The radius of a sphere is increasing at a rate of 2mm//s. How fast is the volume increasing when the diameter is 40mm ?

Find Volume Formula: First, we need to find the formula that relates the rate of change of the volume of a sphere to the rate of change of its radius. The volume V of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.Differentiate Volume Formula: Next, we need to differentiate the volume formula with respect to time t to find the rate of change of volume with respect to time, which is dV/dt. Using the chain rule, we get dV/dt = 4πr^2 * dr/dt, where dr/dt is the rate of change of the radius with respect to time.Given Rates and Radius: We are given that the rate of change of the radius dr/dt is 2mm/s. We also know that the diameter of the sphere is 40mm, which means the radius r is half of that, so r = 40mm / 2 = 20mm.Substitute Values: Now we can substitute the values of r and dr/dt into the differentiated volume formula to find dV/dt. So, dV/dt = 4π(20mm)^2 * 2mm/s.Calculate Rate of Increase: Calculating the value, we have dV/dt = 4π(400mm^2) * 2mm/s = 3200π mm^3/s. This is the rate at which the volume of the sphere is increasing when the diameter is 40mm.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The radius of a sphere is increasing at a rate of $2\,\text{mm/s}$ . How fast is the volume increasing when the diameter is $40\,\text{mm}$ ?

View step-by-step help

Home

Math Problems

Grade 6

Area of quadrilaterals and triangles: word problems

Full solution

Q. The radius of a sphere is increasing at a rate of

2\,\text{mm/s}

. How fast is the volume increasing when the diameter is

40\,\text{mm}

Find Volume Formula: First, we need to find the formula that relates the rate of change of the volume of a sphere to the rate of change of its radius. The volume $V$ of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius of the sphere.
Differentiate Volume Formula: Next, we need to differentiate the volume formula with respect to time $t$ to find the rate of change of volume with respect to time, which is $\frac{dV}{dt}$ . Using the chain rule, we get $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$ , where $\frac{dr}{dt}$ is the rate of change of the radius with respect to time.
Given Rates and Radius: We are given that the rate of change of the radius $\frac{dr}{dt}$ is $2\,\text{mm/s}$ . We also know that the diameter of the sphere is $40\,\text{mm}$ , which means the radius $r$ is half of that, so $r = \frac{40\,\text{mm}}{2} = 20\,\text{mm}$ .
Substitute Values: Now we can substitute the values of $r$ and $\frac{dr}{dt}$ into the differentiated volume formula to find $\frac{dV}{dt}$ . So, $\frac{dV}{dt} = 4\pi(20\,\text{mm})^2 \times 2\,\text{mm/s}.$
Calculate Rate of Increase: Calculating the value, we have $\frac{dV}{dt} = 4\pi(400\text{mm}^2) \times 2\text{mm/s} = 3200\pi \text{mm}^3/\text{s}$ . This is the rate at which the volume of the sphere is increasing when the diameter is $40\text{mm}$ .