Question

martina tracked the number of minutes she spent exercising and the number of calories she burned during her exercise. she then created a scatterplot of her data, shown below. use the scatterplot to answer questions 8 - 10.
8 which of the following best represents the trend line shown on the graph?
a $c = \frac{1}{2}m$
b $c = \frac{1}{60}m$
c $c = 60 + m$
d $c = 60m$
9 use the trend line to predict how many calories you would burn by doing three hours of exercise.
a 90
b 120
c 180
d 300
10 approximately how many hours of exercise would be required to burn 400 calories?

Explanation:

Question 8

Step1: Analyze the trend line's slope

The trend line passes through points like (30, 0.5) (since calories are in hundreds, 0.5 hundred is 50, but let's use the equation form). Let's take two points: when \( m = 30 \), \( c = 0.5 \) (in hundreds, so \( c = 50 \) actual, but in the equation, \( c \) is in hundreds? Wait, no, the y-axis is "Calories Burned (in hundreds)". Wait, no, looking at the graph: at \( m = 30 \), the point is at \( c = 0.5 \) (since the grid lines: each major grid is 1 (hundred), so 30 minutes, calories burned is 0.5 (hundred) = 50? Wait, no, maybe the y-axis is "Calories Burned" with each grid line as 100? Wait, the label is "Calories Burned (in hundreds)"? Wait, the y-axis label says "Calories Burned (in hundreds)", so each unit on y is 100 calories. So at \( m = 30 \), the point is at \( c = 0.5 \) (so 50 calories? No, that doesn't make sense. Wait, maybe the y-axis is just "Calories Burned" with each grid line as 100? Wait, the first point at 30 minutes: the dot is at y=0.5? No, looking at the graph, the y-axis has 1, 2, 3, ..., 9, with each grid line between them. Wait, the trend line: let's take two points. At \( m = 30 \), the trend line is at \( c = 0.5 \) (if y is in hundreds, that's 50, but that's low). Wait, maybe the y-axis is "Calories Burned" with each unit as 100? Wait, no, the problem says "Calories Burned (in hundreds)"? Wait, the label is "Calories Burned (in hundreds)", so c is in hundreds. So at \( m = 30 \), c = 0.5 (hundred) = 50 calories? No, that seems low. Wait, maybe the y-axis is just "Calories Burned" with each grid line as 100. Let's check the options. Option B: \( c = \frac{1}{60}m \). Let's plug m=30: \( c = \frac{30}{60} = 0.5 \). Which matches the point at (30, 0.5) (since c is in hundreds? Wait, no, if c is in hundreds, then 0.5 would be 50, but maybe the y-axis is just calories, and the label is a mistake. Wait, let's check the options. Option B: \( c = \frac{1}{60}m \). So for m=60, c=1. Which matches the point at (60, 1). Yes! At m=60, the dot is at c=1. So that's correct. So the trend line passes through (30, 0.5), (60, 1), (90, 1.5), etc. So the slope is \( \frac{1 - 0.5}{60 - 30} = \frac{0.5}{30} = \frac{1}{60} \). So the equation is \( c = \frac{1}{60}m \). Let's check option B: \( c = \frac{1}{60}m \). Yes, that fits. Option A: \( c = \frac{1}{2}m \) would give c=15 at m=30, which is way too high. Option C: \( c = 60 + m \) would be 90 at m=30, too high. Option D: \( c = 60m \) is way too high. So the correct equation is B.

Step2: Verify with another point

At m=60, option B gives \( c = \frac{60}{60} = 1 \), which matches the trend line at (60, 1). At m=120, \( c = \frac{120}{60} = 2 \), which matches the trend line at (120, 2). So yes, option B is correct.

Step1: Convert 3 hours to minutes

3 hours = 3 * 60 = 180 minutes.

