Question

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the two cubes are similar in shape. compare the volume of the two and determine the scale factor from the smaller cube to the larger cube. enter your response as a fraction
(1 point)
the scale factor from the smaller cube to the larger cube is
check answer remaining attempts : 3

Explanation:

Step1: Recall the formula for scale factor of volumes

If the scale factor of two similar solids is \(k\), then the ratio of their volumes \(V_1\) and \(V_2\) is \(V_1:V_2 = k^3\). Let \(V_{small}=512\) and \(V_{large}=1728\). So, \(\frac{V_{small}}{V_{large}}=k^3\), where \(k\) is the scale factor from the smaller to the larger cube.

Step2: Calculate the ratio of volumes

\(\frac{V_{small}}{V_{large}}=\frac{512}{1728}\). Simplify \(\frac{512}{1728}\) by dividing numerator and denominator by their greatest - common divisor. The GCD of 512 and 1728 is 64. \(\frac{512\div64}{1728\div64}=\frac{8}{27}\).

Step3: Find the scale factor

Since \(\frac{V_{small}}{V_{large}} = k^3=\frac{8}{27}\), then \(k=\sqrt[3]{\frac{8}{27}}\). We know that \(\sqrt[3]{8}=2\) and \(\sqrt[3]{27}=3\), so \(k = \frac{2}{3}\).

Answer:

\(\frac{2}{3}\)