Answer :
To find the radius of the spherical grape, follow these steps:
1. Calculate the Volume:
The volume \( V \) of the grape can be calculated using the formula:
[tex]\[ V = \frac{m}{d} \][/tex]
where \( m \) is the mass and \( d \) is the density.
Given:
- Mass (\( m \)) = 8.4 grams
- Density (\( d \)) = 2 grams per cubic centimeter
Substitute these values into the formula:
[tex]\[ V = \frac{8.4}{2} = 4.2 \text{ cubic centimeters} \][/tex]
2. Use the Volume of a Sphere Formula:
The formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
To find the radius \( r \), rearrange the formula to solve for \( r \):
[tex]\[ r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}} \][/tex]
3. Substitute the Volume into the Rearranged Formula:
Using \( V = 4.2 \) cubic centimeters:
[tex]\[ r = \left( \frac{3 \times 4.2}{4 \pi} \right)^{\frac{1}{3}} \][/tex]
Simplify the expression inside the parentheses first:
[tex]\[ \frac{3 \times 4.2}{4 \pi} = \frac{12.6}{4 \pi} \][/tex]
Now, finding the value:
[tex]\[ r = \left( \frac{12.6}{4 \pi} \right)^{\frac{1}{3}} \][/tex]
4. Calculate the Radius and Round:
From the calculation:
[tex]\[ r \approx 1.00089125259248 \text{ centimeters} \][/tex]
Finally, round this value to the nearest tenth of a centimeter:
[tex]\[ r \approx 1.0 \text{ centimeters} \][/tex]
Therefore, the radius of the grape, rounded to the nearest tenth of a centimeter, is:
[tex]\[ \boxed{1.0 \text{ cm}} \][/tex]
1. Calculate the Volume:
The volume \( V \) of the grape can be calculated using the formula:
[tex]\[ V = \frac{m}{d} \][/tex]
where \( m \) is the mass and \( d \) is the density.
Given:
- Mass (\( m \)) = 8.4 grams
- Density (\( d \)) = 2 grams per cubic centimeter
Substitute these values into the formula:
[tex]\[ V = \frac{8.4}{2} = 4.2 \text{ cubic centimeters} \][/tex]
2. Use the Volume of a Sphere Formula:
The formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
To find the radius \( r \), rearrange the formula to solve for \( r \):
[tex]\[ r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}} \][/tex]
3. Substitute the Volume into the Rearranged Formula:
Using \( V = 4.2 \) cubic centimeters:
[tex]\[ r = \left( \frac{3 \times 4.2}{4 \pi} \right)^{\frac{1}{3}} \][/tex]
Simplify the expression inside the parentheses first:
[tex]\[ \frac{3 \times 4.2}{4 \pi} = \frac{12.6}{4 \pi} \][/tex]
Now, finding the value:
[tex]\[ r = \left( \frac{12.6}{4 \pi} \right)^{\frac{1}{3}} \][/tex]
4. Calculate the Radius and Round:
From the calculation:
[tex]\[ r \approx 1.00089125259248 \text{ centimeters} \][/tex]
Finally, round this value to the nearest tenth of a centimeter:
[tex]\[ r \approx 1.0 \text{ centimeters} \][/tex]
Therefore, the radius of the grape, rounded to the nearest tenth of a centimeter, is:
[tex]\[ \boxed{1.0 \text{ cm}} \][/tex]
