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------------------------------------------------ Student Guide (continued)

Now let's get started!

Step 1: Calculating Distance at Maximum Speed

- If the cheetah sprinted at maximum speed, how far would the cheetah have traveled?

a) Set up an equation to represent the distance the cheetah covered in terms of [tex] t [/tex] minutes running at maximum speed. Remember, units of distance and time must agree. Use the conversion information from the warm-up to write a rate in miles per minute.

b) Let [tex] t = 5 [/tex] and solve for the distance the cheetah covered in 10 minutes.

Step 2: Calculating Distance Using Varied Speeds

- Suppose the cheetah sprinted at maximum speed for 8 minutes and then slowed to 40 mph for the next 8 minutes.

a) How far would the cheetah have traveled in the first 7 minutes? Show how you arrived at your answer.

b) How far would the cheetah have traveled in the next 7 minutes? Show how you arrived at your answer.

c) How much farther did the cheetah travel in the first 7 minutes than in the second 7 minutes?

d) The cheetah traveled 1.75 times faster for the first 7 minutes than it did for the second 7 minutes. Was the distance traveled during the first 7 minutes 1.75 times greater than the distance traveled during the second 7 minutes? Show the calculation to justify your answer.

Sure! Let's go through the problem step-by-step:

### Step 1: Calculating Distance at Maximum Speed

a) Equation for Distance at Maximum Speed:
- The maximum speed of the cheetah is 60 miles per hour (mph). To find the speed in miles per minute, divide by 60 (since there are 60 minutes in an hour):
[tex]\[
\text{Maximum speed in miles per minute} = \frac{60}{60} = 1 \text{ mile per minute}
\][/tex]
- The distance covered in time [tex]$ t $[/tex] minutes is given by the formula:
[tex]\[
\text{Distance} = \text{Speed} \times \text{Time}
\][/tex]
So, the equation is:
[tex]\[
\text{Distance} = 1 \times t = t \text{ miles}
\][/tex]

b) Distance in 10 minutes at Maximum Speed:
- If [tex]$ t = 10 $[/tex] minutes, then:
[tex]\[
\text{Distance} = 1 \times 10 = 10 \text{ miles}
\][/tex]

### Step 2: Calculating Distance Using Varied Speeds

a) Distance in the First 7 Minutes at Maximum Speed:
- Using the speed of 1 mile per minute, in 7 minutes, the distance is:
[tex]\[
\text{Distance} = 1 \times 7 = 7 \text{ miles}
\][/tex]

b) Distance in the Next 7 Minutes at Reduced Speed:
- The cheetah slows to 40 mph. Convert this to miles per minute:
[tex]\[
\text{Reduced speed in miles per minute} = \frac{40}{60} \approx 0.6667 \text{ miles per minute}
\][/tex]
- For 7 minutes at this speed, the distance is:
[tex]\[
\text{Distance} = 0.6667 \times 7 \approx 4.67 \text{ miles}
\][/tex]

c) How Much Farther in the First 7 Minutes:
- The difference in distances is:
[tex]\[
7 - 4.67 \approx 2.33 \text{ miles}
\][/tex]

d) Comparing the Distances:
- The problem states the cheetah traveled 1.75 times faster in the first 7 minutes than the second 7 minutes. Let's check if the distance covered is also in that ratio:
- Calculate the ratio of the two distances:
[tex]\[
\text{Ratio} = \frac{7}{4.67} \approx 1.5
\][/tex]
Since the ratio calculated is approximately 1.5, not 1.75, the distance is not 1.75 times greater.

Hopefully, this breakdown helps clarify the problem! If you have any more questions or need further explanation, feel free to ask.