Spheres - Math

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Question

If the volume of a sphere is $120\ $ft^{3}$$ , what is the approximate length of its diameter?

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

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Answer

The correct answer is 6.12 ft.

Plug the value of into the equation so that

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

Multiply both sides by 3 to get

$360=4\pi $r^{3}$$

Then divide both sides by $4\pi$ to get

Then take the 3rd root of both sides to get 3.06 ft for the radius. Finally, you have to multiply by 2 on both sides to get the diameter. Thus

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Question

The volume of a sphere is . What is its radius?

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Answer

The formula for the volume of a sphere is: $$\frac{4}{3}$ \pi $r^3$ = V$

The only given information in the problem is the sphere's final volume. If the volume is , the formula for volume can be used to calculate the sphere's radius.

In this case, , the radius, is the only unknown variable that needs to be solved for.

$$\frac{4}{3}$ \pi $r^3$ = 57$

$\frac{3}{4}$\cdot $\frac{1}{\pi}$\cdot$\frac{4}{3}$ \pi $r^3$ = 57 \cdot$\frac{3}{4}$\cdot $\frac{1}{\pi}$

$r=\sqrt[3]{13.608}$

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Question

The area of a sphere is . What is its radius?

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Answer

The only information given is the area of .

This problem may be approached "backwards," where the area formula for a sphere can be used to solve for the radius. This is possible because the formula for area is $Area = 4 \cdot \pi \cdot $r^2$$ , where (the radius) is what we're looking for. After is substituted in for the area, the goal is to solve for by getting it by itself on one side of the equals sign.

$87=4 \cdot \pi \cdot $r^2$$

$\frac{87}{4\cdot\pi}$=\frac{4 \cdot \pi \cdot $r^2$}{4\cdot\pi}$

$$\sqrt{\frac{87}{4 \cdot \pi}$} = $$\sqrt{r^2$}$ = r$

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Question

If the volume of a sphere is , what is the sphere's exact radius?

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Answer

Write the formula for the volume of a sphere:

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

Plug in the given volume and solve for the radius, .

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

Start by multiplying each side of the equation by $\frac{3}{4}$ :

$\frac{3}{4}$\cdot1=\frac{4}{3}$\pi $r^3$\cdot$\frac{3}{4}$

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

Now, divide each side of the equation by $\pi$ :

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

Finally, take the cubed root of each side of the equation:

$r=\sqrt[3]{$\frac{3}{4\pi}$}$

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Question

Given the volume of a sphere is , what is the radius?

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Answer

The equation for the volume of a sphere is:

$V=\frac{4}{3}$\pi $r^3$$ , where is the length of the sphere's radius.

Plug in the given volume and solve for to calculate the sphere's radius:

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

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Question

If the volume of a sphere is , what is the radius of the sphere?

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Answer

The formula for the volume of a sphere is:

$V=\frac{4}{3}$\pi $r^3$$ , where is the sphere's radius.

Plug in the volume and solve for , the sphere's radius:

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

$3=\pi $r^3$$

$r=\sqrt[3]{$\frac{3}{\pi}$}:m$

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Question

Find the radius of a sphere if the surface area is .

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Answer

The formula for the surface area of a sphere is:

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

Substitute the given value for the sphere's surface area into the equation and solve for to find the radius:

$8\pi=4\pi $r^2$$

$r=\sqrt2:cm$

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Question

Find the radius of a sphere if its surface area is .

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Answer

The surface area formula for a sphere is:

$A=4\pi $r^2$$ , where is the sphere's radius.

Substitute the given value for the sphere's area into the equation and solve for to find the radius:

$8=4\pi $r^2$$

$\frac{2}{\pi}$= $r^2$

$r=$\sqrt{\frac{2}{\pi}$}:in$

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Question

In terms of $\pi$ , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.

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Answer

feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the surface area formula, and solve for :

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

$A =4\pi \cdot $120^{2}$$

$A =4\pi \cdot 14,400$

$A = 57,600 \pi$

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Question

Which is the greater quantity?

(a) The surface area of a sphere with radius 1

(b) 12

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Answer

The surface area of a sphere can be found using the formula

$A = 4\pi $r^{2}$$ .

