To find the radius of the grape, we will follow a step-by-step approach starting from calculating the volume and proceeding to find the radius.
Given:
- Mass ([tex]\(m\)[/tex]) = 8.4 grams
- Density ([tex]\(d\)[/tex]) = 2 grams per cubic centimeter
First, calculate the volume ([tex]\(V\)[/tex]) of the spherical grape using the formula:
[tex]\[ V = \frac{m}{d} \][/tex]
Plug in the values:
[tex]\[ V = \frac{8.4 \, \text{grams}}{2 \, \text{grams/cm}^3} \][/tex]
[tex]\[ V = 4.2 \, \text{cm}^3 \][/tex]
Next, we use the formula for the volume of a sphere to find the radius:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
We already have the volume ([tex]\(V\)[/tex]):
[tex]\[ 4.2 = \frac{4}{3} \pi r^3 \][/tex]
Solve for [tex]\(r^3\)[/tex]:
[tex]\[ r^3 = \frac{4.2}{\frac{4}{3} \pi} \][/tex]
[tex]\[ r^3 = \frac{4.2 \times 3}{4 \pi} \][/tex]
[tex]\[ r^3 \approx 1.0027 \][/tex]
Now, solve for [tex]\(r\)[/tex]:
[tex]\[ r = (1.0027)^{1/3} \][/tex]
[tex]\[ r \approx 1.0009 \][/tex]
Finally, round the radius to the nearest tenth of a centimeter:
[tex]\[ r \approx 1.0 \, \text{cm} \][/tex]
Thus, the radius of the grape is closest to [tex]\( \boxed{1.0 \, \text{cm}} \)[/tex].
