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The volume of a sphere is given by the formula $V=\frac{4}{3} \pi r^{3}$, where $r$ is the radius of the sphere. Which of the following gives the radius of the sphere in terms of the volume of the sphere?

SAT · Math · previous-year question

  1. A.$\frac{4 \pi}{3 V}$
  2. B.$\frac{3 V}{4 \pi}$
  3. C.$\sqrt[3]{\frac{4 \pi}{3 V}}$
  4. D.$\sqrt[3]{\frac{3 V}{4 \pi}}$correct

Answer

D. $\sqrt[3]{\frac{3 V}{4 \pi}}$

Explanation

Choice D is correct. Dividing both sides of the equation $V=\frac{4}{3} \pi r^{3}$ by $\frac{4}{3} \pi$ results in $\frac{3 V}{4 \pi}=r^{3}$. Taking the cube root of both sides produces $\sqrt[3]{\frac{3 V}{4 \pi}}=r$. Therefore, $\sqrt[3]{\frac{3 V}{4 \pi}}$ gives the radius of the sphere in terms of the volume of the sphere.Choice $A$ is incorrect. This expression is equivalent to the reciprocal of $r^{3}$. Choice $\mathrm{B}$ is incorrect. This expression is equivalent to $r^{3}$. Choice $\mathrm{C}$ is incorrect. This expression is equivalent to the reciprocal of $r$.

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