The volume
𝑉
of a cylinder of height 4 cm and radius
𝑟
is given by:

𝑉
=
4
𝜋
𝑟
2

What is the relationship between
𝑑
𝑉
𝑑
𝑡
and
𝑑
𝑟
𝑑
𝑡
?

(Hint: Differentiate both sides of
𝑉
=
4
𝜋
𝑟
2
with respect to
𝑡
.)

Question

The volume 
𝑉
  of a cylinder of height 4 cm and radius 
𝑟
  is given by:

𝑉
=
4
𝜋
𝑟
2

What is the relationship between 
𝑑
𝑉
𝑑
𝑡
 and 
𝑑
𝑟
𝑑
𝑡
 ?

(Hint: Differentiate both sides of 
𝑉
=
4
𝜋
𝑟
2
 with respect to 
𝑡
.)

BlackTom AI · Accepted Answer

The problem asks us to differentiate the volume formula of a cylinder with respect to time and relate dV/dt to dr/dt.

Option 1: dV/d r = 8 π r r′
This option mixes variables improperly. V is given as a function of r (V = 4π r^2), so dV/dr would be 8π r. The presence of r′ (dr/dt) here is inappropriate because dV/d r denotes differentiation with respect to r, not with respect to t. Therefore, this option misapplies the variables and is incorrect.

Option 2: dV/dt = 8 π r
If we differentiate V = 4π r^2 with respect to t, we must apply the chain rule: dV/dt = dV/dr · dr/dt. Since dV/dr = 8π r, we get dV/dt = 8π r · dr/dt. The expression 8π r alone misses the factor dr/dt, so this is incomplete and incorrect.

Option 3: dV/dt = 4 π r^2 dr/dt
This option uses the wrong derivative form. Starting from V = 4π r^2, applying d/dt yields dV/dt = 8π r · dr/dt, not 4π r^2 · dr/dt. The coefficient and the dependence on r are both incorrect here, making this option false.

Option 4: r = V / (4 π)
This is a rearrangement of the original formula V = 4π r^2, not a statement about the rate of change. While it algebraically relates r and V, it does not express any differentiation with respect to time. Since the question asks for a relationship between dV/dt and dr/dt, this option does not address the required derivative and is not correct.

Option 5: dV/dt = 8 π r dr/dt
This is the correct application of the chain rule to V = 4π r^2. First, compute dV/dr = 8π r. Then multiply by dr/dt to get dV/dt = (8π r)(dr/dt), which matches the form 8π r dr/dt. The factors and variables align with the given V(r) and the chain rule, so this option is correct.

Option 6: dr/dt = V / (4 π) dV/dt
From dV/dt = 8π r dr/dt, solving for dr/dt gives dr/dt = (dV/dt) / (8π r). The expression in this option would require dividing by 4π and multiplying by V, which does not align with the algebra. Therefore, this option misstates the relationship and is incorrect.

In summary, the correct differentiation result is obtained by applying the chain rule to V = 4π r^2, yielding dV/dt = 8π r dr/dt. All other options either misuse differentiation, omit necessary factors, or misrepresent the algebraic relationship between V, r, and their time derivatives.

MAT135H5_F25_ALL SECTIONS 4.1 Preparation Check

The volume 𝑉 of a cylinder of height 4 cm and radius 𝑟 is given by: 𝑉 = 4 𝜋 𝑟 2 What is the relationship between 𝑑 𝑉 𝑑 𝑡 and 𝑑 𝑟 𝑑 𝑡 ? (Hint: Differentiate both sides of 𝑉 = 4 𝜋 𝑟 2 with respect to 𝑡 .)

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