Question

write as a power function:
y varies directly with the cube root
of x and y is 10 when x is 125.
? = \sqrt3{}

Explanation:

Step1: Recall direct variation formula

For direct variation, if \( y \) varies directly with \( \sqrt[3]{x} \), the formula is \( y = k\sqrt[3]{x} \), where \( k \) is the constant of variation.

Step2: Find the constant \( k \)

We know \( y = 10 \) when \( x = 125 \). Substitute these values into the formula:
\( 10 = k\sqrt[3]{125} \).
Since \( \sqrt[3]{125}=5 \) (because \( 5^3 = 125 \)), we have \( 10 = k \times 5 \).
Solving for \( k \), divide both sides by 5: \( k=\frac{10}{5}=2 \).

Step3: Write the power function

Substitute \( k = 2 \) back into the direct variation formula: \( y = 2\sqrt[3]{x} \).

Answer:

\( y = 2\sqrt[3]{x} \) (So the blanks are filled as: \( \boldsymbol{y} = \boldsymbol{2}\sqrt[3]{\boldsymbol{x}} \))