frac{\theta \text{ degrees}}{45 \text{ seconds}} \] We cross-multiply to find \(\theta\).
Degree Measurement
Degree measurement is a way to quantify angles. A full circle is divided into 360 equal parts, called degrees.
Our everyday clocks use this measurement. For instance:
  • A quarter of a circle is 90 degrees.
  • A half-circle is 180 degrees.
  • A full circle is 360 degrees.
When we say the second hand moves in degrees, we are talking about how far it travels around the circle of the clock face.
In the problem, the second hand moves 360 degrees in one full minute.
By breaking this movement into smaller segments, we can calculate how many degrees it moves in any amount of seconds.
Time Calculation
Time calculation involves figuring out how time spans relate to events or movements. When solving problems:
Consider how time breaks into parts and use that to determine the result.
In our exercise, the clock's second hand moves evenly:
  • It travels 360 degrees in 60 seconds.
  • This means it travels 6 degrees per second (since 360 divided by 60 equals 6).
So for 45 seconds:
Multiply 6 degrees by 45 seconds:\br> 6 \( \times \) 45 = 270 degrees.
This tells us that the second hand moves 270 degrees in 45 seconds.
Pre-Algebra
Pre-algebra involves understanding basic mathematical concepts like operations, fractions, and ratios. It's foundational for solving more complex algebraic problems.
Let's break it down with the clock exercise:
  • First, understand the basic operation: the hand moves 360 degrees in 60 seconds.
  • Set up the ratio based on this information.
  • Use cross-multiplication to solve for the unknown.
By using these fundamental skills, we were able to determine that the second hand moved 270 degrees in 45 seconds. This kind of problem-solving teaches students how to methodically approach and solve real-world problems using basic math principles.

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Most popular questions from this chapter

In the Expanding Your Skills of Section \(8.1,\) we converted U.S. Customary units of area. We use the same procedure to convert metric units of area. This procedure involves multiplying by two unit ratios of length. Example: Converting area Convert \(1000 \mathrm{mm}^{2}\) to square centimeters. $$\text { Solution: } \frac{1000 \mathrm{mm}^{2}}{1} \cdot \frac{1 \mathrm{cm}}{10 \mathrm{mm}} \cdot \frac{1 \mathrm{cm}}{10 \mathrm{mm}}=\frac{1000 \mathrm{mm}^{2}}{1} \cdot \frac{1 \mathrm{cm}^{2}}{100 \mathrm{mm}^{2}}=\frac{1000 \mathrm{cm}^{2}}{100}=10 \mathrm{cm}^{2}$$ convert the units of area, using two factors of the given unit ratio. $$65,000,000 \mathrm{m}^{2}=\quad \mathrm{km}^{2}$$ $$\left(\text { Use } \frac{1 \mathrm{km}}{1000 \mathrm{m}}\right)$$

Determine the surface area of the object described. Use 3.14 for \(\pi\) when necessary. A cylinder with radius \(90 \mathrm{mm}\) and height \(75 \mathrm{mm}\)

A contractor plans to construct a cement patio for one of the houses that he is building. The patio will be a square, 25 ft by 25 ft. After the contractor builds the frame for the cement, he checks to make sure that it is square by measuring the diagonals. Use the Pythagorean theorem to determine what the length of the diagonals should be if the contractor has constructed the frame correctly. Round to the nearest hundredth of a foot.

In the U.S. Customary System of measurement, 1 ton \(=2000\) Ib. In the metric system, 1 metric ton \(=1000 \mathrm{kg}\). Use this information to answer Exercises \(69-72\). The average mass of a blue whale (the largest mammal in the world) is approximately 108 metric tons. How many pounds is this?

Convert the units of mass. $$2011 \mathrm{g}= _____ kg$$
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