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Chapter 1: Problem 2

The table below shows the number of calories used per minute as a function of an individual's body weight for three sports: $$\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Activity } & 100 \mathrm{lb} & 120 \mathrm{lb} & 150 \mathrm{lb} & 170 \mathrm{lb} & 200 \mathrm{lb} & 220 \mathrm{lb} \\ \hline \text { Walking } & 2.7 & 3.2 & 4 & 4.6 & 5.4 & 5.9 \\ \hline \text { Bicycling } & 5.4 & 6.5 & 8.1 & 9.2 & 10.8 & 11.9 \\ \hline \text { Swimming } & 5.8 & 6.9 & 8.7 & 9.8 & 11.6 & 12.7 \\ \hline \end{array}$$ a) Determine the number of calories that a 200 lb person uses in one half-hour of walking. b) Who uses more calories, a \(120 \mathrm{lb}\) person swimming for one hour, or a \(220 \mathrm{lb}\) person bicycling for a half-hour? c) Does the number of calories of a person swimming increase or decrease as weight increases?

Short Answer

Expert verified
a) 162 calories. b) 120 lb person swimming uses 414 calories, 220 lb person bicycling uses 357 calories. Swimming increases calories as weight increases.

Step by step solution

01

Identify calories per minute for 200 lb person walking

From the table, find the number of calories used per minute for a 200 lb person walking. The value is 5.4 calories per minute.
02

Calculate total calories for 200 lb person walking for half an hour

Multiply the calories per minute by the number of minutes in half an hour (30 minutes). Thus, total calories = 5.4 * 30.
03

Identify calories used by 120 lb person swimming per minute

From the table, find the number of calories used per minute for a 120 lb person swimming. The value is 6.9 calories per minute.
04

Calculate total calories for 120 lb person swimming for one hour

Multiply the calories per minute by the number of minutes in one hour (60 minutes). Thus, total calories = 6.9 * 60.
05

Identify calories used by 220 lb person bicycling per minute

From the table, find the number of calories used per minute for a 220 lb person bicycling. The value is 11.9 calories per minute.
06

Calculate total calories for 220 lb person bicycling for half an hour

Multiply the calories per minute by the number of minutes in half an hour (30 minutes). Thus, total calories = 11.9 * 30.
07

Compare the calories used by 120 lb person swimming and 220 lb person bicycling

Compare the total calories calculated for the 120 lb person swimming (414) and the 220 lb person bicycling (357).
08

Determine trend for calories used in swimming as weight increases

Observe the table and note the values for swimming as the weight increases. As the weight goes from 100 lbs to 220 lbs, the calories used per minute increase from 5.8 to 12.7.

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Key Concepts

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

sports energy expenditure
Sports and activities require different amounts of energy, which translates into calories burned. This variation depends on the type of activity and the individual's body weight. For instance, walking, bicycling, and swimming each have different calorie-burning rates. Understanding energy expenditure helps us make informed choices about our physical activities and their benefits.
In the exercise, walking for a 200 lb person burns 5.4 calories per minute while bicycling for someone weighing 220 lbs burns 11.9 calories per minute. Swimming has a similar pattern, with a 120 lb person burning 6.9 calories per minute. By knowing how much energy each sport uses, we can compare activities and choose the best ones for our fitness goals.
weight and calories
Body weight significantly influences calorie consumption in physical activities. Heavier individuals typically burn more calories per minute compared to lighter individuals because they expend more energy to move their body mass. Understanding this relationship helps tailor fitness plans for effective weight management.
For instance, in the given table, a 100 lb person walking uses 2.7 calories per minute, whereas a 200 lb person uses 5.4 calories per minute. This doubling factor illustrates how weight proportionally impacts energy expenditure.
Activities like walking, bicycling, and swimming highlight these differences. Knowing these details enables one to better plan workouts to achieve personal health goals, ensuring the activities chosen are both enjoyable and beneficial.
activity comparison
Comparing different physical activities can help determine which ones are more efficient for calorie burning, depending on the available time and effort. In our example, we compare walking, bicycling, and swimming. For a clear comparison, consider the caloric burn for different weights and time spans.
For example, a 120 lb person swimming for one hour burns 414 calories, more than a 220 lb person bicycling for half an hour, who burns 357 calories. Despite the person's heavier weight and the intensity of bicycling, the duration of the activity plays a critical role.
Such comparisons help in deciding which sports are more effective for specific fitness goals, such as weight loss or endurance improvement. Remember, the best activity is one that you enjoy and can maintain consistently over time.

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Most popular questions from this chapter

A bungee jumper dives from a tower at time \(t=0\). Her height \(s\) in feet at time \(t\) in seconds is given by \(s(t)=100 \cos (0.75 t) \cdot e^{-0.2 t}+100\). a. Write an expression for the average velocity of the bungee jumper on the interval \([1,1+\) \(h]\) b. Use computing technology to estimate the value of the limit as \(h \rightarrow 0\) of the quantity you found in (a). c. What is the meaning of the value of the limit in (b)? What are its units?

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For each of the following prompts, give an example of a function that satisfies the stated criteria. A formula or a graph, with reasoning, is sufficient for each. If no such example is possible, explain why. a. A function \(f\) that is continuous at \(a=2\) but not differentiable at \(a=2\). b. A function \(g\) that is differentiable at \(a=3\) but does not have a limit at \(a=3\). c. A function \(h\) that has a limit at \(a=-2,\) is defined at \(a=-2,\) but is not continuous at \(a=-2\) d. A function \(p\) that satisfies all of the following: \- \(p(-1)=3\) and \(\lim _{x \rightarrow-1} p(x)=2\) $$\text { - } p(0)=1 \text { and } p^{\prime}(0)=0$$ \- \(\lim _{x \rightarrow 1} p(x)=p(1)\) and \(p^{\prime}(1)\) does not exist

Suppose that the population, \(P\), of China (in billions) can be approximated by the function \(P(t)=1.15(1.014)^{t}\) where \(t\) is the number of years since the start of \(1993 .\) a. According to the model, what was the total change in the population of China between January 1,1993 and January \(1,2000 ?\) What will be the average rate of change of the population over this time period? Is this average rate of change greater or less than the instantaneous rate of change of the population on January \(1,2000 ?\) Explain and justify, being sure to include proper units on all your answers. b. According to the model, what is the average rate of change of the population of China in the ten-year period starting on January 1, \(2012 ?\) c. Write an expression involving limits that, if evaluated, would give the exact instantaneous rate of change of the population on today's date. Then estimate the value of this limit (discuss how you chose to do so) and explain the meaning (including units) of the value you have found. d. Find an equation for the tangent line to the function \(y=P(t)\) at the point where the \(t\) -value is given by today's date.
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