The surface area of the given sphere can be found by substituting :

$A = 4\pi \cdot $1^{2}$ = 4\pi$

$\pi > 3$ so $4 \times \pi >4 \times 3$ , or $4\pi > 12$

This makes (a) greater.

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Question

Sphere A has volume $972 \pi \textrm{ $in}^{3}$$ . Sphere B has surface area $400 \pi \textrm{ $in}^{2}$$ . Which is the greater quantity?

(a) The radius of Sphere A

(b) The radius of Sphere B

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Answer

(a) Substitute $V = 972 \pi$ in the formula for the volume of a sphere:

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

$$\frac{4}{3}$\pi $r^{3}$= 972 \pi$

$$\frac{4}{3}$\pi $r^{3}$ \div $\frac{4}{3}$\pi = 972 \pi \div $\frac{4}{3}$\pi$

$\frac{4}{3}$\pi $r^{3}$ \cdot $\frac{3}{4\pi}$ = 972 \pi\cdot $\frac{3}{4\pi}$

$r =\sqrt[3]{ 729} = 9$ inches

(b) Substitute $A = 400 \pi$ in the formula for the surface area of a sphere:

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

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

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

$r = $\sqrt{100}$ = 10$ inches

(b) is greater.

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Question

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

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Answer

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

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

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

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Question

In terms of $\pi$ , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.

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Answer

feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the surface area formula, and solve for :

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

$A =4\pi \cdot $120^{2}$$

$A =4\pi \cdot 14,400$

$A = 57,600 \pi$

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Question

Which is the greater quantity?

(a) The surface area of a sphere with radius 1

(b) 12

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Answer

The surface area of a sphere can be found using the formula

$A = 4\pi $r^{2}$$ .

The surface area of the given sphere can be found by substituting :

$A = 4\pi \cdot $1^{2}$ = 4\pi$

$\pi > 3$ so $4 \times \pi >4 \times 3$ , or $4\pi > 12$

This makes (a) greater.

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Question

Sphere A has volume $972 \pi \textrm{ $in}^{3}$$ . Sphere B has surface area $400 \pi \textrm{ $in}^{2}$$ . Which is the greater quantity?

(a) The radius of Sphere A

(b) The radius of Sphere B

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Answer

(a) Substitute $V = 972 \pi$ in the formula for the volume of a sphere:

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

$$\frac{4}{3}$\pi $r^{3}$= 972 \pi$

$$\frac{4}{3}$\pi $r^{3}$ \div $\frac{4}{3}$\pi = 972 \pi \div $\frac{4}{3}$\pi$

$\frac{4}{3}$\pi $r^{3}$ \cdot $\frac{3}{4\pi}$ = 972 \pi\cdot $\frac{3}{4\pi}$

$r =\sqrt[3]{ 729} = 9$ inches

(b) Substitute $A = 400 \pi$ in the formula for the surface area of a sphere:

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

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

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

$r = $\sqrt{100}$ = 10$ inches

(b) is greater.

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Question

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

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Answer

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

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

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

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Question

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

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Answer

36 inches = $36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ feet, is the radius. Set $r = $\frac{3}{2}$$ , substitute in the volume formula, and solve for :

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

$V = $\frac{4}{3}$ \pi\left \cdot \left( $\frac{3}{2}$ \right ) ^{3}$

$V = $\frac{4}{3}$\cdot $\frac{3}{2}$ \cdot $\frac{3}{2}$ \cdot $\frac{3}{2}$ \cdot \pi\left$

$V = $\frac{1}{1}$\cdot $\frac{1}{1}$ \cdot $\frac{3}{1}$ \cdot $\frac{3}{2}$ \cdot \pi\left$

$V = $\frac{9}{2}$ \pi$

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Question

Which is the greater quantity?

(a) The volume of a sphere with radius

(b) The volume of a cube with sidelength

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Answer

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

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Question

Which is the greater quantity?

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

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

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Answer

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.

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Question

Which is the greater quantity?

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

(b) $864 \textrm{ $in}^{3}$$

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Answer

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

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

$V = $\frac{4}{3}$ \pi \cdot 6 ^{3}$

$V = $\frac{4}{3}$ \pi \cdot 216$

$V = 288 \pi \approx 288 \cdot 3.14 = 904.32$ cubic inches,

which is greater than $864 \textrm{ $in}^{3}$$ .

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