Answer
$2.4\; footcandles $.
Work Step by Step
Step 1:- Translate the statement to form an equation.
Let the illumination be $I$
and the distance from the headlight be $D$.
Because $I$ varies inversely as $D^2$ we have:
$\Rightarrow I=\frac{k}{D^2}$ ...... (1)
Step 2:- Substitute the first set of values into the equation to find the value of $k$.
The given values are $I=3.75\; footcandles$
and $D=40\; feet$.
Substitute into the equation (1).
$\Rightarrow 3.75=\frac{k}{40^2}$
$\Rightarrow 3.75=\frac{k}{1600}$
Multiply both sides by $1600$.
$\Rightarrow 1600\cdot 3.75=1600\cdot \frac{k}{1600}$
Simplify.
$\Rightarrow 6000=k$
Step 3:- Substitute the value of $k$ into the original equation.
Substitute $k=6000$ into the equation (1).
$\Rightarrow I=\frac{6000}{D^2}$ ...... (2)
Step 4:- Solve the equation to find the required value.
Substitute $D=50\;feet$ into the equation (2).
$\Rightarrow I=\frac{6000}{(50)^2}$
Simplify.
$\Rightarrow I=2.4$
Hence, the illumination is $2.4\; footcandles $.
