Question

a solid has volume 4 cubic units. the equation ( k = sqrt3{\frac{v}{4}} ) represents the scale factor of ( k ) by which the solid must be dilated to obtain an image with volume ( v ) cubic units. list 2 points which are on the graph representing this equation.

Explanation:

Step1: Choose a value for \( V \)

Let's first choose \( V = 4 \). Substitute \( V = 4 \) into the equation \( k=\sqrt[3]{\frac{V}{4}} \).
\( k=\sqrt[3]{\frac{4}{4}}=\sqrt[3]{1} = 1 \). So one point is \( (4, 1) \).

Step2: Choose another value for \( V \)

Let's choose \( V = 32 \). Substitute \( V = 32 \) into the equation \( k=\sqrt[3]{\frac{V}{4}} \).
\( k=\sqrt[3]{\frac{32}{4}}=\sqrt[3]{8}=2 \). So another point is \( (32, 2) \).

Answer:

Two points on the graph are \( (4, 1) \) and \( (32, 2) \) (other valid points can also be found by choosing different \( V \) values, for example, if \( V = 0.5 \), \( k=\sqrt[3]{\frac{0.5}{4}}=\sqrt[3]{\frac{1}{8}}=\frac{1}{2} \), so \( (0.5, \frac{1}{2}) \) is also valid).