ANSWER:
a.
[tex]\frac{ds}{dt}=\frac{12x+3y}{\sqrt{x^2+y^2}}[/tex]
b. 9.6 ft/sec
STEP-BY-STEP EXPLANATION:
We can see that a right triangle is formed, where we can apply the Pythagorean theorem, let s be the hypotenuse, we obtain the following:
[tex]s^2=x^2+y^2[/tex]
a.
We derive the expression with respect to time so we can calculate ds/dt, like this:
[tex]\begin{gathered} 2s\cdot\frac{ds}{dt}=2x\cdot\frac{dx}{dt}+2y\cdot\frac{dy}{dt} \\ \\ \frac{ds}{dt}=\frac{2x}{2s}\cdot\frac{dx}{dt}+\frac{2y}{2s}\cdot\frac{dy}{dt} \\ \\ \frac{ds}{dt}=\frac{x}{s}\frac{dx}{dt}+\frac{y}{s}\frac{dy}{dt}=\:\frac{1}{s}\left(x\frac{dx}{dt}+y\frac{dy}{dt}\right) \\ \\ \frac{dx}{dt}=12 \\ \\ \frac{dy}{dt}=3 \\ \\ s=\sqrt{x^2+y^2} \\ \\ \text{ We replacing:} \\ \\ \frac{ds}{dt}=\frac{1}{\sqrt{x^2+y^2}}(12x+3y) \\ \\ \frac{ds}{dt}=\frac{12x+3y}{\sqrt{x^2+y^2}} \end{gathered}[/tex]
b.
We calculate the values of x and y to be able to calculate determine ds/dt
[tex]\begin{gathered} x=12\text{ ft/sec}\cdot6\text{ sec}=72\text{ ft} \\ \\ y=78\text{ ft}+3\text{ ft/sec}\cdot6\text{ sec}=96\text{ ft} \\ \\ \text{ We replacing:} \\ \\ \frac{ds}{dt}=\frac{12\cdot72+3\cdot96}{\sqrt{72^2+96^2}}=\frac{1152}{\sqrt{14400}}=\frac{1152}{120}=9.6\text{ ft/sec} \end{gathered}[/tex]
Therefore, s(t) is increasing by 9.6 ft/sec
