Answer
$112$ decibels.
Work Step by Step
Step 1:- Translate the statement to form an equation.
Let the loudness of a stereo speaker be $L$.
and the distance from the speaker be $D$.
Because $L$ varies inversely as $D^2$, we have:
$\Rightarrow L=\frac{k}{D^2}$ ...... (1)
Step 2:- Substitute the first set of values into the equation to find the value of $k$.
The given values are $D=8$ feet and $L=28$ decibels.
Substitute into the equation (1).
$\Rightarrow 28=\frac{k}{8^2}$
$\Rightarrow 28=\frac{k}{64}$
Multiply both sides by $64$.
$\Rightarrow 64\cdot 28=64\cdot \frac{k}{64}$
Simplify.
$\Rightarrow 1792=k$
Step 3:- Substitute the value of $k$ into the original equation.
Substitute $k=1056$ into the equation (1).
$\Rightarrow L=\frac{1792}{D^2}$ ...... (2)
Step 4:- Solve the equation to find the required value.
Substitute $D=4$ feet into the equation (2).
$\Rightarrow L=\frac{1792}{4^2}$
Simplify.
$\Rightarrow L=\frac{1792}{16}$
$\Rightarrow L=112$
Hence, the loudness of the speaker is $112$ decibels.
