Question

a child inflates a balloon, admires it for a while and then lets the air out. if v(t) gives the volume of the balloon at time t, then the figure below shows v(t) as a function of t. at what time does the child: a. begin to inflate the balloon? t = b. finish inflating the balloon? t = c. begin to let the air out? t = (what would the graph of v(t) look like if the child had alternated between pinching and releasing the open end of the balloon, instead of letting the air out at constant rate?)

Explanation:

Step1: Recall the meaning of $V'(t)$

$V'(t)$ is the rate of change of the volume. When $V'(t)>0$, the volume is increasing (inflating), when $V'(t) = 0$, the volume is constant, and when $V'(t)<0$, the volume is decreasing (de - flating).

Step2: Find when inflation starts

The child begins to inflate the balloon when $V'(t)$ changes from $0$ to a positive value. Looking at the graph, $V'(t)$ changes from $0$ to positive at $t = 5$.

Step3: Find when inflation ends

The child finishes inflating the balloon when $V'(t)$ changes from a positive value to $0$. From the graph, this occurs at $t = 10$.

Step4: Find when air - release starts

The child begins to let the air out when $V'(t)$ changes from $0$ to a negative value. From the graph, this happens at $t = 15$.

Answer:

A. $t = 5$
B. $t = 10$
C. $t = 15$