In a physics model, the intensity of a signal at a distance from a point source is given by the formula above, where is a positive constant. Which of the following expresses in terms of and , assuming is positive?

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SAT Nonlinear Equations & Systems Question 216 | Free SAT Prep | Aniko

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Question 216·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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$I = \frac{k}{d^2}$

In a physics model, the intensity $I$ of a signal at a distance $d$ from a point source is given by the formula above, where $k$ is a positive constant. Which of the following expresses $d$ in terms of $I$ and $k$ , assuming $d$ is positive?

When asked to "express one variable in terms of others," systematically isolate that variable: clear fractions by multiplying both sides, collect all instances of the target variable on one side, then undo powers or roots with inverse operations. For equations like $I = \frac{k}{d^2}$ , think in two clear moves—first eliminate the denominator ( $Id^2 = k$ ), then solve the resulting one-step equation for $d^2$ and take the appropriate root, remembering to consider whether the problem restricts the variable to positive values.

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Remove the denominator first

You see $d^2$ in the denominator. What equation do you get if you multiply both sides of $I = \frac{k}{d^2}$ by $d^2$ ?

Solve for $d^2$

Once you have $Id^2 = k$ , what can you divide both sides by so that $d^2$ is alone on one side?

Go from $d^2$ to $d$

After you have an expression for $d^2$ in terms of $I$ and $k$ , what operation lets you solve for $d$ ? How does the fact that $d$ is positive affect your choice?

Desmos Guide

Pick simple values and compute $I$

Choose easy positive numbers for $k$ and $d$ , such as $k = 20$ and $d = 2$ . In Desmos, type 20/(2^2) to compute the corresponding intensity $I$ . Note the value of $I$ that Desmos gives you.

Test each answer choice numerically

Using the same $k$ and the $I$ you just found, type in Desmos the expressions for $d$ from each answer choice: for example, sqrt(20/I), 20/sqrt(I), sqrt(I/20), and I/sqrt(20). These correspond to the four options.

See which expression reproduces $d$

Compare the numerical results from Desmos with your original $d$ (in this example, $d = 2$ ). The expression that gives you back the original distance is the one that correctly solves the equation for $d$ . Use that matching structure to identify the correct choice in the problem.

Step-by-step Explanation

Clear the denominator

Start with the given equation:

I = \frac{k}{d^2}

Multiply both sides by $d^2$ to remove the denominator:

Id^2 = k

Isolate $d^2$

Now get $d^2$ by itself by dividing both sides of the equation by $I$ :

Id^2 = k \quad \Rightarrow \quad d^2 = \frac{k}{I}

Now you have $d^2$ expressed in terms of $I$ and $k$ .

Take the positive square root

To solve for $d$ , take the square root of both sides:

\sqrt{d^2} = \sqrt{\frac{k}{I}}

Since $d$ is given to be positive, $\sqrt{d^2} = d$ . So

d = \sqrt{\frac{k}{I}}

This matches choice A.

Strategy

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation