Question

martina is running. the number of minutes she has run varies directly with the number of calories she has burned. see the graph below.
(a) how many calories is martina burning per minute?
\boxed{} calorie(s)
(b) what is the slope of the graph?
\boxed{}

Explanation:

Step1: Analyze Direct Variation

Since minutes (y) vary directly with calories (x), \( y = kx \), where \( k \) is the constant of proportionality (calories per minute).

Step2: Identify Points on Graph

From the graph, when \( y = 5 \) minutes, \( x = 25 \) calories (approximate, or check intercepts). Wait, actually, let's take a clear point. Suppose when \( y = 10 \) minutes, \( x = 50 \) calories? Wait, no, let's check the axes. The x-axis is calories burned, y-axis is minutes. So when y=5, x=25? Wait, no, maybe when y=20 minutes, x=100 calories? Wait, let's find the slope. Slope \( m=\frac{\Delta y}{\Delta x} \), but since it's direct variation, slope is \( \frac{x}{y} \) for calories per minute? Wait, no: minutes (y) vs calories (x), so the rate of calories per minute is \( \frac{x}{y} \). Let's take two points: (0,0) and (100, 20) (assuming from graph: when calories x=100, minutes y=20). Then calories per minute is \( \frac{100}{20}=5 \). So for part (a), 5 calories per minute. For part (b), slope of the graph (y vs x) is \( \frac{\Delta y}{\Delta x}=\frac{20}{100}=0.2 \)? Wait, no, wait: y is minutes, x is calories. So slope is \( \frac{y}{x}=\frac{20}{100}=0.2 \) minutes per calorie? But the question for (a) is calories per minute, which is \( \frac{x}{y}=\frac{100}{20}=5 \). So let's confirm:

(a) Calories burned per minute: take x=100 calories, y=20 minutes. So \( \frac{100}{20}=5 \) calories per minute.

(b) Slope of the graph (y as minutes, x as calories): slope \( m = \frac{\Delta y}{\Delta x} \). Using (0,0) and (100,20), \( m=\frac{20 - 0}{100 - 0}=0.2 \)? Wait, no, that's minutes per calorie. But maybe the graph has x as minutes? Wait, no, the label says: x-axis "Calories burned", y-axis "Number of minutes". So y is minutes, x is calories. So direct variation: \( y = kx \), so \( k = \frac{y}{x} \), which is minutes per calorie. But the question (a) is calories per minute, which is \( \frac{1}{k} \). Wait, maybe I mixed up. Let's re-express:

Let \( y \) = minutes, \( x \) = calories. Direct variation: \( y = kx \), so \( k = \frac{y}{x} \) (minutes per calorie). Then calories per minute is \( \frac{1}{k}=\frac{x}{y} \).

From the graph, let's pick a point: when y=5 minutes, x=25 calories (since 5 on y-axis, 25 on x-axis). Then \( \frac{x}{y}=\frac{25}{5}=5 \) calories per minute. So that's part (a): 5.

For part (b), slope of the graph (y vs x) is \( k = \frac{y}{x}=\frac{5}{25}=0.2 \) minutes per calorie? But the slope formula is \( \frac{\Delta y}{\Delta x} \), so with (0,0) and (25,5), slope is \( \frac{5 - 0}{25 - 0}=0.2 \). Wait, but the question for (b) is "what is the slope of the graph?" The graph has y (minutes) on vertical, x (calories) on horizontal. So slope is \( \frac{\Delta y}{\Delta x}=\frac{5}{25}=0.2 \). But wait, maybe the graph is x (minutes) and y (calories)? Wait, the label says: "Number of minutes" on y-axis, "Calories burned" on x-axis. So no, y is minutes, x is calories. So slope is 0.2 minutes per calorie. But the calories per minute is 5, which is the reciprocal. Let's check with another point: y=10 minutes, x=50 calories. Then \( \frac{50}{10}=5 \) calories per minute, and slope \( \frac{10}{50}=0.2 \) minutes per calorie. So that's consistent.

So (a) 5 calories per minute, (b) slope is 0.2? Wait, no, wait: maybe I made a mistake. Let's see the graph again. The x-axis is calories, y-axis is minutes. Let's assume the graph has a point at (25,5): calories x=25, minutes y=5. Then calories per minute: \( 25/5=5 \). Slope of the graph (y vs x) is \( 5/25=0.2 \) minutes per calorie. So that'…

Answer:

(a) \(\boxed{5}\)
(b) \(\boxed{0.2}\) (or \(\frac{1}{5}\), but 0.2 is decimal)

Wait, but maybe the slope is calculated as \( \frac{x}{y} \) for calories per minute, but no, slope of the graph (y as dependent variable, x as independent) is \( \frac{\Delta y}{\Delta x} \). So if y is minutes, x is calories, slope is minutes per calorie. But the question (a) is calories per minute, which is \( \frac{\Delta x}{\Delta y} \). So for (a), using (25,5): \( \frac{25}{5}=5 \). For (b), slope is \( \frac{5}{25}=0.2 \) (minutes per calorie). So the answers are 5 for (a) and 0.2 (or 1/5) for (b).