Question 110·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The period $T$ (in seconds) of a simple pendulum of length $L$ (in meters) is given by

T = 2\pi \sqrt{\frac{L}{g}}

where $g$ is the acceleration due to gravity, in meters per second squared.

What is $L$ in terms of $T$ and $g$ ?

The phrase “in terms of” signals a formula rearrangement: get the requested letter alone while leaving the other letters in the answer. Use algebra rather than a Desmos sample: a numerical check cannot produce a formula that works for every value. Undo the outside operation first. Here, dividing when you need to multiply would put $g$ on the wrong side of the fraction.

Hints

Hint 1
The requested length $L$ is the target. “In terms of $T$ and $g$ ” means those two letters stay in your formula. What multiplies the entire square root, and how can you undo that multiplication?
Hint 2
Once the square root is alone, square both whole sides to remove it. The denominator on the left is all of $2\pi$ , so what does it become when squared?
Hint 3
After squaring, $L$ is still divided by $g$ . Multiply both sides by $g$ to leave $L$ alone; dividing by $g$ would move it farther from the target.

Step-by-step

Undo the operations around $L$
Step 1
Get the square root alone
No Desmos needed. The goal is a formula for every $T$ and $g$ , not a numerical sample.

The outside factor $2\pi$ multiplies the entire square root. Divide both sides by $2\pi$ before changing anything inside it:
$\frac{T}{2\pi}=\sqrt{\frac{L}{g}}.$
Step 2
Remove the square root
Square both whole sides to undo the square root. Squaring the entire denominator $2\pi$ gives $4\pi^2$ , not $2\pi^2$ . Square both sides:
$\frac{T^2}{4\pi^2}=\frac{L}{g}.$
Step 3
Isolate the pendulum length
Now $L$ is divided by $g$ . Multiply both sides by $g$ so that division cancels:
$L=\frac{gT^2}{4\pi^2}.$
So this expression gives the pendulum’s length in meters. Choice C.

Lessons that teach this

SAT Advanced AlgebraIntermediateCoreRearrange nonlinear formulas

Question 110·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this