How to find the diameter of a sphere - Intermediate Geometry

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Question

The surface area of a sphere is $100\pi ; in^{2}$ . What is the diameter of the sphere?

Answer

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

So the equation to sovle becomes $4\pi r^{2}=100\pi$ or $r^{2}=25$ so

To answer the question we need to find the diameter:

$d=2r=2\cdot 5=10; in$

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Question

A company wants to construct an advertising balloon spherical in shape. It can afford to buy 28,000 square meters of material to make the balloon. What is the largest possible diameter of this balloon (nearest whole meter)?

Answer

This is equivalent to asking the diameter of a balloon with surface area 28,000 square meters.

The relationship between the surface area and the radius is:

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

To find the radius, substitute for the surface area, then solve:

$28,000=4\pi r^{2}$

$r^{2}=\frac{28,000}{4\pi }$

$r=\sqrt{\frac{28,000}{4\pi }}\approx47$

To find the diameter , double the radius—this is 94.

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Question

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

Answer

Write the formula for the volume of a sphere:

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

Plug in the volume and find the radius by solving for :

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

Start solving for by multiplying both sides of the equation by :

$2\cdot3=\frac{4}{3}\pi r^3\cdot3$

$6=4\pi r^3$

Now, divide each side of the equation by $4\pi$ :

$\frac{6}{4\pi}=\frac{4\pi r^3}{4\pi}$

$\frac{6}{4\pi}=r^3$

Reduce the left side of the equation:

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

Finally, take the cubed root of both sides of the equation:

$\sqrt[3]{\frac{3}{2\pi}}=r$

Keep in mind that you've solved for the radius, not the diameter. The diameter is double the radius, which is: $2\sqrt[3]{\frac{3}{2\pi}}$ .

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Question

The circumference of a sphere is . Find the radius.

Answer

If the circumference of a sphere is , that means we can easily solve for the diameter by using $C=\pi \cdot d$ . This is the equation to find the circumference.

By substituting in the value for circumference (), we can solve for the missing variable :

$84.823:in= \pi \cdot d$

$\frac{84.823:in}{ \pi}= d$

To find the radius of the sphere, we need to divide this value by :

$r=\frac{d}{2}=\frac{27:in}{2}=13.5:in$

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Question

Find the diameter of a sphere if the volume is $\pi$ .

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.

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

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

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

The diameter is double the radius.

$d=2\sqrt[3]{\frac{3}{4}}$

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Question

Find the diameter of a sphere if the volume is $\pi^2$ .

Answer

Write the volume for the sphere.

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

Substitute the volume and solve for the radius.

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

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

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

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

Double the radius to find diameter.

$d=2\sqrt[3]{\frac{3}{4}\pi}$

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Question

Find the diameter of a sphere with the volume listed below.

$V=288\pi$

Answer

In order to solve, we must the given volume into the volume formula for a sphere.

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

$288\pi =\frac{4}{3}\pi(r^{3})$

Divide both sides by pi.

$288 =\frac{4}{3}\(r^{3})$

Multiply both sides by 3.

$216 =(r^{3})$

Take the cube root of both sides to find the radius.

Double the radius to get the diameter, diameter is 12.

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Question

Find the diameter of a sphere if it has a volume of $16\pi$ .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(16\pi)}{\pi}}=4.58$

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Question

Find the diameter of a sphere if it has a volume of $24\pi$ .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(24\pi)}{\pi}}=5.24$

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Question

Find the diameter of the sphere if it has a volume of $39\pi$ .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(39\pi)}{\pi}}=6.16$

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Question

Find the diameter of a sphere if it has a volume of .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(100)}{\pi}}=5.76$

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Question

Find the diameter of a sphere if it has a volume of .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(32)}{\pi}}=3.94$

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Question

Find the diameter of a sphere if its volume is .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(60)}{\pi}}=4.86$

Make sure to round to places after the decimal.

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Question

Find the diameter of the sphere if it has a volume of .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(90)}{\pi}}=5.56$

Make sure to round to places after the decimal.

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Question

Find the diameter of a sphere if it has a volume of .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(908)}{\pi}}=12.01$

Make sure to round to places after the decimal.

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Question

Find the diameter of a sphere if it has a volume of $106\pi$ .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(106\pi)}{\pi}}=8.60$

Make sure to round to places after the decimal.

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Question

Find the diameter of a sphere that has a volume of $69\pi$ .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(69\pi)}{\pi}}=7.45$

Make sure to round to places after the decimal.

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Question

Find the diameter of a sphere if it has a volume of $650\pi$ .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(650\pi)}{\pi}}=15.74$

Make sure to round to places after the decimal.

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Question

Find the diameter of a sphere if it has a volume of $887\pi$ .

Answer

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$ , where is the radius of the sphere.

Now, since the radius is half the diameter, the equation for the volume of a sphere can be rewritten as thus:

$\text{Volume of Sphere}=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^3$ , where is the diameter of the sphere.

Rewrite the equation to solve for .

$d^3=\frac{6(\text{Volume of Sphere})}{\pi}$

$d=\sqrt[3]{\frac{6(\text{Volume of Sphere})}{\pi}}$

Now, plug in the volume of the sphere to find the diameter.

$d=\sqrt[3]{\frac{6(887\pi)}{\pi}}=17.46$

Make sure to round to places after the decimal.

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