Question

You ordered a large block of wood with length 5, width 2, and height 1 (each in feet). Unfortunately, the manufacturer can only guarantee to meet these measurements up to an error of 0.1 excess feet per side. Use a suitable gradient to approximate the most total excess volume that may occur. Compute the exact maximal excess volume.

Answer 1

If each side is 0.1 feet extra,
The volume will be 5.1*2.1*1.1= about 11.781.
Perhaps this helps.

Answer 2

"1.7 ft³" will be the exact maximal excess volume.

According to the question,

Length,

• 5

Breadth,

• 2

Height,

• 1

Error feet per side,

• 0.1

→ $Volume = Length\times Breadth\times Height$

                $= 5\times 2\times 1$

                $= 10 \ ft^3$

→ $\frac{dV}{dx} = BH\frac{dL}{dx} + LH\frac{dB}{dx}+BL\frac{dH}{dx}$

By substituting the values, we get

        $= 0.1\times 2\times 1+0.1\times 1\times 5+ 0.1\times 2\times 5$

        $= 1.3 \ ft^3$ (Total excess volume)

hence,

→ The exact maximal excess volume will be:

= $5.1\times 2.1\times 1.1-(5\times 2\times 1)$

= $11.7-10$

= $1.7 \ ft^3$

Thus the above answer is correct.

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