Question

Suppose a spherical balloon grows in such a way that after t seconds, its volume is V = 4 sqrt(t) cm3. What is the volume of the balloon at time t = 169 seconds?: V =? . How fast is the volume changing after 169 seconds?

Answer 1

:  You need to know the derivative of the sqrt function. Remember that sqrt(x) = x^(1/2), and that (d x^a)/(dx) = a x^(a-1). So (d sqrt(x))/(dx) = (d x^(1/2))/(dx) = (1/2) x^((1/2)-1) = (1/2) x^(-1/2) = 1/(2 x^(1/2)) = 1/(2 sqrt(x)). 

There is a subtle shift in meaning in the use of t. If you say "after t seconds", t is a dimensionless quantity, such as 169. Also in the formula V = 4 sqrt(t) cm3, t is apparently dimensionless. But if you say "t = 169 seconds", t has dimension time, measured in the unit of seconds, and also expressing speed of change of V as (dV)/(dt) presupposes that t has dimension time. But you can't mix formulas in which t is dimensionless with formulas in which t is dimensioned. 

Below I treat t as being dimensionless. So where t is supposed to stand for time I write "t seconds" instead of just "t".

Then (dV)/(d(t seconds)) = (d 4 sqrt(t))/(dt) cm3/s = 4 (d sqrt(t))/(dt) cm3/s = 4 / (2 sqrt(t)) cm3/s = 2 / (sqrt(t)) cm3/s. 

Plugging in t = 169 gives 2/13 cm3/s.

Answer 2

The volume at $t=169$ is $V=52\text{ cm}^3$rate of change is $\dfrac{dV}{dt}=\dfrac{2}{13}$ cubic cm per second.

Important information:

• The volume of a spherical balloon after $t$ seconds is $V=4\sqrt{t}\text{ cm}^3$.
• The given time is $t=169$.

We need to find the volume of the balloon at time $t=169$ seconds and how fast is the volume changes after 169 seconds.

Volume of sphere:

Substitute $t=169$ in the given function to find the volume at $t=169$.

$V=4\sqrt{169}\text{ cm}^3$

$V=4(13)\text{ cm}^3$

$V=52\text{ cm}^3$

Now differentiate the given function with respect to time.

$\dfrac{dV}{dt}=4\dfrac{1}{2\sqrt{t}}$

Substitute $t=169$.

$\dfrac{dV}{dt}=2\dfrac{1}{\sqrt{169}}$

$\dfrac{dV}{dt}=\dfrac{2}{13}$

Therefore, the volume at $t=169$ is $V=52\text{ cm}^3$rate of change is $\dfrac{dV}{dt}=\dfrac{2}{13}$ cubic cm per second.

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