Solve the problem.

A rectangular sheet of perimeter 24 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

Question

Solve the problem.

A rectangular sheet of perimeter 24 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

BlackTom AI · Accepted Answer

First, restating the problem in my own words helps set up the relationships: a rectangle with perimeter 24 cm has dimensions x and y, so 2x + 2y = 24, i.e., x + y = 12, which means y = 12 − x. When rolled into a cylinder, the circumference of the base equals x and the height of the cylinder equals y. The volume is V = (area of base) × height = π(r^2) × h, with r = x/(2π) and h = y.

Now evaluate each option in light of these relationships.
Option A: x = 9 cm, y = 3 cm. Check the perimeter constraint: x + y = 12, which is satisfied since 9 + 3 = 12. However, compute the volume scaling: r = x/(2π) = 9/(2π), h = y = 3, so V ∝ x^2 y = 9^2 × 3 = 81 × 3 = 243 (up to the constant 1/(4π)). We will compare with other options, but note this is just a raw product; the maximum occurs where x and y balance to maximize x^2 y given the fixed sum x + y = 12.
Option B: x = 7 cm, y = 5 cm. Here x + y = 12 is also satisfied. The volume proxy x^2 y = 7^2 × 5 = 49 × 5 = 245, which is a bit larger than 243, suggesting this could be competitive, but we must find the true maximum by optimization rather than comparison.
Option C: x = 10 cm, y = 2 cm. Again x + y = 12 holds. The volume proxy is x^2 y = 100 × 2 = 200, which is smaller than the previous values, hinting this is not maximal.
Option D: x = 8 cm, y = 4 cm. Check: x + y = 12, satisfied. The volume proxy is x^2 y = 64 × 4 = 256, which is larger than all the previous ones, indicating this is a strong candidate for the maximum.

To be rigorous, set up the volume (up to a constant factor) as V ∝ x^2 y with y = 12 − x, so V ∝ x^2(12 − x) = 12x^2 − x^3. Differentiate: dV/dx ∝ 24x − 3x^2 = 3x(8 − x). Critical points occur at x = 0 and x = 8. At x = 0 or x = 12, the volume is zero; the meaningful interior critical point is x = 8, which yields y = 4. This confirms the maximum occurs at x = 8, y = 4.

Therefore, among the given options, the pair that gives the largest volume is x = 8 cm and y = 4 cm, matching the choice with those values. The other options either violate the optimization balance (even though they satisfy x + y = 12) or yield a smaller x^2 y value, hence smaller volume.

Solve the problem. A rectangular sheet of perimeter 24 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume? 单项选择题

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