Here are the steps required for Solving Joint Variation Problems:

 Step 1: Write the correct equation. Joint variation problems are solved using the equation y = kxz. When dealing with word problems, you should consider using variables other than x, y, and z, you should use variables that are relevant to the problem being solved. Also read the problem carefully to determine if there are any other changes in the joint variation equation, such as squares, cubes, or square roots. Step 2: Use the information given in the problem to find the value of k, called the constant of variation or the constant of proportionality. Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. When solving word problems, remember to include units in your final answer.

Example 1 – If y varies jointly as x and z, and y = 12 when x = 9 and z = 3, find z when y = 6 and x = 15.

 Step 1: Write the correct equation. Joint variation problems are solved using the equation y = kxz. Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when y = 12, x = 9, and z = 3. Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find z when y = 6 and x = 15.

Example 2 – If p varies jointly as q and r squared, and p = 225 when q = 4 and r = 3, find p when q = 6 and
r = 8.

 Step 1: Write the correct equation. Joint variation problems are solved using the equation y = kxz. In this case, you should use p, q, and r instead of x, y, and z and notice how the word “squared” changes the equation. Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when p = 225, q = 4, and r = 3. Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find p when q = 6 and r = 8.

Example 3 – If a varies jointly as b cubed and c, and a = 36 when b = 4 and c = 6, find a when b = 2 and
c = 14.

 Step 1: Write the correct equation. Joint variation problems are solved using the equation y = kxz. In this case, you should use a, b, and c instead of x, y, and z and notice how the word “cubed” changes the equation. Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when a = 36, b = 4, and r = 6. Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find a when b = 2 and c = 14.

Example 4 – The volume of a cone varies jointly as its height and the square of its radius. A cone with a radius of 6 inches and a height of 10 inches has a volume of 120π cubic inches. Find the volume of a cone having a radius of 15 inches and a height of 7 inches.

 Step 1: Write the correct equation. Joint variation problems are solved using the equation y = kxz. In this case, you should use v, h, and r instead of x, y, and z and notice how the word “square” changes the equation. Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when v = 120π, h = 10, and r = 6. Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find v when h = 7 and r = 15.

Example 5 – Kinetic energy varies jointly as the mass and the square of the velocity. A mass of 8 grams and a velocity of 5 centimeters per second has a kinetic energy of 100 ergs. Find the kinetic energy for a mass of 6 grams and a velocity of 9 centimeters per second.

 Step 1: Write the correct equation. Joint variation problems are solved using the equation y = kxz. In this case, you should use e, m, and v instead of x, y, and z and notice how the word “square” changes the equation. Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when e = 100, m = 8, and v = 5. Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2. Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. In this case, you need to find e when m = 6 and v = 9.