I hope someone can help me solve the following problem: A balloon (assume a perfect sphere) is blown up at a rate of $0.1$ L/s. How much does the radius increase when it is $20$ cm?
I have been told to use this formula, but I am not sure why:$$\dfrac{dV}{dt} = \dfrac{dV}{dr} \cdot \dfrac{dr}{dt}$$
Could someone explain what this formula is used for and how I should use it to solve the problem above?
This is my guess:
the differential equation for the volume of a balloon that is expanding. It relates the rate of change of the volume of the balloon, represented by $\dfrac{dV}{dt}$, to the rate of change of the radius of the balloon, represented by $\dfrac{dr}{dt}$, and the relationship between the volume and radius of the balloon, represented by $\dfrac{dV}{dr}$.
To use this formula to solve the problem above, you need to know the value of $\dfrac{dV}{dr}$, which represents the change in volume of the balloon for a given change in its radius. In the case of a perfect sphere, the volume of the balloon is given by the formula:
$V = \dfrac{4}{3} \pi r^3$
Where $r$ is the radius of the balloon. Differentiating this formula with respect to $r$ gives:
$\dfrac{dV}{dr} = 4 \pi r^2$
Substituting this value of $\dfrac{dV}{dr}$ into the original differential equation gives:
$\dfrac{dV}{dt} = 4 \pi r^2 \cdot \dfrac{dr}{dt}$
In order to solve for $\dfrac{dr}{dt}$, you need to know the value of $\dfrac{dV}{dt}$, which represents the rate at which the volume of the balloon is changing. In this case, you are told that the balloon is being blown up at a rate of $0.1 L/s$, so $\dfrac{dV}{dt} = 0.1$. Substituting this value into the equation above gives:
$0.1 = 4 \pi r^2 \cdot \dfrac{dr}{dt}$
Solving for $\dfrac{dr}{dt}$ gives:
$\dfrac{dr}{dt} = \dfrac{0.1}{4 \pi r^2}$
In order to find the increase in radius of the balloon when it is 20 cm, you need to know the value of $r$. In this case, you are told that the radius of the balloon is 20 cm, so $r = 20$ cm. Substituting this value into the equation above gives:
$\dfrac{dr}{dt} = \dfrac{0.1}{4 \pi (20)^2} = 0.00000785$ cm/s
This is the rate at which the radius of the balloon is increasing. To find the total increase in radius, you can simply multiply this rate by the time over which the balloon is being blown up. For example, if the balloon is being blown up for 1 minute, the total increase in radius would be:
$\dfrac{dr}{dt} \cdot t = 0.00000785 \cdot 60 = 0.00469$ cm
Thank you in advance for any help!