Question

write as a power function:
y varies directly with the fourth root
of x and y is 14 when x is 16.

y = ?\sqrt4{}

Explanation:

Step1: Recall direct variation formula

For direct variation, if \( y \) varies directly with the fourth root of \( x \), the formula is \( y = k\sqrt[4]{x} \), where \( k \) is the constant of variation.

Step2: Substitute known values

We know \( y = 14 \) when \( x = 16 \). Substitute these into the formula: \( 14 = k\sqrt[4]{16} \).

Step3: Simplify \( \sqrt[4]{16} \)

Since \( 16 = 2^4 \), \( \sqrt[4]{16}=\sqrt[4]{2^4} = 2 \). So the equation becomes \( 14 = k\times2 \).

Step4: Solve for \( k \)

Divide both sides by 2: \( k=\frac{14}{2}=7 \).

Step5: Write the power function

Substitute \( k = 7 \) back into \( y = k\sqrt[4]{x} \), we get \( y = 7\sqrt[4]{x} \).

Answer:

\( y = 7\sqrt[4]{x} \) (so the first box is 7 and the second box is \( x \))