4. a rectangular well is 6 feet long, 4 feet wide, and 8 feet deep. if water is running into the well at the…

Explanation:

Step1: Write the formula for the volume of the rectangular well

The volume (V) of a rectangular prism (well) is given by (V = l\times w\times h), where (l = 6) ft (length, constant), (w=4) ft (width, constant), and (h) is the height (depth) of the water. So (V=6\times4\times h = 24h).

Step2: Differentiate the volume formula with respect to time (t)

Using the chain - rule (\frac{dV}{dt}=\frac{dV}{dh}\times\frac{dh}{dt}). Since (V = 24h), then (\frac{dV}{dh}=24). We know that (\frac{dV}{dt}=3) ft³/sec. Substitute into the chain - rule formula: (3=24\times\frac{dh}{dt}).

Step3: Solve for (\frac{dh}{dt})

To find (\frac{dh}{dt}), we use the equation (\frac{dh}{dt}=\frac{\frac{dV}{dt}}{\frac{dV}{dh}}). Substitute (\frac{dV}{dt} = 3) and (\frac{dV}{dh}=24) into the formula: (\frac{dh}{dt}=\frac{3}{24}=\frac{1}{8}) ft/sec.

Answer:

The water level is rising at a rate of (\frac{1}{8}) ft/sec.