PSAT Math - How to find the volume of a sphere

A foam ball has a volume of 2 units and has a diameter of x. If a second foam ball has a radius of 2x, what is its volume?

128 units

16 units

8 units

4 units

2 units

Explanation

Careful not to mix up radius and diameter. First, we need to identify that the second ball has a radius that is 4 times as large as the first ball. The radius of the first ball is (1/2)x and the radius of the second ball is 2x. The volume of the second ball will be 43, or 64 times bigger than the first ball. So the second ball has a volume of 2 * 64 = 128.

The surface area of a sphere is $36\pi\ mm^2$ . Find the volume of the sphere in cubic millimeters.

$36\pi\ mm^3$

$4\pi\ mm^3$

$9\pi\ mm^3$

$3\pi\ mm^3$

$6\pi\ mm^3$

Explanation

$SA=4\pi r^2$

$36\pi=4\pi r^2$

$V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\pi(3)^3$

$V=\frac{4}{3}(27)\pi$

$V=4(9)\pi$

$V=36\pi$

A solid hemisphere has a radius of length r. Let S be the number of square units, in terms of r, of the hemisphere's surface area. Let V be the number of cubic units, in terms of r, of the hemisphere's volume. What is the ratio of S to V?

$\frac{9}{2r}$

$3r$

$\frac{4r}{3}$

$\frac{9r}{2}$

$\frac{3}{r}$

Explanation

First, let's find the surface area of the hemisphere. Because the hemisphere is basically a full sphere cut in half, we need to find half of the surface area of a full sphere. However, because the hemisphere also has a circular base, we must then add the area of the base.

$S = \frac{1}{2}\cdot$ (surface area of sphere) + (surface area of base)

The surface area of a sphere with radius r is equal to $4\pi r^2$ . The surface area of the base is just equal to the surface area of a circle, which is $\pi r^2$ .

$S=\frac{1}{2}\cdot 4\pi r^2+\pi r^2=2\pi r^2+\pi r^2=3\pi r^2$

The volume of the hemisphere is going to be half of the volume of an entire sphere. The volume for a full sphere is $\frac{4}{3}\pi r^3$ .

$V = \frac{1}{2}\cdot$ (volume of sphere)

$V = \frac{1}{2}\cdot \frac{4}{3}\pi r^3=\frac{2}{3}\pi r^3$

Ultimately, the question asks us to find the ratio of S to V. To do this, we can write S to V as a fraction.

$\frac{S}{V}=\frac{3\pi r^2}{\frac{2}{3}\pi r^3}$

In order to simplify this, let's multiply the numerator and denominator both by 3.

$\frac{S}{V}=\frac{3\pi r^2}{\frac{2}{3}\pi r^3}$ = $\frac{9\pi r^2}{2\pi r^3}=\frac{9}{2r}$

The answer is $\frac{9}{2r}$ .

At x = 3, the line y = 4_x_ + 12 intersects the surface of a sphere that passes through the xy-plane. The sphere is centered at the point at which the line passes through the x-axis. What is the volume, in cubic units, of the sphere?

None of the other answers

2040_π_√(7)

816_π_√(11)

4896_π_

4896_π_√(17)

Explanation

We need to ascertain two values: The center point and the point of intersection with the surface. Let's do that first:

The center is defined by the x-intercept. To find that, set the line equation equal to 0 (y = 0 at the x-intercept):

0 = 4x + 12; 4_x_ = –12; x = –3; Therefore, the center is at (–3,0)

Next, we need to find the point at which the line intersects with the sphere's surface. To do this, solve for the point with x-coordinate at 3:

y = 4 * 3 + 12; y = 12 + 12; y = 24; therefore, the point of intersection is at (3,24)

Reviewing our data so far, this means that the radius of the sphere runs from the center, (–3,0), to the edge, (3,24). If we find the distance between these two points, we can ascertain the length of the radius. From that, we will be able to calculate the volume of the sphere.

