Multiple choice

The radius of a cylindrical box is $8$ cm and the height is $3$cm. The number of inches that may be added to either the radius or the height to give the same non-zero increase in volume is:

  1. $1$
  2. $5\dfrac{1}{3}$

  3. any number

  4. non-existent

  5. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the increase be x. Equating the volume increases gives 3π(8 + x)^2 - 3π(8)^2 = 64πx, which simplifies to x(3x - 16) = 0. Since the increase is non-zero, x = 16/3 = 5 1/3.

AI explanation

Using the volume formula for a cylinder, the original volume is pi * (8)^2 * 3 = 192 * pi cubic cm. Let x be the added length; adding x to the radius gives a volume of pi * (8 + x)^2 * 3, and adding x to the height gives a volume of pi * (8)^2 * (3 + x). Setting the increases equal to each other, 3 * (64 + 16x + x^2) - 192 * pi equals 192 + 64x - 192 * pi, which simplifies to 48x + 3x^2 = 64x; solving 3x^2 = 16x for a non-zero x yields x = 16/3 or 5 1/3.

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