volume of a sphere homework
***show all work***
1)
image of a sphere with radius labeled r
what is the volume of the sphere above, to the nearest tenth of a cubic inch, if the radius (r) equals 5 inches?
$v = \frac{4}{3}\pi r^3$
2) find, in terms of $\pi$, the volume of a sphere whose radius is 3 centimeters.
$v = \frac{4}{3}\pi r^3$
3) find, in terms of $\pi$, the volume of a sphere whose diameter is 18 inches.
$v = \frac{4}{3}\pi r^3$
4) find, to the nearest cubic millimeter, the volume of a sphere whose diameter is 4 millimeters. use $\pi = 3.14$

Question

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Recall the volume formula for a sphere.
The formula for the volume $ V $ of a sphere is $ V = \frac{4}{3}\pi r^3 $, where $ r $ is the radius of the sphere.
## Step2: Substitute the given radius into the formula.
Given that the radius $ r = 5 $ inches. Substitute $ r = 5 $ into the formula:
$$
V=\frac{4}{3}\pi(5)^3
$$
## Step3: Calculate $ 5^3 $.
$ 5^3 = 5\times5\times5 = 125 $. So the formula becomes:
$$
V=\frac{4}{3}\pi\times125
$$
## Step4: Multiply $ \frac{4}{3} $ and $ 125 $.
$ \frac{4}{3}\times125=\frac{500}{3}\approx166.666\cdots $
## Step5: Multiply by $ \pi $ and round to the nearest tenth.
$ V=\frac{500}{3}\pi\approx166.666\cdots\times3.14159\approx523.6 $ (cubic inches)
# Answer:
The volume of the sphere is approximately $ 523.6 $ cubic inches.

### Problem 2:
# Explanation:
## Step1: Recall the volume formula for a sphere.
The formula for the volume $ V $ of a sphere is $ V=\frac{4}{3}\pi r^3 $, where $ r $ is the radius of the sphere.
## Step2: Substitute the given radius into the formula.
Given that the radius $ r = 3 $ centimeters. Substitute $ r = 3 $ into the formula:
$$
V=\frac{4}{3}\pi(3)^3
$$
## Step3: Calculate $ 3^3 $.
$ 3^3=3\times3\times3 = 27 $. So the formula becomes:
$$
V=\frac{4}{3}\pi\times27
$$
## Step4: Multiply $ \frac{4}{3} $ and $ 27 $.
$ \frac{4}{3}\times27 = 36 $
# Answer:
The volume of the sphere in terms of $ \pi $ is $ 36\pi $ cubic centimeters.

### Problem 3:
# Explanation:
## Step1: Find the radius from the diameter.
The diameter $ d = 18 $ inches, and the radius $ r=\frac{d}{2} $. So $ r=\frac{18}{2}=9 $ inches.
## Step2: Recall the volume formula for a sphere.
The formula for the volume $ V $ of a sphere is $ V = \frac{4}{3}\pi r^3 $.
## Step3: Substitute the radius into the formula.
Substitute $ r = 9 $ into the formula:
$$
V=\frac{4}{3}\pi(9)^3
$$
## Step4: Calculate $ 9^3 $.
$ 9^3=9\times9\times9 = 729 $
## Step5: Multiply $ \frac{4}{3} $ and $ 729 $.
$ \frac{4}{3}\times729 = 972 $
# Answer:
The volume of the sphere in terms of $ \pi $ is $ 972\pi $ cubic inches.

### Problem 4:
# Explanation:
## Step1: Find the radius from the diameter.
The diameter $ d = 4 $ millimeters, and the radius $ r=\frac{d}{2} $. So $ r=\frac{4}{2}=2 $ millimeters.
## Step2: Recall the volume formula for a sphere.
The formula for the volume $ V $ of a sphere is $ V=\frac{4}{3}\pi r^3 $, and we are given $ \pi = 3.14 $.
## Step3: Substitute the radius and $ \pi $ into the formula.
Substitute $ r = 2 $ and $ \pi = 3.14 $ into the formula:
$$
V=\frac{4}{3}\times3.14\times(2)^3
$$
## Step4: Calculate $ 2^3 $.
$ 2^3 = 8 $
## Step5: Multiply the numbers.
First, $ \frac{4}{3}\times3.14\times8=\frac{4\times3.14\times8}{3}=\frac{100.48}{3}\approx33.5 $ (rounded to the nearest cubic millimeter)
# Answer:
The volume of the sphere to the nearest cubic millimeter is $ 34 $ cubic millimeters (Note: The more accurate calculation: $ \frac{4}{3}\times3.14\times8=\frac{100.48}{3}\approx33.493\approx33 $ if we calculate precisely, but maybe a miscalculation in step 5 earlier. Let's recalculate:

$ \frac{4}{3}\times3.14\times8 = \frac{4\times3.14\times8}{3}=\frac{100.48}{3}\approx33.49\approx33 $ cubic millimeters. But let's do it step by step:

$ (2)^3 = 8 $

$ \frac{4}{3}\times3.14 = \frac{12.56}{3}\approx4.1867 $

$ 4.1867\times8 = 33.4936\approx33 $ cubic millimeters. Maybe the initial rounding was wrong. Let's correct it.

So the correct calculation:

$ V=\frac{4}{3}\times3.14\times8=\frac{100.48}{3}\approx33.5 $, but to the nearest cubic millimeter, it is $ 33 $ or $ 34 $? Wait, $ 33.4936 $ is closer to $ 33 $ than $ 34 $. So the correct answer is $ 33 $ cubic millimeters. But let's check again:

$ r = 2 $ mm.

$ V=\frac{4}{3}\times3.14\times2^3=\frac{4}{3}\times3.14\times8=\frac{100.48}{3}\approx33.49\approx33 $ cubic millimeters.

# Answer:
The volume of the sphere to the nearest cubic millimeter is $ 33 $ cubic millimeters (or if we consider the initial miscalculation, maybe $ 34 $, but the precise calculation gives $ 33 $).

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QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Recall the volume formula for a sphere.

Step2: Substitute the given radius into the formula.

Step3: Calculate \( 5^3 \).

Step4: Multiply \( \frac{4}{3} \) and \( 125 \).

Step5: Multiply by \( \pi \) and round to the nearest tenth.

Step1: Recall the volume formula for a sphere.

Step2: Substitute the given radius into the formula.

Step3: Calculate \( 3^3 \).

Step4: Multiply \( \frac{4}{3} \) and \( 27 \).

Step1: Find the radius from the diameter.

Step2: Recall the volume formula for a sphere.

Step3: Substitute the radius into the formula.

Step4: Calculate \( 9^3 \).

Step5: Multiply \( \frac{4}{3} \) and \( 729 \).

Answer:

Problem 2: