Question

the graph below shows the relationship between the volume in ml, v, and the height in cm, h, for cylinders a and b. which cylinder has a greater rate of change in the height of the water, for a given change in volume?
height in cm
0 25 50 75 100 125 150 175 200 225 250
(50, 0.5)
(100, 1)
a
b
slope =

Explanation:

Step1: Recall the formula for slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the rate of change of height \( h \) with respect to volume \( V \), we consider the slope of the line \( h \) vs \( V \).

Step2: Calculate slope for Cylinder A

Take two points for Cylinder A: \((50, 0.5)\) and \((100, 1)\).
Slope \( m_A=\frac{1 - 0.5}{100 - 50}=\frac{0.5}{50}=0.01 \)

Step3: Calculate slope for Cylinder B

Take two points for Cylinder B: Let's assume the line passes through \((0,0)\) and \((150, 5)\) (from the graph, when \( V = 150 \), \( h = 5 \)).
Slope \( m_B=\frac{5 - 0}{150 - 0}=\frac{5}{150}\approx0.033 \) (or using another pair, like from origin to \((50, \frac{5}{3})\) but simpler to see the steepness). Alternatively, from the graph, for Cylinder B, when \( V = 50 \), \( h \) seems to be around \( \frac{5}{3}\approx1.666 \)? Wait, no, looking at the graph, the line for B goes from (0,0) to (150,5) (since at V=150, h=5). So slope \( m_B=\frac{5}{150}=\frac{1}{30}\approx0.033 \), and \( m_A = 0.01=\frac{1}{100} \). Since \( 0.033>0.01 \), Cylinder B has a greater slope (rate of change of h with respect to V).

Answer:

Cylinder B has a greater rate of change in the height of the water for a given change in volume.