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Chapter 12: Problem 27

Solve. If \(L\) varies inversely as the square of \(h,\) and \(L=8\) when \(h=3,\) find \(L\) when \(h=2\)

Short Answer

Expert verified
When \(h = 2\), the value of \(L\) is 18.

Step by step solution

01

Set up the inverse variation equation

Since \(L\) varies inversely as the square of \(h\), we can write the relationship as an equation of the form: \[L = \frac{k}{h^2}\] where k is the constant of proportionality.
02

Find the constant of proportionality (k)

We are given the initial values for \(L\) and \(h\) as \(L = 8\) and \(h = 3\). We can substitute these values into our equation to solve for k: \[8 = \frac{k}{3^2}\] \[8 = \frac{k}{9}\] Now, we need to solve for k: \[k = 8 \cdot 9\] \[k = 72\]
03

Find the value of L when h = 2

Now that we have our constant of proportionality, k = 72, we can find the value of \(L\) when \(h = 2\). Substitute the values for k and h back into our equation: \[L = \frac{72}{2^2}\] \[L = \frac{72}{4}\] Now, divide 72 by 4 to find the value of L: \[L = 18\]
04

State the final answer

When \(h = 2\), the value of \(L\) is 18.

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