Question

firework shell diameter vs. height at burst
what is the rate of change?
enter your answer in the box.

Explanation:

Step1: Identify two points

From the graph, when diameter \( x_1 = 1 \) in, height \( y_1 = 25 \) m (approx). When \( x_2 = 5 \) in, \( y_2 = 120 \) m (approx). Or use \( (1,25) \) and \( (5,120) \).

Step2: Calculate rate of change

Rate of change (slope) \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{120 - 25}{5 - 1}=\frac{95}{4}=23.75 \)? Wait, no, maybe better points. Wait, at \( x=1 \), \( y=25 \); \( x=2 \), \( y=50 \)? Wait, no, let's check the grid. At \( x=1 \), y is 25 (between 20 and 30). At \( x=2 \), y is 50? Wait, no, the line goes from (1,25) to (5,120)? Wait, no, let's take (1,25) and (5,120). Wait, or (1,25) and (2,50): slope \( \frac{50 - 25}{2 - 1}=25 \). Wait, maybe the correct points: when x=1, y=25; x=5, y=125? Wait, the top is 120? Wait, maybe the graph has (1,25) and (5,125). Wait, let's recalculate. The rate of change is \( \frac{\Delta y}{\Delta x} \). Let's take two clear points: (1, 25) and (5, 125). Then \( \Delta y = 125 - 25 = 100 \), \( \Delta x = 5 - 1 = 4 \). Wait, no, 125 -25=100, 5-1=4, 100/4=25. Wait, maybe the correct slope is 25. Wait, let's check the graph again. At x=1, y=25; x=2, y=50 (25+25); x=3, y=75; x=4, y=100; x=5, y=125. So the slope is \( \frac{125 - 25}{5 - 1}=\frac{100}{4}=25 \). So rate of change is 25 m per inch.

Answer:

25