If the rate at which water vapor condenses onto a spherical raindrop is proportional to the surface area of the raindrop, show that the radius of the raindrop will increase at a constant rate?
Let us setup the following variables:
{
(r, "Radius of raindrop at time t","(cm)"), (S, "Surface area of raindrop at time t", "(cm"^3")"), (V, "Volume of raindrop at time t", "(cm"^2")"), (t, "time", "(sec)") :} #
The standard formula for Surface Area of a sphere, and the volume are:
We are given that the rate at which water vapor condenses onto a spherical raindrop is proportional to the surface area of the raindrop, Thus:
And os applying the chain rule, we have:
By signing up, you agree to our Terms of Service and Privacy Policy
The rate of change of the radius of the raindrop ((r)) is directly proportional to the rate at which water vapor condenses onto its surface ((dV/dt)). According to the given information, the rate of condensation ((dV/dt)) is proportional to the surface area of the raindrop ((4\pi r^2)). Therefore, the rate of change of the radius ((dr/dt)) is constant.
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.
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.
- The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?
- How do you maximize and minimize #f(x,y)=e^-x+e^(-3y)-xy# subject to #x+2y<7#?
- How do you use the linear approximation to #f(x, y)=(5x^2)/(y^2+12)# at (4 ,10) to estimate f(4.1, 9.8)?
- How do you find the linear approximation of the function #g(x)=root5(1+x)# at a=0?
- How do you minimize and maximize #f(x,y)=x-y/(x-y/(x-y))# constrained to #1<yx^2+xy^2<16#?
- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7