Answered

The volume of an object is equal to the ratio of its mass to density, [tex]V=\frac{m}{d}[/tex]. The mass of a spherical grape is 8.4 grams, and its density is 2 grams per cubic centimeter.

What is the radius of the grape? Round to the nearest tenth of a centimeter.

A. [tex]1.0 \text{ cm}[/tex]
B. [tex]1.5 \text{ cm}[/tex]
C. [tex]1.9 \text{ cm}[/tex]
D. [tex]2.1 \text{ cm}[/tex]



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]