ISEE Upper Level Quantitative : How to find the volume of a sphere

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

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Example Questions

Example Question #1 : How To Find The Volume Of A Sphere

In terms of , give the volume, in cubic feet, of a spherical tank with diameter 36 inches.

Possible Answers:

Correct answer:

Explanation:

36 inches =  feet, the diameter of the tank. Half of this, or  feet, is the radius. Set , substitute in the volume formula, and solve for :

Example Question #2 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The volume of a sphere with radius 

(b) The volume of a cube with sidelength 

Possible Answers:

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

Correct answer:

(b) is greater

Explanation:

A sphere with radius  has diameter  and can be inscribed inside a cube of sidelength . Therefore, the cube in (b) has the greater volume.

Example Question #3 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The volume of a cube with sidelength  inches.

(b) The volume of a sphere with radius  inches.

Possible Answers:

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

You do not need to calculate the volumes of the figures. All you need to do is observe that a sphere with radius  inches has diameter  inches, and can therefore be inscribed inside the cube with sidelength  inches. This give the cube larger volume, making (a) the greater quantity.

Example Question #4 : How To Find The Volume Of A Sphere

Which is the greater quantity? 

(a) The volume of a sphere with diameter one foot

(b) 

Possible Answers:

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

The radius of the sphere is one half of its diameter of one foot, which is six inches, so substitute :

 cubic inches,

which is greater than .

Example Question #5 : How To Find The Volume Of A Sphere

 is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length 

(B) The volume of a sphere with radius 

Possible Answers:

(B) is greater

(A) and (B) are equal 

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

(A) is greater

Correct answer:

(A) is greater

Explanation:

No calculation is really needed here, as a sphere with radius  - and, subsequently, diameter  - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

Example Question #6 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The radius of a sphere with surface area 

(b) The radius of a sphere with volume 

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

Correct answer:

(a) and (b) are equal

Explanation:

The formula for the surface area of a sphere, given its radius , is 

The sphere in (a) has surface area , so 

 

 

The formula for the volume of a sphere, given its radius , is

The sphere in (b) has volume , so 

 

The radius of both spheres is 3.

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