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ISEE Upper Level Quantitative : Spheres

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Example Questions

Example Question #42 : Solid Geometry

$\displaystyle t$ is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius $\displaystyle t$

(B) The surface area of a cube with edges of length $\displaystyle 2t$

Possible Answers:

(B) is greater

It is impossible to determine which is greater from the information given

(A) is greater

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

The surface area of a sphere is $4 \pi$ $\displaystyle 4 \pi$ times the square of its radius, which here is $\displaystyle t$ ; the surface area of the sphere in (A) is $4 \pi t^{2}$ $\displaystyle 4 \pi t^{2}$ .

The area of one face of a cube is the square of the length of an edge, which here is $\displaystyle 2t$ , so the area of one face of the cube in (B) is $(2t) ^{2} = 4t^{2}$ $\displaystyle (2t) ^{2} = 4t^{2}$ . The cube has six faces so the total surface area is $6 \cdot 4t^{2} = 24t^{2}$ $\displaystyle 6 \cdot 4t^{2} = 24t^{2}$ .

$\pi < 4$ $\displaystyle \pi < 4$ , so $4 \pi t^{2} < 16 t^{2} < 24 t^{2}$ $\displaystyle 4 \pi t^{2} < 16 t^{2} < 24 t^{2}$ , giving the sphere less surface area. (B) is greater.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Spheres

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #42 : Solid Geometry

All ISEE Upper Level Quantitative Resources

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