Solve each problem. If m varies jointly as x and y, and m=10 when x=2 and y=14, find m when x=21 and y=8.

Verified step by step guidance

1. Understand that the phrase "m varies jointly as x and y" means that m is directly proportional to the product of x and y. This can be written as the equation: \(m = k \cdot x \cdot y\), where \(k\) is the constant of proportionality.
2. Use the given values \(m=10\), \(x=2\), and \(y=14\) to find the constant \(k\). Substitute these values into the equation: \$10 = k \cdot 2 \cdot 14$.
3. Solve the equation for \(k\) by dividing both sides by the product of \$2\( and \)14\(: \)k = \frac{10}{2 \cdot 14}$.
4. Now that you have the value of \(k\), use it to find \(m\) when \(x=21\) and \(y=8\). Substitute these values and \(k\) into the original equation: \(m = k \cdot 21 \cdot 8\).
5. Simplify the expression to find the value of \(m\) for the new values of \(x\) and \(y\).

Key Concepts

Joint Variation

Joint variation describes a relationship where one variable depends on the product of two or more other variables. In this case, m varies jointly as x and y means m = kxy, where k is a constant. Understanding this helps set up the equation to find unknown values.

Finding the Constant of Variation

To solve joint variation problems, you first find the constant k by substituting known values of the variables into the equation m = kxy. This step is crucial because k allows you to relate m, x, and y for any other values.

Substitution to Find Unknown Values

After determining the constant k, substitute the new values of x and y into the equation m = kxy to find the corresponding value of m. This process applies the variation model to solve for unknowns based on given conditions.