An open rectangular box with square ends is fitted with an overlapping lid that covers the top and front face. Determine the maximum volume of the box if 1 square meter of metal is used in its construction.

so would it be 8 meters because it has 8 sides

Answered by skamal2002 | 2024-06-10

Volume is a three-dimensional scalar quantity. The **maximum volume **of the box is 0.8m³.
What is volume?
A **volume **is a scalar number that **expresses **the **amount **of three- dimensional space enclosed by a closed surface.
**Surface **area of the box = 2x² + 3xy
**Surface **area of lid = 2xy
Total area = 2x² + 5xy = 6
y = (6-2x²)/5x
**Volume **of the box, V = x²y
V = x²[(6-2x²)/5x] = (6x/5)-(2x³/5)
Now, for the **volume **to be the maximum,
dV/dx = 0
(6/5)-(6x²/5) = 0
x² = 1
x = ±1
y = 4/5
Thus, the **maximum volume **of the box = x²y = (1)²×(4/5) = 0.8 m³
Hence, the **maximum volume **of the box is 0.8m³.

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Answered by ap8997154 | 2024-06-16

The maximum volume of the open rectangular box with square ends, using 1 square meter of metal, is approximately 0.8 cubic meters. We derive this by establishing a relationship between the dimensions and maximizing the volume using calculus. The approach involves setting up the surface area equation, solving for dimensions, and maximizing the volume expression.
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Answered by ap8997154 | 2024-10-01