Question

Weather balloons are filled with hydrogen and released at various sites to measure and transmit data about conditions such as air pressure and temperature. A weather balloon is filled with hydrogen at the rate of 0.5 ft^3/s. Initially, the balloon has 4 ft^3 of hydrogen. Initially, the balloon has 2 ft^3 of hydrogen.

Required:
a. Find a linear function V that models the volume of hydrogen in the balloon at any time t.
b. If the balloon has a capacity of 15 ft^3, how long does it take to completely fill the balloon?

Answer 1

Answer:

a) The linear function that models the volume of hydrogen in the balloon at any time $t$ is $V(t) = 2 + 0.5\cdot t$.

b) 26 seconds are needed to completely fill the balloon.

Step-by-step explanation:

The statement has a mistake, the correct form is described below:

Weather balloons are filled with hydrogen and released at various sites to measure and transmit data about conditions such as air pressure and temperature. A weather balloon is filled with hydrogen at the rate of $0.5\,\frac{ft^{3}}{s}$. Initially, the balloon has $2\,ft^{3}$ of hydrogen.

a) The volume of weather balloons is increasing linearly in time ($t$), in seconds, since the rate of change of volume ($\dot V$), in cubic feet per second, is stable. The linear function of the volume of the weather balloon in terms of time is:

$V(t) = V_{o} + \dot V\cdot t$ (1)

Where:

$V(t)$ - Current volume, in cubic feet.

$V_{o}$ - Initial volume, in cubic feet.

If we know that $V_{o} = 2\,ft^{3}$ and $\dot V = 0.5\,\frac{ft^{3}}{s}$, then the volume as a function of time is:

$V(t) = 2 + 0.5\cdot t$

b) If we know that $V(t) = 2 + 0.5\cdot t$ and $V(t) = 15\,ft^{3}$, then the time taken to fill the balloon is:

$V(t) = 2 + 0.5\cdot t$

$V(t) - 2 = 0.5\cdot t$

$t = \frac{V(t) - 2}{0.5}$

$t = \frac{15-2}{0.5}$

$t = 26\,s$

26 seconds are needed to completely fill the balloon.