To determine the radius of the spherical grape, we need to use the relationship between the mass, density, and volume, followed by applying the formula for the volume of a sphere. Here is the step-by-step solution:
1. Calculate the volume of the grape:
- The formula for volume (V) based on mass (m) and density (d) is given by:
[tex]\[
V = \frac{m}{d}
\][/tex]
- Substitute the given values: mass [tex]\( m = 8.4 \)[/tex] grams and density [tex]\( d = 2 \)[/tex] grams per cubic centimeter:
[tex]\[
V = \frac{8.4}{2} = 4.2 \text{ cubic centimeters}
\][/tex]
2. Relate the volume of the grape to the volume of a sphere:
- The formula for the volume of a sphere is:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
- To find the radius [tex]\( r \)[/tex], solve for [tex]\( r^3 \)[/tex]:
[tex]\[
r^3 = \frac{3V}{4\pi}
\][/tex]
3. Substitute the volume into the equation:
- Using [tex]\( V = 4.2 \)[/tex] cubic centimeters:
[tex]\[
r^3 = \frac{3 \times 4.2}{4\pi} \approx \frac{12.6}{12.566370614359172} \approx 1.0026761414789407
\][/tex]
4. Calculate the radius by taking the cube root of [tex]\( r^3 \)[/tex]:
- To find [tex]\( r \)[/tex], take the cube root of [tex]\( r^3 \)[/tex]:
[tex]\[
r = (1.0026761414789407)^{\frac{1}{3}} \approx 1.00089125259248 \text{ centimeters}
\][/tex]
5. Round the radius to the nearest tenth:
- The radius is approximately [tex]\( 1.00089125259248 \)[/tex] centimeters.
- Rounding this value to the nearest tenth:
[tex]\[
r \approx 1.0 \text{ centimeters}
\][/tex]
Thus, the radius of the grape is [tex]\( 1.0 \)[/tex] cm when rounded to the nearest tenth of a centimeter.
