Question

a box with a square base

.6 4. Compute x + 3x4 + 2x3 + 1 -da. 24 일 5. Let F(x) = tet-2+tº +1 dt, find F'(2). tt +3 0

Answer 1

Let x be the length,

Let y be the height,

Let z be the width.

The volume of the box is given by,

for square base x = z,

Ꮖ Ꮖ ;

$y = \frac{500}{x^{2}}$

The surface area of the open-top box is given by,

put x = z,

plugin equation for y,

,$S = x^{2} + 4x\left ( \frac{500}{x^{2}} \right )$

$S = x^{2} + \frac{2000}{x}$

diff wrt x,

$\frac{\mathrm{d} S}{\mathrm{d} x} = 2x - \frac{2000}{x^{2}}$

For minimum surface are,

$\frac{\mathrm{d} S}{\mathrm{d} x} =0$

$2x - \frac{2000}{x^{2}} =0$

$x - \frac{1000}{x^{2}} =0$

Now find the y value,

$y = \frac{500}{10^{2}} = 5$

Hence the dimensions of the box are,

I hope this answer helps,
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