# Suppose you are blowing a spherical bubble, filling it in with air at a uniform rate of 400mm^3/s. How fast is the radius of the bubble increasing by the time it is already 20mm long?

Differentiate the two sides in terms of time:

By signing up, you agree to our Terms of Service and Privacy Policy

To find the rate at which the radius of the bubble is increasing, we can use the formula for the volume of a sphere, which is (V = \frac{4}{3}\pi r^3), where (V) is the volume and (r) is the radius.

We are given that the volume is increasing at a rate of (400 , \text{mm}^3/\text{s}). So, we can find the rate of change of the radius, ( \frac{dr}{dt} ), by differentiating the volume formula with respect to time (t), assuming the radius is also a function of time.

[ \frac{dV}{dt} = \frac{d}{dt}\left(\frac{4}{3}\pi r^3\right) ]

Using the chain rule, we get:

[ \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} ]

Given that ( \frac{dV}{dt} = 400 , \text{mm}^3/\text{s} ) and ( r = 20 , \text{mm} ) (when the bubble has a radius of ( 20 , \text{mm} )), we can solve for ( \frac{dr}{dt} ):

[ 400 = 4\pi (20)^2 \frac{dr}{dt} ] [ 400 = 1600\pi \frac{dr}{dt} ] [ \frac{dr}{dt} = \frac{400}{1600\pi} ] [ \frac{dr}{dt} = \frac{1}{4\pi} ]

So, the rate at which the radius of the bubble is increasing when it is 20 mm long is ( \frac{1}{4\pi} ) mm/s.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you use the definition of a derivative to show that if #f(x)=1/x# then #f'(x)=-1/x^2#?
- What is the equation of the line tangent to #f(x)= sqrt(3x^3-2x) # at #x=2#?
- What is the slope of the line normal to the tangent line of #f(x) = tanx+sin(x-pi/4) # at # x= (5pi)/8 #?
- What is the equation of the normal line of #f(x)=2x^3-x^2-3x+9# at #x=-1#?
- How do you find the equation of the line tangent to #y=x^2# at (2,4)?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7