Step2: Use the trend line equation from question 8, which is \( c = \frac{1}{60}m \) (where c is in hundreds? Wait, no, in question 8, the y-axis was "Calories Burned (in hundreds)"? Wait, no, in question 9, the options are 90, 120, 180, 300. So maybe the y-axis in question 8 was just "Calories Burned" (not in hundreds). Wait, maybe I misinterpreted the y-axis. Let's re-examine. The y-axis label: "Calories Burned (in hundreds)"? No, the original graph: the y-axis has numbers 1, 2, 3, ..., 9, with each grid line. So if at m=30, the point is at c=0.5 (but that would be 50, but options for question 9 are 90, 120, etc. So maybe the y-axis is "Calories Burned" with each unit as 100? No, that can't be. Wait, question 9: "Use the trend line to predict how many calories you would burn by doing three hours of exercise." Three hours is 180 minutes. From the trend line equation (question 8 answer: \( c = \frac{1}{60}m \)). Wait, if m=180, then \( c = \frac{180}{60} = 3 \). But the options are 90, 120, 180, 300. Wait, that's a problem. Wait, maybe the y-axis in the graph is "Calories Burned" with each unit as 100? No, that would make c=3 (hundred) = 300. Oh! Wait, the y-axis is "Calories Burned (in hundreds)", so c is in hundreds. So when we calculate c = 3 (from \( c = \frac{1}{60}*180 = 3 \)), that's 3 hundred calories, which is 300. Wait, but the options for question 9 are A.90, B.120, C.180, D.300. So D is 300. Wait, but let's check the trend line. At m=180 minutes, where is the trend line? Looking at the graph, at m=180, the trend line is at c=3 (since the grid lines: 180 minutes, the trend line is at y=3). Since y is "Calories Burned (in hundreds)", that's 3*100=300? No, that would be 300 calories. Wait, but let's check the options. Option D is 300. Wait, but let's re-express the equation. If the y-axis is "Calories Burned" (not in hundreds), then the equation would be \( c = \frac{1}{60}m \), but that would make m=180, c=3, which is too low. So I must have misinterpreted the y-axis. Let's look at the graph again. The y-axis has labels 1, 2, 3, ..., 9. The first point at 30 minutes: the dot is at y=0.5? No, looking at the graph, the first dot at 30 minutes is at y=0.5? No, the grid lines: each major grid is 1, so between 0 and 1, there are two grid lines (so each minor grid is 0.5). So at 30 minutes, the dot is at y=0.5. At 60 minutes, y=1. At 120 minutes, y=2. So the equation is \( c = \frac{1}{60}m \), where c is in hundreds? No, c is the number of hundreds. So c=0.5 means 50 calories? No, that's not right. Wait, the problem says "Calories Burned (in hundreds)" as the y-axis label. So c=0.5 means 50 calories? That seems low, but maybe. But question 9: three hours is 180 minutes. Using the equation \( c = \frac{1}{60}*180 = 3 \) (hundreds of calories), which is 300 calories. So option D is 300.

Step2: Confirm with the graph

At 180 minutes, the trend line is at y=3 (since each unit is 100 calories), so 3*100=300 calories. So the answer is D.

Step1: Understand the trend line equation

From question 8, the trend line is \( c = \frac{1}{60}m \) (where c is in hundreds? No, wait, if we need to burn 400 calories, and the y-axis is "Calories Burned (in hundreds)", then c = 4 (since 4100=400). Wait, no, if the y-axis is "Calories Burned" (not in hundreds), then c=400. Wait, let's clarify. The y-axis label: "Calories Burned (in hundreds)" means that each unit on the y-axis represents 100 calories. So c (the y-value) is the number of hundreds of calories. So to burn 400 calories, c = 4 (since 4100=400).

Step2: Solve for m in the equation \( c = \frac{1}{60}m \)

We have c = 4 (hundreds of calories? No, if c is in hundreds, then 4 (hundred) = 400 calories. So set c = 4 (since 4*100=400) and solve for m:

\( 4 = \frac{1}{60}m \)

Multiply both sides by 60:

\( m = 4 * 60 = 240 \) minutes.

Step3: Convert minutes to hours

240 minutes = \( \frac{240}{60} = 4 \) hours.

Wait, but let's check the graph. The trend line at c=4 (calories burned) would be at m=240 minutes (since \( c = \frac{1}{60}m \), so m=60c). So c=4, m=240 minutes=4 hours.

Answer:

B. \( c = \frac{1}{60}m \)

Question 9