Question

blueberries. now, he is using them to make blueberry scones. there is a linear relationship between the number of batches of scones ian makes, and the amount of blueberries left, in ounces. chart: x-axis: batches of scones (0 - 12), y-axis: amount of blueberries left (ounces, 10 - 120). line starts at (0,90), ends at (12, ~17). describe the rate of change for this relationship. the amount of blueberries left dropdown by dropdown per batch of scones.

Explanation:

Step1: Identify two points on the line

From the graph, when \( x = 0 \) (batches), \( y = 90 \) (ounces of blueberries left). When \( x = 12 \) (batches), \( y = 15 \) (ounces of blueberries left, looking at the end point). Wait, maybe better to take two clear points. Let's take (0, 90) and (12, 15)? Wait no, the line at x=12 is at y=15? Wait the graph: when x=0, y=90. Let's take another point, say x=6, y=55? Wait no, let's calculate the slope. The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take (0, 90) and (12, 15)? Wait no, maybe (0,90) and (12, 15) is not correct. Wait looking at the grid, each square is 10 on y and 1 on x. Wait when x=0, y=90. When x=3, y=70? Wait no, x=3: y is 70? Wait the line goes from (0,90) to (12, 15)? Wait no, let's check the slope. Let's take (0,90) and (12, 15) – no, that would be (90 - 15)/(0 - 12) = 75/(-12) = -6.25? No, that can't be. Wait maybe (0,90) and (12, 15) is wrong. Wait the line at x=12 is at y=15? Wait the graph: the y-axis is amount left, so as x (batches) increases, y decreases. Let's take two points: (0, 90) and (12, 15)? Wait no, let's see, when x=0, y=90. When x=12, y=15? Wait the difference in y is 90 - 15 = 75, over 12 batches. So 75/12 = 6.25? No, slope is (y2 - y1)/(x2 - x1) = (15 - 90)/(12 - 0) = -75/12 = -6.25? Wait that seems high. Wait maybe I misread the graph. Wait the y-axis: at x=0, y=90. At x=12, y=15? Wait no, the end point is at x=12, y=15? Wait the graph shows the line ending at x=12, y=15? Wait maybe the correct two points are (0,90) and (12, 15), so the slope is (15 - 90)/(12 - 0) = -75/12 = -6.25? No, that's -6.25. Wait but maybe the intended points are (0,90) and (12, 15) is wrong. Wait maybe (0,90) and (6, 55)? No, let's do it properly. Wait the problem is about rate of change, which is the slope (rate of change for linear relationship is the slope). So let's take two points: (0, 90) and (12, 15). Then slope is (15 - 90)/(12 - 0) = -75/12 = -6.25? Wait that's -6.25 ounces per batch. Wait but maybe the graph is different. Wait maybe (0,90) and (12, 15) is not correct. Wait let's check the y-intercept is 90, so the equation is y = mx + 90. When x=12, y=15, so 15 = 12m + 90 → 12m = -75 → m = -75/12 = -6.25. So the rate of change is -6.25, meaning the amount of blueberries left decreases by 6.25 ounces per batch. Wait but maybe the graph has different points. Wait maybe I made a mistake. Alternatively, maybe the two points are (0,90) and (6, 55)? No, 90 - 55 = 35, over 6 batches: 35/6 ≈ 5.83. No, that's not. Wait maybe the correct slope is -5? Wait no, let's see the grid. Each x is 1, y is 10. So from (0,90) to (12, 15): 90 - 15 = 75, 12 batches. 75/12 = 6.25. So the rate of change is -6.25 ounces per batch, meaning it decreases by 6.25 ounces per batch. Wait but maybe the problem expects a simpler number. Wait maybe I misread the graph. Wait the line at x=0 is 90, x=12 is 15? Wait the end point is at x=12, y=15? So the change in y is 90 - 15 = 75, over 12 batches. So 75/12 = 6.25. So the rate of change is -6.25, so the amount of blueberries left decreases by 6.25 ounces per batch. Wait but maybe the graph is different. Alternatively, maybe the two points are (0,90) and (12, 15) is correct. So the rate of change is the slope, which is (15 - 90)/(12 - 0) = -75/12 = -6.25. So the amount decreases by 6.25 ounces per batch.

Step2: Interpret the rate of change

The rate of change (slope) is negative, indicating a decrease. The magnitude is 6.25 ounces per batch. So the amount of blueberries left decreases by 6.25 ounces per batch.

Answer:

The amount of blueberries left decreases by 6.25 ounces per batch of scones.