Step 1: | Write the correct equation. Direct variation problems are solved using the equation y = kx. When dealing with word problems, you should consider using variables other than x and y, 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 direct 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 x varies directly as y, and x = 9 when y = 6, find x when y = 15.
Step 1: Write the correct equation. Direct variation problems are solved using the equation y = kx. | |
Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when x = 9 and y = 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 x when y = 15. |
Example 2 –If p varies directly as the square of q, and p = 20 when q = 5, find p when q = 8.
Step 1: Write the correct equation. Direct variation problems are solved using the equation y = kx. In this case, you should use p and q instead of x and y 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 p = 20 and q = 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 p when q = 15. |
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Example 3 – If c varies directly as the square root of d, and c = 6 when d = 256, find c when d = 625.
Step 1: Write the correct equation. Direct variation problems are solved using the equation y = kx. In this case, you should use c and d instead of x and y and notice how the word “square root” 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 c = 6 and d = 256. | |
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 c when d = 625. |
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Example 4 – Hooke’s Law for an elastic spring states that the distance a spring stretches varies directly as the force applied. If a force of 160 newtons stretches a spring 5 cm, how much will a force of 368 newtons stretch the same spring?
Step 1: Write the correct equation. Direct variation problems are solved using the equation y = kx. In this case, you should use d for distance and f for force instead of x and y. | |
Step 2: Use the information given in the problem to find the value of k. In this case, you need to find k when f = 160 and d = 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 d when f = 368. |
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Example 5 – The distance a body falls from rest varies directly as the square of the time it falls (ignoring air resistance). If a ball falls 144 feet in three seconds, how far will the ball fall in seven seconds?
Step 1: Write the correct equation. Direct variation problems are solved using the equation y = kx. In this case, you should use d for distance and t for time instead of x and y 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 d = 144 and t = 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 d when t = 7. |