Answer:
a)
b)
c)
Step-by-step explanation:
In order to solve this problem, we must first picture a cylinder of height h and radius r (see attached picture).
a) So, in order to find the rate at which the area of the circular surface of the dough is increasing with respect to time, we need to start by using the are formula for a circle:
So, to find the rate of change of the area, we can now take the derivative of this formula with respect to the radius r:
and divide both sides into dt so we get:
and now we can substitute:
b) In order to solve part b, we can start with the formula for the volume:
and solve the equation for h, so we get:
So now we can rewrite the equation so we get:
and now we can take its derivative so we get:
we can rewrite the derivative so we get:
we can take the original volume formula and substitute it into our current derivative, so we get:
and simplify:
so now we can go ahead and substitute the values provided by the problem:
Which simplifies to:
c)
Part c was explained as part of part b where we got the expression for the rate of change of the height of the dough with respect to the radius of the dough in terms of the height h and the radius r: