A botanist models the amount of sunlight , in lumens, that reaches the forest floor with the equation where is the average height, in meters, of the tree canopy. Which of the following expresses in terms of ?

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SAT Nonlinear Equations & Systems Question 47 | Free SAT Prep | Aniko

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Question 47·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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A botanist models the amount of sunlight $S$ , in lumens, that reaches the forest floor with the equation

S = \frac{1000}{\left(1 + 0.05h\right)^2}

where $h$ is the average height, in meters, of the tree canopy. Which of the following expresses $h$ in terms of $S$ ?

When a question asks you to "express one variable in terms of another" from a given formula, treat it like solving an equation for that variable: clear any fractions first, isolate the grouped expression (such as a square), undo exponents with roots, and then isolate the variable step by step, carefully tracking signs and reciprocals. Finally, compare your derived expression to the answer choices, watching for common traps like flipped fractions (e.g., $\tfrac{S}{1000}$ vs. $\tfrac{1000}{S}$ ) or incorrect coefficients from dividing by decimals.

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Clear the denominator first

You see $S = \dfrac{1000}{(1 + 0.05h)^2}$ . What can you multiply both sides by to eliminate the fraction and get rid of the denominator?

Solve for the squared quantity

After you clear the denominator, you should have $S(1 + 0.05h)^2 = 1000$ . What should you divide both sides by to get $(1 + 0.05h)^2$ alone?

Undo the square carefully

Once you have $(1 + 0.05h)^2 = \dfrac{1000}{S}$ , what operation undoes a square? After that, how can you isolate $0.05h$ ?

Finish isolating $h$

When you reach an equation like $0.05h = \text{(expression in }S)$ , what do you do with the $0.05$ to solve for $h$ ? Remember that dividing by $0.05$ is the same as multiplying by its reciprocal.

Desmos Guide

Create a slider for S

In Desmos, type S = 500 and press Enter. Desmos will create a slider for $S$ (you can adjust the min and max, for example from $50$ to $1000$ ) so you can test different sunlight values.

Enter the original model

Type the original equation for sunlight as an expression of $h$ :

y = \frac{1000}{(1 + 0.05x)^2}.

Here $x$ represents $h$ , the canopy height. This graph shows $S$ as a function of $h$ .

Define $h$ using each answer choice

For each answer choice, define $h$ in terms of $S$ using a separate expression:

hA = 20*(1 - sqrt(1000/S))
hB = 0.05*(sqrt(S/1000) - 1)
hC = 20*(sqrt(1000/S) - 1)
hD = 0.05*(1 - sqrt(S/1000))

These give four possible values of $h$ for each slider value of $S$ .

Check which $h(S)$ matches the model

For each $h$ expression, compute the sunlight it would predict from the original formula. For example, for hA, type SA = 1000/(1 + 0.05*hA)^2; do similarly for hB, hC, and hD to get SB, SC, and SD.

Move the $S$ slider and compare each computed value (SA, SB, SC, SD) with the slider $S$ . The expression for $h$ that makes its corresponding $S$ -value match the slider $S$ for all tested values is the correct formula.

Step-by-step Explanation

Clear the fraction

Start with the given equation:

S = \frac{1000}{(1 + 0.05h)^2}.

To remove the denominator, multiply both sides by $(1 + 0.05h)^2$ :

S(1 + 0.05h)^2 = 1000.

Isolate the squared expression

Now get $(1 + 0.05h)^2$ by itself by dividing both sides by $S$ :

(1 + 0.05h)^2 = \frac{1000}{S}.

Undo the square with a square root

Take the positive square root of both sides (the expression $1 + 0.05h$ is positive):

1 + 0.05h = \sqrt{\frac{1000}{S}}.

Now the variable $h$ appears only inside a simple linear expression.

Get the term with $h$ alone

Subtract $1$ from both sides to isolate the term with $h$ :

0.05h = \sqrt{\frac{1000}{S}} - 1.

Now $h$ is being multiplied by $0.05$ .

Solve for $h$ completely

To solve for $h$ , divide both sides by $0.05$ . Since $0.05 = \tfrac{1}{20}$ , dividing by $0.05$ is the same as multiplying by $20$ :

\begin{aligned} 0.05h &= \sqrt{\frac{1000}{S}} - 1 \\ h &= \frac{\sqrt{1000/S} - 1}{0.05} = 20\left(\sqrt{\frac{1000}{S}} - 1\right). \end{aligned}

So the correct expression is

h = 20\left(\sqrt{\frac{1000}{S}} - 1\right),

which corresponds to answer choice C.

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Step-by-step Explanation

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