Question

1. cylinders a, b, and c have the same radius but different heights. order the cylinders from least volume to greatest volume.
2. here is a cylinder and a cone with the same base and height. how much more water would you need to fill the cylinder than the cone?

a. 3 times as much
b. 2 times as much
c. 5 times as much
d. 4 times as much
problems 3–4: cylinders p and q have the same height. each starts off filled with different amounts of water. the graph shows the height of the water in each cylinder as the volume of water increases.

1. match lines a and b to cylinders p and q.

cylinder	line
q

1. describe what the slopes of lines a and b represent in this situation,

Explanation:

Step1: Recall cylinder volume formula

Volume of cylinder: $V_{cyl} = \pi r^2 h$

Step2: Recall cone volume formula

Volume of cone: $V_{cone} = \frac{1}{3}\pi r^2 h$

Step3: Calculate volume difference

Difference: $V_{cyl} - V_{cone} = \pi r^2 h - \frac{1}{3}\pi r^2 h = \frac{2}{3}\pi r^2 h$

Step4: Compare difference to cone volume

$\frac{V_{cyl}-V_{cone}}{V_{cone}} = \frac{\frac{2}{3}\pi r^2 h}{\frac{1}{3}\pi r^2 h} = 2$

1. For cylinders with equal radius, volume is directly proportional to height. From the diagram, height of B < C < A, so volume follows the same order.
2. For cylinders of equal height, a wider radius means more volume is needed to increase height. Cylinder P has a larger radius, so height increases more slowly with volume (matches line b). Cylinder Q has a smaller radius, so height increases more quickly with volume (matches line a).
3. Slope is height per unit volume, so it represents the inverse of the cylinder's base area: how much the water height rises for each unit of added volume.

Answer:

B. 2 times as much

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