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.

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: \[ V = \frac{37044}{24} \pi \text{ cubic inches} \quad \text{or} \quad V \approx 1543.5 \pi \text{ cubic inches} \]