Spheres - PSAT Math

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Question

If a sphere has an approximate volume of $904.32\textup{ cubic meters}$ , then what is the approximate diameter of this sphere?

Answer

The formula for the volume of a sphere is

$v=\left ( \frac{4}{3} \right )\pi r^3$

Therefore,

$904.32=\left ( \frac{4}{3} \right )\pi r^3$

Dividing both sides by $\left ( \frac{4}{3} \right )\pi$ , leaves us with $r^3\approx215.89$ . Taking the cube root, we find $r\approx5.999$ , meaning our diameter $d\approx12$ .

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Question

A cube with volume 27 cubic inches is inscribed inside a sphere such that each vertex of the cube touches the sphere. What is the radius, in inches, of the sphere?

Answer

We know that the cube has a volume of 27 cubic inches, so each side of the cube must be ∛27=3 inches. Since the cube is inscribed inside the sphere, the diameter of the sphere is the diagonal length of the cube, so the radius of the sphere is half of the diagonal length of the cube. To find the diagonal length of the cube, we use the distance formula d=√(32+32+32 )=√(3*32 )=3√3, and then divide the result by 2 to find the radius of the sphere, (3√3)/2.

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Question

The surface area of a sphere is 100π square feet. What is the radius in feet?

Answer

S = 4π(r2)

100π = 4π(r2)

100 = 4r2

25 = r2

5 = r

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Question

Find the radius of a sphere whose volume is $\small \small 36\pi$ $\text{in}^3$ .

Answer

Use the equation for the volume of a sphere to find the radius.

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

$\small 36\pi = \frac{4}{3}\pi r^{3}$

$\small \small 36 = \frac{4}{3} r^{3}$

$\small 27 = r^{3}$

$\small r = 3$

So, the radius of the sphere is 3

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Question

A spherical water tank has a surface area of 400 square meters. To the nearest tenth of a meter, give the radius of the tank.

Answer

Given radius , the surface area of a sphere is given by the formula

$S = 4 \pi r^{2}$

Set and solve for :

$4 \pi r^{2} = 400$

$4 \pi r^{2}\div 4 = 400 \div 4$

$\pi r^{2} = 100$

$\pi r^{2} \div \pi = 100 \div \pi$

$r^{2} \approx 100 \div 3.14159 \approx 31.831$

$r \approx \sqrt{31.831} \approx 5.6$ meters.

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Question

The city of Washington wants to build a spherical water tank for the town hall. The tank is to have capacity 120 cubic meters of water.

To the nearest tenth, what will the radius of the tank be?

Answer

Given the radius , the volume of a sphere is given by the formula

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

We find the inner radius by setting :

$\frac{4}{3} \pi r^{3} = 120$

$\frac{3}{4} \cdot \frac{4}{3} \pi r^{3} = \frac{3}{4} \cdot 120$

$\pi r^{3} = 90$

$r^{3} =\frac{ 90}{\pi} \approx 28.65$

$r \approx\sqrt[3]{ 28.65} \approx 3.1$ meters.

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Question

A spherical tank is to hold 50,000 liters of water. Given that one cubic meter is equal to 1,000 liters, give the radius of this tank to the nearest tenth of a meter.

Answer

Given radius , the volume of a sphere is given by the formula

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

Since 1,000 liters = 1 cubic meter, 50,000 liters = 50 cubic meters, and we find by setting :

$\frac{4}{3} \pi r^{3} = 50$

$\frac{3}{4} \cdot \frac{4}{3} \pi r^{3} = \frac{3}{4} \cdot 50$

$\pi r^{3} = 37.5$

$r^{3} =\frac{ 37.5}{\pi} \approx 11.9366$

$r \approx\sqrt[3]{ 11.9366} \approx 2.3$ meters.

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Question

The Kelvin temperature scale is basically the same as the Celsius scale except with a different zero point; to convert degrees Celsius to Kelvins, add 273. Also, by Charles's law, the volume of a given mass of gas varies directly as its temperature, expressed in Kelvins.

A spherical balloon is filled with 10,000 cubic meters of gas in the morning when the temperature is $10^{\circ }C$ . The temperature increases to $25^{\circ }C$ by noon, with no other conditions changing. What is the radius of the balloon at noon, to the nearest tenth of a meter?

