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Chapter 1: Problem 55

In the 2016 Olympics, Usain Bolt of Jamaica won the gold medal in the 100 -meter race with a time of 9.81 seconds. In the 1896 Olympics, Thomas Burke of the United States won the gold medal in the 100-meter race in 12.0 seconds. If they ran in the same race, repeating their respective times, by how many meters would Bolt beat Burke?

Short Answer

Expert verified
Bolt would beat Burke by approximately 18.24 meters.

Step by step solution

01

- Determine Bolt's Speed

Calculate Usain Bolt's speed by dividing the distance he ran by his time. Bolt's speed is given by the formula: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] where the distance is 100 meters and the time is 9.81 seconds. Thus, Bolt's speed is \[ \text{Speed}_{\text{Bolt}} = \frac{100}{9.81} \approx 10.19 \text{ meters/second} \]
02

- Determine Burke's Speed

Calculate Thomas Burke's speed by using the same formula: \[ \text{Speed}_{\text{Burke}} = \frac{100}{12.0} \approx 8.33 \text{ meters/second} \]
03

- Calculate Time Difference

Find the difference in their times by subtracting Bolt's time from Burke's time: \[ \text{Time}_\text{difference} = 12.0 - 9.81 = 2.19 \text{ seconds} \]
04

- Calculate Distance Covered by Burke in Time Difference

Find how many meters Burke would run in the time difference. This is done by multiplying Burke's speed by the time difference: \[ \text{Distance}_\text{Burke} = \text{Speed}_\text{Burke} \times \text{Time}_\text{difference} \] which gives \[ \text{Distance}_\text{Burke} = 8.33 \times 2.19 \approx 18.24 \text{ meters} \]
05

- Calculate Meters Bolt Would Win By

Since we are asked by how many meters Bolt would win, the answer is the distance calculated in the previous step: \[ \text{Bolt wins by } \approx 18.24 \text{ meters} \]

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Key Concepts

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

distance-time relationship
The distance-time relationship in speed calculations is fundamental. Speed is a measure of how fast an object moves and can be determined if you know the distance covered and the time taken. The basic formula for speed is: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\). For instance, if Usain Bolt ran 100 meters in 9.81 seconds, we use the formula to find his speed. This relationship highlights how faster speeds reduce travel time for the same distance.
unit conversion
In speed calculation problems, it's essential to ensure that units are consistent. Typically, distances might be given in meters and times in seconds, so the speed will be in meters per second (m/s). However, if distances were given in different units like kilometers or miles, we would need to convert them to meters first. Consistent units allow us to apply formulas accurately and avoid mistakes in calculations. Remember, always check your units before finalizing your solution.
mathematical comparison
Making mathematical comparisons can help understand how one value relates to another. In our exercise, we compared the speeds of two athletes and their race times. We calculated their speeds and then found the time difference by subtracting one time from the other: \(\text{Time}_\text{difference} = 12.0 \text{ s} - 9.81 \text{ s} = 2.19 \text{ s}\). Finally, using Burke鈥檚 speed and the time difference, we determined how many more meters Bolt would have run: \(\text{Distance}_\text{Burke} = 8.33 \text{ m/s} \times 2.19 \text{ s} \approx 18.24 \text{ meters}\).
linear equations
Linear equations are a straightforward method used to solve problems involving constant speed, distance, and time. In this exercise, you saw how we used linear equations to determine each athlete鈥檚 speed: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\). Then, by finding the time difference and multiplying by Burke's speed, we employed another linear equation to calculate the additional distance Bolt would cover: \(\text{Distance}_\text{Burke} = \text{Speed}_\text{Burke} \times \text{Time}_\text{difference}\). Linear equations are helpful tools in breaking down these problems into manageable steps.

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Most popular questions from this chapter

Elaine can complete a landscaping project in 2 hours with the help of either her husband Brian or both her two daughters. If Brian and one of his daughters work together, it would take them 4 hours to complete the project. Assuming the rate of work is constant for each person, and the two daughters work at the same rate, how long would it take Elaine, Brian, and one of their daughters to complete the project?

The distance to the surface of the water in a well can sometimes be found by dropping an object into the well and measuring the time elapsed until a sound is heard. If \(t_{1}\) is the time (measured in seconds) that it takes for the object to strike the water, then \(t_{1}\) will obey the equation \(s=16 t_{1}^{2}\), where \(s\) is the distance (measured in feet). It follows that \(t_{1}=\frac{\sqrt{s}}{4}\). Suppose that \(t_{2}\) is the time that it takes for the sound of the impact to reach your ears. Because sound waves are known to travel at a speed of approximately 1100 feet per second, the time \(t_{2}\) to travel the distance \(s\) will be \(t_{2}=\frac{s}{1100} .\) See the illustration. Now \(t_{1}+t_{2}\) is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard. We have the equation $$ \text { Total time elapsed }=\frac{\sqrt{s}}{4}+\frac{s}{1100} $$ Find the distance to the water's surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.

\(2+3 i\) is a solution of a quadratic equation with real coefficients. Find the other solution.

Challenge Problem Sharing the cost of a Pizza Judy and Tom agree to share the cost of an 18 dollar pizza based on how much each ate. If Tom ate \(\frac{2}{3}\) the amount that Judy ate, how much should each pay? [Hint: Some pizza may be left.]

Perform the indicated operation, and write each expression in the standard form \(a+\) bi. $$ \frac{2-i}{-2 i} $$
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