Please help with this problem.

Question

You are to design a stainless-steel fermenter, which has a right circular cylindrical shape with a height of 1.2 m and radius of 0.3 m. How fast will the fermenter’s volume change with a small change in radius (e.g. 0.005 increase)?

Answer 1

Ok so always start with drawing a picture of what you’re working with. A right circular cylinder implies a cylinder standing upright on its circular face like a soda can. Now the question is asking for how fast the volume is changing given a small change in the radius, which means we are trying to solve for dV/dt (change in volume per change in time). We know the radius will be changing at a rate of 0.005m per some time interval, which gives us that dr/dt=0.005. First, we will need an initial equation for the volume of a right circular cylinder which is given by google as

V=Pi*r²*h

and then we will have take the derivative with respect to time t. But recognize that the height is staying the same so h is just a constant and can be brought in front with the pi. This will give us

d/dt(V=h*pi*r²) = dV/dt = 2h*pi*r*dr/dt = 2(1.2)*pi*(.3)(+0.005) ≈ 0.011

Notice the positive sign with the changing radius (+0.005), the plus sign is there because the question gives that the radius is increasing in magnitude, if it asked this question with a decreasing radius then the sign would be negative which makes sense since this would yield a negative change in volume or decreasing volume. Also notice the dr/dt at the end of the equation since related rates problems, like this one, always involve implicit differentiation.

Final answer, the fermenter’s volume will change at a rate of approximately +0.011m/sec given a small change in radius of +0.005m (assuming time in seconds).

Steps to solving these types of questions

1) draw a picture

2) figure out what you’re looking to solve for

3) list givens and make sure you clearly identify which variables are changing and which are constant

4) find a formula for the variable you were looking to solve for. Here the question asked for a change in volume so we knew to look for an equation for volume of our object.

5) differentiate the equation

6) isolate the desired variable

7) plug in values from step 3

8) solve/calculate answer

9) review and relate answer back to real-world scenario given.