SOLUTION
(a) From the question, water is flowing through a gallon at the rate of 4 gallons per minute. This means that the rate is
[tex]\begin{gathered} rate=\frac{volume}{time\text{ }}=\frac{V}{t} \\ 4=\frac{V}{t} \\ cross\text{ multiplying, we have } \\ V=4t \end{gathered}[/tex]Comparing the equation we got to equation of a line which is
[tex]\begin{gathered} y=mx+b,\text{ where m is slope, from } \\ V=4t \\ It\text{ appears that m = 4} \end{gathered}[/tex]So, the slope of the function is 4
(b) Equation for change in V over t, means we find the derivative of V with respect to t. from the linear equation V = 4t, we have
[tex]\begin{gathered} V=4t \\ \frac{dV}{dt}=4 \end{gathered}[/tex]Hence the answer is 4
(c) If there were 4 gallons of water initially in the tank. It means that this initial 4 gallons would be added to the formula, we have
[tex]\begin{gathered} V=4t+4 \\ V(t)=4t+4 \end{gathered}[/tex]Hence the answer is 4t + 4
(d) If the tank can hold a total of 17 gallons, time t it would take to fill the tank becomes
[tex]\begin{gathered} V=17\text{ gallons} \\ V(t)=4t+4 \\ 17=4t+4 \\ collecting\text{ like terms } \\ 17-4=4t \\ 13=4t \\ t=\frac{13}{4} \\ t=3.25 \end{gathered}[/tex]Hence the answer is 3.25 mins