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To determine the best estimate for the volume of a basketball with a diameter of 10 inches, let's go through the steps one by one.
1. Understanding the Diameter and Radius:
- The diameter of the basketball is given as 10 inches.
- The radius (r) is half of the diameter:
[tex]\[ r = \frac{d}{2} = \frac{10 \, \text{inches}}{2} = 5 \, \text{inches} \][/tex]
2. Formula for the Volume of a Sphere:
- A basketball is roughly spherical, so we use the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
3. Calculating the Volume:
- Substituting the radius (5 inches) into the formula:
[tex]\[ V = \frac{4}{3} \pi (5 \, \text{inches})^3 \][/tex]
- First, we evaluate [tex]\( 5^3 \)[/tex]:
[tex]\[ 5^3 = 5 \times 5 \times 5 = 125 \][/tex]
- Then, we substitute back into the volume formula:
[tex]\[ V = \frac{4}{3} \pi \times 125 \][/tex]
- Approximating [tex]\(\pi\)[/tex] as 3.14159:
[tex]\[ V \approx \frac{4}{3} \times 3.14159 \times 125 \][/tex]
- Doing the multiplication:
[tex]\[ V \approx \frac{4 \times 3.14159 \times 125}{3} \][/tex]
[tex]\[ V \approx \frac{1570.796}{3} \approx 523.5987755982989 \, \text{cubic inches} \][/tex]
4. Converting the Volume to Scientific Notation:
- We express the volume in scientific notation to match the given choices:
[tex]\[ V \approx 5.24 \times 10^2 \, \text{cubic inches} \][/tex]
5. Matching the Best Estimate:
- The calculated volume [tex]\( \approx 523.60 \, \text{cubic inches} \)[/tex], which is closest to [tex]\( 5 \times 10^2 \)[/tex] in scientific notation.
- Comparing all given options:
[tex]\[ 5 \times 10^1 \, \text{cubic inches} \quad \text{(50 cubic inches)} \][/tex]
[tex]\[ 5 \times 10^2 \, \text{cubic inches} \quad \text{(500 cubic inches)} \][/tex]
[tex]\[ 5 \times 10^3 \, \text{cubic inches} \quad \text{(5000 cubic inches)} \][/tex]
[tex]\[ 5 \times 10^4 \, \text{cubic inches} \quad \text{(50000 cubic inches)} \][/tex]
- The value [tex]\( 523.60 \, \text{cubic inches} \)[/tex] is closest to [tex]\(\boldsymbol{5 \times 10^2 \, \text{cubic inches}}\)[/tex].
Therefore, the best estimate for the volume of a basketball based on the given diameter is:
[tex]\[ \boxed{5 \times 10^2 \, \text{cubic inches}} \][/tex]
1. Understanding the Diameter and Radius:
- The diameter of the basketball is given as 10 inches.
- The radius (r) is half of the diameter:
[tex]\[ r = \frac{d}{2} = \frac{10 \, \text{inches}}{2} = 5 \, \text{inches} \][/tex]
2. Formula for the Volume of a Sphere:
- A basketball is roughly spherical, so we use the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
3. Calculating the Volume:
- Substituting the radius (5 inches) into the formula:
[tex]\[ V = \frac{4}{3} \pi (5 \, \text{inches})^3 \][/tex]
- First, we evaluate [tex]\( 5^3 \)[/tex]:
[tex]\[ 5^3 = 5 \times 5 \times 5 = 125 \][/tex]
- Then, we substitute back into the volume formula:
[tex]\[ V = \frac{4}{3} \pi \times 125 \][/tex]
- Approximating [tex]\(\pi\)[/tex] as 3.14159:
[tex]\[ V \approx \frac{4}{3} \times 3.14159 \times 125 \][/tex]
- Doing the multiplication:
[tex]\[ V \approx \frac{4 \times 3.14159 \times 125}{3} \][/tex]
[tex]\[ V \approx \frac{1570.796}{3} \approx 523.5987755982989 \, \text{cubic inches} \][/tex]
4. Converting the Volume to Scientific Notation:
- We express the volume in scientific notation to match the given choices:
[tex]\[ V \approx 5.24 \times 10^2 \, \text{cubic inches} \][/tex]
5. Matching the Best Estimate:
- The calculated volume [tex]\( \approx 523.60 \, \text{cubic inches} \)[/tex], which is closest to [tex]\( 5 \times 10^2 \)[/tex] in scientific notation.
- Comparing all given options:
[tex]\[ 5 \times 10^1 \, \text{cubic inches} \quad \text{(50 cubic inches)} \][/tex]
[tex]\[ 5 \times 10^2 \, \text{cubic inches} \quad \text{(500 cubic inches)} \][/tex]
[tex]\[ 5 \times 10^3 \, \text{cubic inches} \quad \text{(5000 cubic inches)} \][/tex]
[tex]\[ 5 \times 10^4 \, \text{cubic inches} \quad \text{(50000 cubic inches)} \][/tex]
- The value [tex]\( 523.60 \, \text{cubic inches} \)[/tex] is closest to [tex]\(\boldsymbol{5 \times 10^2 \, \text{cubic inches}}\)[/tex].
Therefore, the best estimate for the volume of a basketball based on the given diameter is:
[tex]\[ \boxed{5 \times 10^2 \, \text{cubic inches}} \][/tex]
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