Question

theodores reasoning
we should buy the cone - shaped cup because it has a larger volume. since the radii of the cone and cylinder are the same, but the cone has a height of 4 times the height of the cylinder, its volume is bigger.
kyleighs reasoning
we can buy either cup because their volumes are the same. the radii of the cone and the cylinder are the same, but the cone has a height that is exactly 4 times the height of the cylinder so the volumes are equal.
which statement about theodores and kyleighs reasoning is true?

• theodore is correct. the volume of the cone - shaped cup is larger.
• kyleigh is correct. the volumes of the two cups are the same.
• neither student is correct. the volume of the cylinder - shaped cup is larger.
• both students are correct.

Explanation:

Step1: Recall volume formulas

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

Step2: Substitute given height relation

Given $H = 4h$, substitute into cone volume:
$V_{cone} = \frac{1}{3}\pi r^2 (4h) = \frac{4}{3}\pi r^2 h$

Step3: Compare volumes

Compare to $V_{cyl} = \pi r^2 h$:
$\frac{4}{3}\pi r^2 h > \pi r^2 h$, so $V_{cone} > V_{cyl}$

Step4: Evaluate student reasoning

Theodore claims cone volume is larger (correct). Kyleigh claims volumes are equal (incorrect).

Answer:

Theodore is correct. The volume of the cone-shaped cup is larger.