Question 1042·Medium·Math

No Desmos needed. Algebra keeps $S$ symbolic, so the result works for every possible sunlight amount. Multiply both sides by the whole squared denominator:

The model gives a positive sunlight amount $S$, so divide both sides by $S$ to leave the squared expression alone:

Because $h$ is a height, $h\ge0$, so $1+0.05h\ge1$. The expression inside the square is positive, so keep the positive square root. Take that positive root:

Undo the addition by subtracting $1$ from both sides:

Undo multiplication by $0.05$ by dividing both sides by $0.05$:

Since $0.05=1/20$, dividing by $0.05$ multiplies the entire expression by $20$. Rewrite the height:

This gives the canopy height $h$ from the sunlight amount $S$. Choice C.

For a multiple-choice check, pick the sample height $h=20$: then $0.05(20)=1$, making the model quick to evaluate. Type the model as a function, a rule that gives sunlight for a height. Desmos shows $f(20)=250$, so a height of $20$ meters gives $S=250$ lumens.

Type $s=250$, then the four proposed expressions in their listed order. Desmos prints $-20$, $-0.025$, $20$, and $0.025$. Only the third expression returns the height you started with. One sample doesn't prove a formula for every $S$, but it rules out the other three choices. So the expression for the canopy height is $h=20\left(\sqrt{\frac{1000}{S}}-1\right)$. Choice C.