Certainly! Let's solve the problem step-by-step, just as you would if you were tackling it yourself.
### Given Information:
- Mass ([tex]\( m \)[/tex]): 8.4 grams
- Density ([tex]\( d \)[/tex]): 2 grams per cubic centimeter
### Step 1: Calculate the Volume
The formula for the volume [tex]\( v \)[/tex] is:
[tex]\[ v = \frac{m}{d} \][/tex]
Plugging in the given mass and density:
[tex]\[ v = \frac{8.4 \, \text{grams}}{2 \, \text{grams per cubic centimeter}} \][/tex]
[tex]\[ v = 4.2 \, \text{cubic centimeters} \][/tex]
### Step 2: Use the Volume to Find the Radius of the Sphere
The volume [tex]\((V)\)[/tex] of a sphere is given by the formula:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
We need to rearrange this formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\[ r^3 = \frac{3V}{4\pi} \][/tex]
Substitute [tex]\( V = 4.2 \, \text{cubic centimeters} \)[/tex]:
[tex]\[ r^3 = \frac{3 \times 4.2}{4 \pi} \][/tex]
Calculate the value inside the parentheses:
[tex]\[ r^3 = \frac{12.6}{4 \pi} \][/tex]
Next, solve for [tex]\( r \)[/tex] by taking the cube root of both sides:
[tex]\[ r = \left( \frac{12.6}{4 \pi} \right)^{1/3} \][/tex]
By evaluating the expression:
[tex]\[ r \approx 1.00089125259248 \, \text{centimeters} \][/tex]
### Step 3: Round the Radius to the Nearest Tenth
To find the radius to the nearest tenth:
[tex]\[ r \approx 1.0 \, \text{centimeters} \][/tex]
### Final Answer
The radius of the grape is:
- 1.0 cm
So, the correct choice is [tex]\( \boxed{1.0 \, \text{cm}} \)[/tex].