The distance between these two points is defined by the distance formula:

d = √( (_x_1 – _x_0)2 + (_y_1 – _y_0)2 )

For our data, that is:

√( (3 + 3)2 + (24 – 0)2 ) = √( 62 + 242 ) = √(36 + 576) = √612 = √(2 * 2 * 3 * 3 * 17) = 6√(17)

Now, the volume of a sphere is defined by: V = (4/3)_πr_3

For our data, that would be: (4/3)π * (6√(17))3 = (4/3) * 63 * 17√(17) * π = 4 * 2 * 62 * 17√(17) * π = 4896_π_√(17)

A cube with sides of 4” each contains a floating sphere with a radius of 1”. What is the volume of the space outside of the sphere, within the cube?

59.813 in3

64 in3

4.187 in3

11.813 in3

Explanation

Volume of Cube = side3 = (4”)3 = 64 in3

Volume of Sphere = (4/3) * π * r3 = (4/3) * π * 13 = (4/3) * π * 13 = (4/3) * π = 4.187 in3

Difference = Volume of Cube – Volume of Sphere = 64 – 4.187 = 59.813 in3

A cross-section is made at the center of a sphere. The area of this cross-section is 225_π_ square units. How many cubic units is the total volume of the sphere?

None of the other answers

1687.5_π_

3375_π_

13500_π_

4500_π_

Explanation

The solution to this is simple, though just take it step-by-step. First find the radius of the circular cross-section. This will give us the radius of the sphere (since this cross-section is at the center of the sphere). If the cross-section has an area of 225_π_, we know its area is defined by:

A = 225_π_ = _πr_2

Solving for r, we get r = 15.

From here, we merely need to use our formula for the volume of a sphere:

V = (4 / 3)_πr_3

For our data this is: (4 / 3)π * 153 = 4_π_ * 152 * 5 = 4500_π_

A water tank takes the shape of a sphere whose exterior has radius 16 feet; the tank is six inches thick throughout. To the nearest hundred, how many cubic feet of water does the tank hold?

$15,600\textrm{ ft}^{3}$

$14,100\textrm{ ft}^{3}$

$3,100\textrm{ ft}^{3}$

$2,800\textrm{ ft}^{3}$

$17,100\textrm{ ft}^{3}$

Explanation

Six inches is equal to 0.5 feet, so the radius of the interior of the tank is

feet.

The amount of water the tank holds is the volume of the interior of the tank, which is

$V = \frac{4}{3} \pi r^{3}$

$V \approx \frac{4}{3} \cdot 3.14 \cdot 15.5^{3} \approx 15,591$ ,

which rounds to 15,600 cubic feet.

The surface area of a sphere is 676_π_ in2. How many cubic inches is the volume of the same sphere?

8788_π_

2028_π_

(2028_π)_/3

(2197_π_)/3

(8788_π)_/3

Explanation

To begin, we must solve for the radius of our sphere. To do this, recall the equation for the surface area of a sphere: A = 4_πr_2

For our data, that is: 676_π_ = 4_πr_2; 169 = _r_2; r = 13

From this, it is easy to solve for the volume of the sphere. Recall the equation:

V = (4/3)_πr_3

For our data, this is: V = (4/3)π * 133 = (4_π_ * 2197)/3 = (8788_π)_/3

What is the difference between the volume and surface area of a sphere with a radius of 6?

144_π_

288_π_

216_π_

133_π_

720_π_

Explanation

Surface Area = 4_πr_2 = 4 * π * 62 = 144_π_

Volume = 4_πr_3/3 = 4 * π * 63 / 3 = 288_π_

Volume – Surface Area = 288_π_ – 144_π_ = 144_π_

A sphere is perfectly contained within a cube that has a volume of 216 units. What is the volume of the sphere?

$9\pi$

$12\pi$

$48\pi$

$288\pi$

$36\pi$

Explanation

To begin, we must determine the dimensions of the cube. This is done by solving the simple equation:

We know the volume is 216, allowing us to solve for the length of a side of the cube.

Taking the cube root of both sides, we get s = 6.

The diameter of the sphere will be equal to side of the cube, since the question states that the sphere is perfectly contained. The diameter of the sphere will be 6, and the radius will be 3.

We can plug this into the equation for volume of a sphere:

$V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\pi (3)^3$

We can cancel out the 3 in the denominator.

$V=4\pi (3)^2$

Simplify.

$V=4\pi (9)$

$V=36\pi$

How to find the volume of a sphere

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