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?

To rearrange a nonlinear formula with the target in a squared denominator, clear the denominator first. Then get the square alone, take a square root, and use any sign condition to choose the valid result. The square root must cover the entire fraction; rooting only its denominator changes the formula.

Hints

Hint 1
To clear a denominator, multiply both sides of the equation by it. Since the distance is positive, its square isn't zero. What happens to the fraction when you multiply by that square?
Hint 2
Once the squared distance is multiplied by the intensity, divide by the whole multiplier to get the square alone. The intensity is positive because the constant and the squared distance are positive.
Hint 3
Squaring hides a number's sign, so undoing a square can give two signs. Which sign fits a positive distance?

Step-by-step

Approach 1: Rearrange the formula
Step 1
Clear the denominator
No Desmos needed. Rearranging by hand gives a formula in letters, rather than one numerical result. Since $d$ is positive, $d^2$ isn't zero. Multiply both sides by $d^2$ to clear the denominator:
$Id^2=k$
Step 2
Get the square alone
The given positive $k$ and positive $d$ make $I$ positive, so you can divide by it. Divide both sides by $I$ to isolate $d^2$ :
$d^2=\frac{k}{I}$
Step 3
Undo the square
Squaring hides the sign: $d$ and $-d$ have the same square. Take the square root of both sides, keeping both possible signs for now:
$d=\pm\sqrt{\frac{k}{I}}$
Step 4
Choose the positive distance
The question says $d$ is positive, so keep the positive root:
$d=\sqrt{\frac{k}{I}}$
This gives the positive signal distance in terms of $I$ and $k$ . Choice D.

Approach 2: Check the choices with sample values
Step 1
Find a sample distance
As a check in Desmos, choose positive sample values $k=36$ and $I=4$ ; their ratio is a perfect square. Type $k=36$ , $I=4$ , then $I\sim\frac{k}{d_1^2}\{d_1>0\}$ . The $\sim$ tells Desmos to find $d_1$ , and the restriction keeps the positive solution. Under PARAMETERS, Desmos shows $d_1=3$ .
Step 2
Compare the expressions
Type each proposed expression using those same values. Desmos shows $18$ , $\frac13$ , $\frac23$ , and $3$ , respectively. Only the last expression matches the distance from the original formula. A sample can catch a wrong choice; the algebra above shows why the match holds for every allowed value. Choice D.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

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

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

Hints

Step-by-step

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