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2 years ago

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A Rectangular box with square base and open at the top is to have a capacity of 16,823 cu.cm. Find the height of the box that re

quires minimum amount of the material required.
a. 14.12cm

b. 16.14cm

c. 12.13cm

d. 10.36cm

1 answer:

Marat540 [252]2 years ago

4 0

<span>let the side of base =x and height=h

$hx^2=16,823$ so $x=\sqrt{\frac{16,823}{h}}$

surface area of the box = S(x)= x</span>²<span> +4xh

to find the minimum we calculate the first derivative
S'(x) = 2x + 4h
this is minimum for S'(x) =0

2x + 4h = 0 replace x by its value..
</span> $2\sqrt{\frac{16,823}{h}}+4h=0$ <span>
$4h=-2\sqrt{ \frac{16,823}{h} }$ square both sides

$16h^2=4\frac{16,823}{h}$ cross multiply

$h^3=4\frac{16,823}{16}$ <span>

so </span> $h=\sqrt[3]{\frac{16,823}{4}} =16.14 \text{ cm}$ </span>

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