The volume of an object is equal to the ratio of its mass to density, [tex]V = \frac{\%}{6}[/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 length of a centimeter.
1.0 cm
1.5 cm
2.1 cm
1.9 cm
Answer (1)
Calculate the volume of the grape using the formula V = d m .
Apply the formula for the volume of a sphere V = 3 4 π r 3 and set it equal to the calculated volume.
Solve for the radius r by isolating r 3 and then taking the cube root.
Round the radius to the nearest tenth of a centimeter: 1.0 cm .
Explanation
Problem Analysis We are given that the volume of an object is equal to the ratio of its mass to density, V = d m . The mass of a spherical grape is m = 8.4 grams and its density is d = 2 grams per cubic centimeter. We need to find the radius of the grape, rounded to the nearest tenth of a centimeter.
Calculate the Volume First, we calculate the volume of the grape using the given mass and density: V = d m = 2 grams/cm 3 8.4 grams = 4.2 cm 3
Apply the Volume of Sphere Formula Next, we use the formula for the volume of a sphere, which is V = 3 4 π r 3 , where r is the radius. We set the calculated volume equal to the sphere volume formula and solve for r :
4.2 = 3 4 π r 3
Isolate r^3 To solve for r , we first multiply both sides by 4 π 3 :
r 3 = 4 π 3 × 4.2 = 4 π 12.6 ≈ 1.00268
Calculate r Now, we take the cube root of both sides to find r :
r = 3 1.00268 ≈ 1.00089
Round to Nearest Tenth Finally, we round the radius to the nearest tenth of a centimeter: r ≈ 1.0 cm
Examples
Imagine you're designing a water tank in the shape of a sphere. Knowing the material's density and the desired mass of water the tank should hold, you can calculate the required radius using the same principles. This ensures your tank is perfectly sized for its intended purpose, preventing over- or under-sizing.
