21. Glass Vase You are taking a glass-blowing class and have created a vase in the shape of a sphere. The vase will have a hole in the top so you can put flowers in it and it will sit on a stand. The radius of your vase is $$ \frac{21}{2} $$ inches. Use the formula $$ V=\frac{4}{3} \pi r^{3} $$ to write an expression for the volume of your vase.

Question

UpStudy AI · Accepted Answer

To find the volume of the vase, we will use the formula for the volume of a sphere, which is given by:

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

where $$ V $$ is the volume and $$ r $$ is the radius of the sphere.

### Step 1: Identify the radius
The radius of the vase is given as $$ \frac{21}{2} $$ inches.

### Step 2: Substitute the radius into the volume formula
We will substitute $$ r = \frac{21}{2} $$ into the volume formula:

$$
V = \frac{4}{3} \pi \left( \frac{21}{2} \right)^3
$$

### Step 3: Calculate $$ \left( \frac{21}{2} \right)^3 $$
First, we need to calculate $$ \left( \frac{21}{2} \right)^3 $$:

$$
\left( \frac{21}{2} \right)^3 = \frac{21^3}{2^3} = \frac{9261}{8}
$$

### Step 4: Substitute back into the volume formula
Now we substitute this back into the volume formula:

$$
V = \frac{4}{3} \pi \cdot \frac{9261}{8}
$$

### Step 5: Simplify the expression
Now we can simplify the expression:

$$
V = \frac{4 \cdot 9261 \pi}{3 \cdot 8} = \frac{37044 \pi}{24}
$$

### Step 6: Further simplify
Now we simplify $$ \frac{37044}{24} $$:

$$
\frac{37044}{24} = 1543.5
$$

Thus, the volume of the vase can be expressed as:

$$
V = 1543.5 \pi \text{ cubic inches}
$$

### Final Expression
The expression for the volume of your vase is:

$$
V = \frac{37044}{24} \pi \text{ cubic inches} \quad \text{or} \quad V \approx 1543.5 \pi \text{ cubic inches}
$$
The volume of the vase is $$ V = \frac{37044}{24} \pi $$ cubic inches or approximately $$ 1543.5 \pi $$ cubic inches.

Glass Vase You are taking a glass-blowing class and have created a vase in the
shape of a sphere. The vase will have a hole in the top so you can put flowers in it
and it will sit on a stand. The radius of your vase is inches. Use the formula
to write an expression for the volume of your vase.

Upstudy AI Solution

Answer

Solution

Step 1: Identify the radius

Step 2: Substitute the radius into the volume formula

Step 3: Calculate

Step 4: Substitute back into the volume formula

Step 5: Simplify the expression

Step 6: Further simplify

Final Expression

Mind Expander

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Glass Vase You are taking a glass-blowing class and have created a vase in the shape of a sphere. The vase will have a hole in the top so you can put flowers in it and it will sit on a stand. The radius of your vase is inches. Use the formula to write an expression for the volume of your vase.