Explanation
Step 1: Write down the formula for the volume of a cylinder
The volume V of a cylinder with radius r and height h is given by the formula:
V=πr2h
Step 2: Use the given conditions
Constant Volume: The volume V is kept constant, which means its derivative with respect to time t is zero:
dtdV=0
Rates of Change Relationship: The rate of change of radius (dtdr) and height (dtdh) are equal in magnitude but opposite in sign:
dtdr=−dtdh
Step 3: Differentiate the volume equation with respect to time (t)
Using the Product Rule for differentiation on V=πr2h:
dtdV=π[dtd(r2)⋅h+r2⋅dtdh]
Applying the chain rule to dtd(r2):
dtdV=π[2rdtdr⋅h+r2dtdh]
Step 4: Substitute the given conditions into the differentiated equation
Since dtdV=0:
π[2rhdtdr+r2dtdh]=0
Divide both sides by π (since π=0):
2rhdtdr+r2dtdh=0
Now, substitute dtdr=−dtdh into the equation:
2rh(−dtdh)+r2dtdh=0
−2rhdtdh+r2dtdh=0
Step 5: Simplify and find the relation
Factor out rdtdh from the equation:
rdtdh(−2h+r)=0
Since the dimensions of a real cylinder are changing, r=0 and dtdh=0. Therefore, we can divide both sides by rdtdh:
−2h+r=0
r=2h
Correct Answer:
(a) r = 2h