Answer

First, we use the variation equation to figure out the volume of the balloon at noon. First, we add 273 to each of the temperatures.

The initial temperature is $273 + 10 = 283 \textup{ K}$

The final (noon) temperature is $273 + 25= 298 \textup{ K}$

Since volume varies directly as temperature, we can set up the equation

$\frac{V'}{T'} = \frac{V}{T}$

$\frac{V'}{298} = \frac{10,000}{283}$

$V' = \frac{10,000}{283} \cdot 298 \approx 10,530$

The volume of a sphere is given by the formula

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

so set and solve for :

$\frac{4}{3} \pi r^{3} = 10,530$

$\frac{3}{4} \cdot \frac{4}{3} \pi r^{3} = \frac{3}{4} \cdot 10,530$

$\pi r^{3} \approx 7,897.5$

$r^{3} \approx 7,897.5 \div \pi \approx 2,513.8$

$r \approx\sqrt[3]{ 2,513.8} \approx 13.6$ meters.

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Question

The Kelvin temperature scale is basically the same as the Celsius scale except with a different zero point; to convert degrees Celsius to Kelvins, add 273. Also, by Charles's law, the volume of a given mass of gas varies directly as its temperature, expressed in Kelvins.

In the early morning, when the temperature is $5^{\circ }C$ , a spherical balloon is filled with helium until its radius is 100 meters. At 2:00 PM, the temperature is $20^{\circ }C$ . To the nearest tenth of a meter, what is the radius of the balloon at this time?

Answer

The volume of the balloon is given by the formula

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

Also, by the variation relationship,

$\frac{V'}{T'} = \frac{V}{T}$

We do not need to calculate the original volume of the balloon; we can simply replace the volume with the formula:

$\frac{ \frac{4}{3} \pi r'^{3}}{T'} = \frac{ \frac{4}{3} \pi r^{3}}{T}$

Dividing both sides by $\frac{4}{3} \pi$ yields a new equation:

$\frac{ r'^{3}}{T'} = \frac{ r^{3}}{T}$

The initial radius is

The initial temperature is $T = 273 + 5 = 278 \textup{ K}$

The final (noon) temperature is $T ' = 273 + 20= 293 \textup{ K}$

Substitute to find the final radius:

$\frac{ r'^{3}}{293} = \frac{ 100^{3}}{278}$

$r'^{3} = \frac{ 1,000,000}{278} \cdot 293 \approx 1,053,957$

$r' = \approx\sqrt[3]{ 1,053,957} \approx 101.8$ meters.

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Question

If a sphere has a volume of $36\pi\textup{ cubic inches}$ , then what is the radius of the sphere?

Answer

The volume of a sphere is equal to

$v=\left ( \frac{4}{3} \right )\pi r^3$

Therefore,

$r=\sqrt[3]{(3/4)v/\pi}=\sqrt[3]{(3/4)36\pi/\pi}=\sqrt[3]{27}=3$

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Question

A spherical orange fits snugly inside a small cubical box such that each of the six walls of the box just barely touches the surface of the orange. If the volume of the box is 64 cubic inches, what is the surface area of the orange in square inches?

Answer

The volume of a cube is found by V = s3. Since V = 64, s = 4. The side of the cube is the same as the diameter of the sphere. Since d = 4, r = 2. The surface area of a sphere is found by SA = 4π(r2) = 4π(22) = 16π.

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Question

A solid sphere is cut in half to form two solid hemispheres. What is the ratio of the surface area of one of the hemispheres to the surface area of the entire sphere before it was cut?

Answer

The surface area of the sphere before it was cut is equal to the following:

surface area of solid sphere = 4_πr_2, where r is the length of the radius.

Each hemisphere will have the following shape:

In order to determine the surface area of the hemisphere, we must find the surface area of the flat region and the curved region. The flat region will have a surface area equal to the area of a circle with radius r.

area of flat part of hemisphere = _πr_2

The surface area of the curved portion of the hemisphere will equal one-half of the surface area of the uncut sphere, which we established to be 4_πr_2.

area of curved part of hemisphere = (1/2)4_πr_2 = 2_πr_2

The total surface area of the hemisphere will be equal to the sum of the surface areas of the flat part and curved part of the hemisphere.

total surface area of hemisphere = _πr_2 + 2_πr_2 = 3_πr_2

Finally, we must find the ratio of the surface area of the hemisphere to the surface area of the uncut sphere.

ratio = (3_πr_2)/(4_πr_2) = 3/4

The answer is 3/4.

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Question

The volume of a sphere is 2304_π_ in3. What is the surface area of this sphere in square inches?

Answer

To solve this, we must first begin by finding the radius of the sphere. To do this, recall that the volume of a sphere is:

V = (4/3)_πr_3

For our data, we can say:

2304_π_ = (4/3)_πr_3; 2304 = (4/3)_r_3; 6912 = 4_r_3; 1728 = _r_3; 12 * 12 * 12 = _r_3; r = 12

Now, based on this, we can ascertain the surface area using the equation:

A = 4_πr_2

For our data, this is:

A = 4_π_*122 = 576_π_

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Question

A sphere has its center at the origin. A point on its surface is found on the x-y axis at (6,8). In square units, what is the surface area of this sphere?

Answer

To find the surface area, we must first find the radius. Based on our description, this passes from (0,0) to (6,8). This can be found using the distance formula:

62 + 82 = _r_2; _r_2 = 36 + 64; _r_2 = 100; r = 10

It should be noted that you could have quickly figured this out by seeing that (6,8) is the hypotenuse of a 6-8-10 triangle (which is a multiple of the "easy" 3-4-5).

The rest is easy. The surface area of the sphere is defined by:

A = 4_πr_2 = 4 * 100 * π = 400_π_

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Question

A sphere is perfectly contained within a cube that has a surface area of 726 square units. In square units, what is the surface area of the sphere?

Answer

To begin, we must determine the dimensions of the cube. To do this, recall that the surface area of a cube is made up of six squares and is thus defined as: A = 6_s_2, where s is one of the sides of the cube. For our data, this gives us:

726 = 6_s_2; 121 = _s_2; s = 11

Now, if the sphere is contained within the cube, that means that 11 represents the diameter of the sphere. Therefore, the radius of the sphere is 5.5 units. The surface area of a sphere is defined as: A = 4_πr_2. For our data, that would be:

A = 4_π_ * 5.52 = 30.25 * 4 * π = 121_π_

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Question

The area of a circle with radius 4 divided by the surface area of a sphere with radius 2 is equal to:

Answer

The surface area of a sphere is 4_πr_2. The area of a circle is _πr_2. 16/16 is equal to 1.

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Question

What is the ratio of the surface area of a cube to the surface area of a sphere inscribed within it?

Answer

Let's call the radius of the sphere r. The formula for the surface area of a sphere (A) is given below:

A = 4_πr_2

Because the sphere is inscribed inside the cube, the diameter of the sphere is equal to the side length of the cube. Because the diameter is twice the length of the radius, the diameter of the sphere is 2_r_. This means that the side length of the cube is also 2_r_.

The surface area for a cube is given by the following formula, where s represents the length of each side of the cube:

surface area of cube = 6_s_2

The formula for surface area of a cube comes from the fact that each face of the cube has an area of _s_2, and there are 6 faces total on a cube.

Since we already determined that the side length of the cube is the same as 2_r_, we can replace s with 2_r_.

surface area of cube = 6(2_r_)2 = 6(2_r_)(2_r_) = 24_r_2.

We are asked to find the ratio of the surface area of the cube to the surface area of the sphere. This means we must divide the surface area of the cube by the surface area of the sphere.

ratio = (24_r_2)/(4_πr_2)

The _r_2 term cancels in the numerator and denominator. Also, 24/4 simplifes to 6.

ratio = (24_r_2)/(4_πr_2) = 6/π

The answer is 6/π.

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Question

What is the surface area of a hemisphere with a diameter of $4\ cm$ ?

Answer

A hemisphere is half of a sphere. The surface area is broken into two parts: the spherical part and the circular base.

The surface area of a sphere is given by $SA = 4\pi r^{2}$ .

So the surface area of the spherical part of a hemisphere is $SA = 2\pi r^{2}$ .

The area of the circular base is given by $A = \pi r^{2}$ . The radius to use is half the diameter, or 2 cm.

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Question

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?

Answer

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

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Question

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

Answer

$SA=4\pi r^2$

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

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

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

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

$V=4(9)\pi$

$V=36\pi$

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