### Math Notes

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#### Differential Calculus Solutions

##### Topics || Problems

A box is to be constructed from a piece of zinc 20 inches square by cutting equal squares from each corner and burning up the zinc to form the side. What is the volume of the largest box that can be constructed?

Answer: $$592.59 ~in^3$$

Solution:

Volume of the box, $$V = lwh$$

$$V = (20-2x)(20-2x)x$$

$$V = (400 + 4x^2 - 80x)(x)$$

$$V = 400x + 4x^3 - 80x^2$$

Maximize:

$$\frac{dV}{dx} = 400 + 12x^2 - 160x$$

$$0 = 400 + 12x^2 - 160x$$

$$x = \frac{-(-160)- \pm \sqrt{160^2 - 4(12)(400)}}{2(12)}$$

$$x = \frac{160 \pm 180}{24}$$

$$x_1 = 10$$

$$x_2 \approx 3.333$$

10 is not a possible value. Since if $$x = 10$$ then $$20-2x = 0$$, thus the the box has no width.

Also: $$V'' = 24x-160$$

At $$x = 10$$; $$V '' >0$$, thus the volume is at minimum.

At $$x \approx 3.33$$; $$V '' <0$$, thus the volume is at maximum.

$$V = (20-2(3.333))(20-2(3.333))(3.333)$$

$$V = 592.593 ~in^3$$