Question

The volume of a sphere is \(4 \pi r^{3} / 3,\) where \(r\) is the radius. One student measured the radius to be \(4.30 \mathrm{~cm}\). Another measured the radius to be \(4.33 \mathrm{~cm} .\) What is the difference in volume between the two measurements?

Short answer

Answer: The difference in volume between the two spheres is approximately 6.26 cm³.

Step-by-step solution

Write down the volume formula of a sphere

The volume formula for a sphere is given as \(V = \dfrac{4}{3} \pi r^3\).

Calculate the volume for the first radius

The first radius measurement is \(4.30\,\text{cm}\). To find the volume of the sphere, plug this value into the formula: \(V_1 = \dfrac{4}{3} \pi (4.30)^3\) Now calculate the volume: \(V_1 \approx 333.23\,\text{cm}^3\)

Calculate the volume for the second radius

The second radius measurement is \(4.33\,\text{cm}\). To find the volume of the sphere, plug this value into the formula: \(V_2 = \dfrac{4}{3} \pi (4.33)^3\) Now calculate the volume: \(V_2 \approx 339.49\,\text{cm}^3\)

Calculate the difference in volume

To find the difference in volume between the two spheres, subtract the smaller volume from the larger volume: \(\Delta V = V_2 - V_1 \approx 339.49 - 333.23\) Now calculate the difference: \(\Delta V \approx 6.26\,\text{cm}^3\) The difference in volume between the two measurements is approximately \(6.26\,\text{cm}^3\).