Question

§4.6: use the following information to find each.
9.) a variable a varies directly with the cube of b and inversely with the root of c.

a.) if a = 32 when b = 2 and c = 7, find the constant of variation.

b.) find the value of a when b = \sqrt5{7} and c = 28.

Explanation:

Part (a)

Step 1: Write the variation equation

Since \( a \) varies directly with the cube of \( b \) and inversely with the square root of \( c \) (assuming "root" here means square root, as it's a common root in such problems), the variation equation is \( a = k \cdot \frac{b^3}{\sqrt{c}} \), where \( k \) is the constant of variation.

Step 2: Substitute the given values

We know \( a = 32 \), \( b = 2 \), and \( c = 7 \). Substitute these into the equation: \( 32 = k \cdot \frac{2^3}{\sqrt{7}} \).

Step 3: Solve for \( k \)

First, calculate \( 2^3 = 8 \). So the equation becomes \( 32 = k \cdot \frac{8}{\sqrt{7}} \). To solve for \( k \), multiply both sides by \( \frac{\sqrt{7}}{8} \): \( k = 32 \cdot \frac{\sqrt{7}}{8} = 4\sqrt{7} \).

Step 1: Use the variation equation with the found \( k \)

From part (a), we have \( a = 4\sqrt{7} \cdot \frac{b^3}{\sqrt{c}} \). Now substitute \( b = \sqrt[3]{7} \) and \( c = 28 \).

Step 2: Simplify \( b^3 \)

First, calculate \( b^3 = (\sqrt[3]{7})^3 = 7 \).

Step 3: Substitute and simplify

Substitute \( b^3 = 7 \) and \( c = 28 \) into the equation: \( a = 4\sqrt{7} \cdot \frac{7}{\sqrt{28}} \). Simplify \( \sqrt{28} = 2\sqrt{7} \), so the equation becomes \( a = 4\sqrt{7} \cdot \frac{7}{2\sqrt{7}} \).

Step 4: Cancel out common terms

The \( \sqrt{7} \) terms cancel out, and \( \frac{4}{2} = 2 \), so we have \( a = 2 \cdot 7 = 14 \).

Answer:

The constant of variation \( k \) is \( 4\sqrt{7} \).

Part (b)