Consider the following. The variable r is directly proportional to the square of t, and r = 144 when t = 108. Write the equation that expresses the relationship between the variables. (Let k represent the variation constant.) r = 1t281 Using the given data, solve for the variation constant k. k =
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Since r is directly proportional to the square of t, we can express this relationship as: \[ r = k t^2 \] where k is the variation constant. Show more…
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Recall the definition of a direct proportion. If y varies directly as x, or y is directly proportional to x, then y = kx, where k is the variation constant. The variable r is directly proportional to the square of t. Since we see the relationship 'directly proportional,' the equation of variation for this relationship will have a format similar to y = kx. First, represent 'the square of t' as an algebraic expression involving t. Next, write the equation that expresses the relationship between the variables: r is directly proportional to the square of t. (Use k for the variation constant.)
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Write a formula that expresses the relationship described by each statement. Use k for the constant of variation. See Examples 1 and 2. $t$ is inversely proportional to the square of $x .$
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Determine the constant (k) of proportionality: T varies as the square root of L. If L = 81, then T = 10.
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