Question

a superhero action figure has volume 1500 cm³. the figure is dilated to create a store display. the volume of the dilated figure is 45,000 cm³. what was the approximate scale factor of the dilation? if a cube has a volume of 125 units³ and doubled the volume to 250 units³. is the scale factor between the two cubes also doubled? explain your reasoning.

Explanation:

Step1: Find volume scale factor

$\text{Volume Scale Factor} = \frac{45000}{1500} = 30$

Step2: Relate to linear scale factor

For dilation, linear scale factor $k$ satisfies $k^3 = \text{Volume Scale Factor}$. So:
$k = \sqrt[3]{30} \approx 3.11$
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Step3: Analyze cube volume change

Original cube volume $V_1=125$, new volume $V_2=250$. Volume scale factor:
$\frac{V_2}{V_1} = \frac{250}{125} = 2$

Step4: Calculate cube scale factor

Linear scale factor $k = \sqrt[3]{2} \approx 1.26$, which is not 2.

Answer:

1. The approximate scale factor of the dilation is $\boldsymbol{\approx 3.11}$
2. No, the scale factor is not doubled. The volume scale factor is 2, but the linear scale factor is the cube root of the volume scale factor, so it is $\sqrt[3]{2} \approx 1.26$, not 2. Volume scales with the cube of the linear scale factor, not linearly.