Question

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lesson 26-3 practice

1. how is the relationship between the formula for the volume of a cone and the formula for the volume of a cylinder related to the relationship between the formula for the volume of a pyramid and the formula for the volume of a prism?
2. find the volume of a beach ball with a diameter of 12 inches. round to the nearest tenth.
3. a glass jar has a height of 5 inches and a radius of 2.5 inches. vanessa wants to fill the jar with beads that cost $0.12 per cubic inch. how much will it cost for her to fill the jar?
4. a. the figure shows the dimensions of a paper cone that will be filled with popcorn. the popcorn costs $0.02 per cubic inch. what is the cost of filling the cone with popcorn?

b. popcorn is also sold in cylindrical tubs that have a diameter of 8 in and a height of 10 in. what is the cost of filling the cylinder with popcorn?
c. explain how you could determine the answer to part b without using the formula for volume of a cylinder.

1. critique the reasoning of others. jason states that the volume of the cone shown below must be one-third of the volume of the cylinder since the two solids have the same height. do you agree or disagree? justify your response.

Explanation:

Step1: Calculate cone volume

First, find radius: $r = \frac{8}{2} = 4$ in. Use cone volume formula:
$$V_{cone} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (4)^2 (10) = \frac{160}{3}\pi \approx 167.55 \text{ cubic inches}$$

Step2: Calculate cone fill cost

Multiply volume by cost per cubic inch:
$$\text{Cost}_a = 167.55 \times 0.02 \approx 3.35$$

Step3: Calculate cylinder volume

Cylinder has same radius ($r=4$ in) and height ($h=10$ in). Use cylinder volume formula:
$$V_{cylinder} = \pi r^2 h = \pi (4)^2 (10) = 160\pi \approx 502.65 \text{ cubic inches}$$

Step4: Calculate cylinder fill cost

Multiply volume by cost per cubic inch:
$$\text{Cost}_b = 502.65 \times 0.02 \approx 10.05$$

Step5: Explain part c logic

A cylinder with same $r,h$ as a cone has 3x the cone's volume. So multiply part a's cost by 3:
$$\text{Cost}_b = 3.35 \times 3 \approx 10.05$$

Step6: Evaluate Jason's reasoning

Jason is incorrect: The cone's volume is $\frac{1}{3}$ of the cylinder's only if they have the same radius and height. The figure shows different base sizes, so the relationship does not hold.

Answer:

12a. $\$3.35$
12b. $\$10.05$
12c. Since a cylinder with identical radius and height has 3 times the volume of a cone, multiply the cost from part (a) by 3 to get the cylinder's fill cost.

1. Disagree. The volume of a cone is only one-third the volume of a cylinder if both solids have the same radius (or base area) and height. The figures have different base sizes, so this relationship does not apply.