Multiple choice

The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is $1cm$ the altitude is $6 cm$. When the radius is $6cm$, then volume is increasing at the rate of $1$ Cu $cm/sec$. When the radius is $36cm$, the volume is increasing at a rate of $n$ cu. $cm/sec$. The value of '$n$' is equal to:

  1. $12$
  2. $22$
  3. $30$
  4. $33$
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D Correct answer
Explanation

r = r(t), h = 3r + c. At r=1, h=6, so 6 = 3(1) + c => c=3. h = 3r + 3. V = pi*r^2*h = pi*r^2(3r+3) = 3pi(r^3 + r^2). dV/dt = 3pi(3r^2 + 2r) * dr/dt. At r=6, dV/dt = 1: 1 = 3pi(3*36 + 12) * dr/dt => 1 = 3pi(120) * dr/dt => dr/dt = 1/(360pi). At r=36, dV/dt = 3pi(3*36^2 + 2*36) * (1/(360pi)) = 3pi(3888 + 72) / 360pi = 3960 / 120 = 33.

AI explanation

The volume of a cylinder is V = pi * r squared * h. We are given that the altitude is a linear function of r with dh/dt = 3 * dr/dt, meaning h = 3r + c. When r = 1, h = 6, so the constant c = 3, making the formula h = 3r + 3. Differentiating the volume with respect to time gives dV/dt = pi * r squared * dh/dt + 2 * pi * r * h * dr/dt. Substituting dh/dt = 3k and dr/dt = k into the derivative yields dV/dt = 3 * pi * r squared * k + 2 * pi * r * (3r + 3) * k. Using the given condition at r = 6, 1 = 3 * pi * 36 * k + 12 * pi * 21 * k, we find 1 = 108 * pi * k + 252 * pi * k, so 1 = 360 * pi * k. When r = 36, the rate is n = 3 * pi * 1296 * k + 72 * pi * 111 * k = (3888 + 7992) * pi * k = 11880 * pi * k. Substituting pi * k = 1/360 gives n = 11880 / 360, which results in n = 33.

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