Question

crystal fills two cylinders with water and records h, the height of the water in cm for different volumes, v, of the water in ml. the two cylinders do not have the same radius. when cylinder a is filled with 50 ml of water, the height of the water is 1 cm. when cylinder b is filled with 50 ml of water, the height of the water is \boxed{} cm.

Explanation:

Step1: Analyze Cylinder A's graph

For Cylinder A, when \( V = 50 \, \text{mL} \), \( h = 1 \, \text{cm} \). Looking at the graph, the line for A passes through \( (50, 1) \).

Step2: Analyze Cylinder B's graph

We need to find the height of Cylinder B when \( V = 50 \, \text{mL} \). From the graph, the line for B (steeper line) at \( V = 50 \, \text{mL} \), we can see the height. Let's check the slope or the points. Alternatively, since at \( V = 50 \), looking at the y - axis (height) for line B, when \( V = 50 \), the height is 2 cm? Wait, no, wait. Wait, Cylinder A: 50 mL gives 1 cm. Let's check the grid. The x - axis is volume (0 - 100 mL), y - axis is height (0 - 2 cm). For Cylinder B, the line goes from (0,0) to, say, when V = 50, what's h? Let's see the graph: the two lines, A and B. When V = 50, for B, the height is 2? Wait, no, wait the problem says "the two cylinders do not have the same radius". Wait, Cylinder A: 50 mL → 1 cm. Cylinder B: let's see the graph. The line for B: when V = 50, h is 2? Wait, no, maybe I misread. Wait, the first statement: "When Cylinder A is filled with 50 mL of water, the height of the water is 1 cm." So on the graph, the line for A passes through (50,1). Then the line for B: let's see, when V = 50, what's the y - value? Looking at the graph, the line B (the steeper one) at x = 50 (V = 50), the y (h) is 2? Wait, no, wait the grid: each square on y - axis: from 0 to 2, with 0.2 increments? Wait, no, the y - axis is labeled 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2. So each major grid line is 0.2? Wait, no, the distance between 0 and 1 is 5 grid lines? Wait, no, the graph has a grid where each small square is, say, 10 mL on x - axis (since x goes from 0 to 100, with 10 - unit intervals: 10,20,30,40,50,60,...) and on y - axis, each small square is 0.2 cm? Wait, no, the y - axis labels are 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2. So from 0 to 1, there are 5 intervals (0.2 each). Now, for Cylinder B, the line: when V = 50 (x = 50), what's the y - coordinate? Let's see the two lines: A is the less steep, B is steeper. At x = 50, for B, the y is 2? Wait, no, wait the first point: Cylinder A: 50 mL → 1 cm. So the line for A has a slope of \( \frac{1}{50} \) (h per V). The line for B: let's see, when V = 50, h is 2? Wait, no, maybe I made a mistake. Wait, the problem says "when Cylinder A is filled with 50 mL of water, the height of the water is 1 cm". So on the graph, the point (50,1) is on line A. Then line B: let's look at the graph. The line B (the upper line) at x = 50 (V = 50), what's the y - value? Looking at the y - axis, when x = 50, the line B reaches y = 2? Wait, no, the y - axis goes up to 2. Wait, maybe the answer is 2? Wait, no, wait let's check again. Wait, the grid: x - axis: 0,10,20,30,40,50,60,... y - axis: 0,0.2,0.4,0.6,0.8,1,1.2,1.4,1.6,1.8,2. So for line B, when x = 50 (V = 50), the y (h) is 2? Wait, but let's see the slope. For Cylinder A: \( h=\frac{1}{50}V \). For Cylinder B, if at V = 50, h = 2, then slope is \( \frac{2}{50}=\frac{1}{25} \), which is steeper, which matches the graph (B is steeper than A). So when V = 50 mL, Cylinder B's height is 2 cm? Wait, no, wait the problem's graph: let's see the two lines. Line A: (50,1), line B: (50,2)? Wait, maybe I misread. Wait, the user's graph: "the two cylinders do not have the same radius". So Cylinder B has a smaller radius (since for same volume, smaller radius means higher height, because volume of cylinder \( V=\pi r^{2}h \), so \( h = \frac{V}{\pi r^{2}} \), so smaller r means larger h for same V). So Cylinder B has smaller radius, so higher height. Cylinder A: 50 mL → 1 cm. So Cylinder B: 50 mL → higher height. From the graph, at V = 50, B's height is 2 cm. Wait, but let's check the graph again. The two lines: A (less steep) and B (steeper). At x = 50 (V = 50), A is at y = 1, B is at y = 2. So the height for Cylinder B when V = 50 mL is 2 cm? Wait, no, wait the y - axis labels: 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2. So each major tick is 0.2? No, wait 0 to 1 is 5 intervals (0.2 each), 1 to 2 is another 5. So when x = 50, for B, the y - value is 2? Wait, maybe the answer is 2. Wait, but let's confirm. The problem says "the two cylinders do not have the same radius". So volume of cylinder is \( V=\pi r^{2}h \), so \( h=\frac{V}{\pi r^{2}} \). For Cylinder A: \( 1=\frac{50}{\pi r_{A}^{2}} \), so \( \pi r_{A}^{2}=50 \). For Cylinder B: \( h=\frac{50}{\pi r_{B}^{2}} \). Since \( r_{B}<r_{A} \) (because B is steeper, so for same V, h is larger, so \( r_{B} \) is smaller), so \( \pi r_{B}^{2}<50 \), so \( h=\frac{50}{\pi r_{B}^{2}}>1 \). From the graph, at V = 50, B's height is 2. So the answer is 2.

Answer:

2