Solve the problem.

A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.

Question

Solve the problem.

A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of  What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.

BlackTom AI · Accepted Answer

We are given an open-top square-based tank with volume V = 44.5 ft^3. Let the base be a square of side length s (ft) and the height be h (ft).

- The volume constraint is: V = s^2 * h, so h = V / s^2.
- The surface area to minimize (open top) is: SA = base area + lateral area = s^2 + 4sh.
  Substituting h from the volume constraint gives SA(s) = s^2 + 4s*(V/s^2) = s^2 + 4V/s.

To find the minimum, differentiate with respect to s and set to zero:
- d(SA)/ds = 2s - 4V/s^2 = 0.
- Multiply both sides by s^2: 2s^3 - 4V = 0 → s^3 = 2V → s = (2V)^(1/3).

Plugging V = 44.5 ft^3:
- 2V = 89.0, so s ≈ 89.0^(1/3) ≈ 4.46 ft.
- Then h = V / s^2 ≈ 44.5 / (4.46^2) ≈ 44.5 / 19.9 ≈ 2.24 ft.

Thus the dimensions that yield the minimum surface area are approximately s × s × h ≈ 4.46 ft × 4.46 ft × 2.24 ft. When rounded to the nearest tenth, this is about 4.5 ft × 4.5 ft × 2.2 ft.

Now evaluate each option:
- Option 1: 4.5 ft × 4.5 ft × 2.2 ft
  This matches the rounded results derived from the optimization and is consistent with the volume constraint (4.5^2 × 2.2 ≈ 44.5 ft^3). The base area is 4.5^2 = 20.25 ft^2 and the lateral area is 4×4.5×2.2 = 39.6 ft^2, giving SA ≈ 59.85 ft^2.
- Option 2: 3.5 ft × 3.5 ft × 3.5 ft
  All sides equal would imply V = s^2 h = 3.5^2 × 3.5 ≈ 42.875 ft^3, which does not match 44.5 ft^3, and the height is not aligned with the optimal ratio.
- Option 3: 9.4 ft × 9.4 ft × 0.5 ft
  Large base with very small height gives V ≈ 9.4^2 × 0.5 ≈ 44.18 ft^3, close to 44.5, but the ratio does not satisfy the minimization condition (h should be roughly V/s^2 ≈ 44.5 / 88.36 ≈ 0.50, yet the optimization yields h ≈ 2.24 when s ≈ 4.46, so this is not the minimizing configuration).
- Option 4: 5.1 ft × 5.1 ft × 1.7 ft
  This yields V ≈ 5.1^2 × 1.7 ≈ 44.2 ft^3, which is near the target volume but its s and h do not correspond to the ratio that minimizes SA for the given V, so it is not the optimal solution.

In summary, the dimensions that best satisfy the volume and minimize surface area closely correspond to the first option, given the rounded values.

Solve the problem. A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary. 单项选择题

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