Question

Solve each problem. Let a be directly proportional to m and n2, and inversely proportional to y3. If a=9when m=4, n=9, and y=3, find a when m=6, n=2, and y=5.

Accepted Answer

Identify the relationship given: $$ a  is $$directly proportional to $$ m $$ and $$ n^2 $$, and inversely proportional to $$ y^3 . $$This can be written as the equation $$ a = k \cdot \frac{m \cdot n^2}{y^3} $$, where $$ k  is $$the constant of proportionality. Use the given values $$ a=9 $$, $$ m=4 $$, $$ n=9 $$, and $$ y=3  to $$find the constant $$ k . $$Substitute these values into the equation: $$ 9 = k \cdot \frac{4 \cdot 9^2}{3^3} . $$Simplify the expression on the right side to solve for $$ k . $$Calculate $$ 9^2 $$ and $$ 3^3 $$, then isolate $$ k  by $$dividing both sides of the equation by the resulting fraction. Once $$ k  is $$found, use it to find $$ a $$ when $$ m=6 $$, $$ n=2 $$, and $$ y=5 . $$Substitute these values and $$ k $$ into the original formula: $$ a = k \cdot \frac{6 \cdot 2^2}{5^3} . $$Simplify the expression on the right side to find the new value of $$ a . $$Calculate $$ 2^2 $$ and $$ 5^3 $$, then multiply and divide accordingly to express $$ a  in $$terms of $$ k .$$

Solve each problem. Let a be directly proportional to m and n², and inversely proportional to y³. If a=9when m=4, n=9, and y=3, find a when m=6, n=2, and y=5.

Verified video answer for a similar problem:

Key Concepts

Direct and Inverse Proportionality

Formulating Proportionality Equations

Solving for the Constant of Proportionality