Mathematics: A container manufacturer plans to make rectangular boxes whose bottom and top measure 3x by 4x. The container must contain 48 cm^3. The top and the bottom will cost $3.50 per square centimeter, while the four sides will cost $4.40 per square centimeter. What should the height of the container be so as to minimize cost? Round your answer to the nearest hundredth.

A container manufacturer plans to make rectangular

ashleyzamarripa08

22.01.2022

The height of the container that minimize cost is 3.08 cm

Step-by-step explanation:

Let

h ---> the height of the container

we know that

The volume of the box is equal to

V=(3x)(4x)h

V=12x^2h

$V=48\ cm^3$

substitute

48=12x^2h

$h=\frac{4}{x^2}$

The function cost is equal to

C=3.50(2)(12x^2)+4.40(14x)(h)

$C=84x^2+61.6x\frac{4}{x^2}$

$C=84x^2+\frac{246.4}{x}$

To find out the minimum cost determine the first derivative

$\frac{dC}{dx}=168x-\frac{246.4}{x^2}$

equate to zero

$168x=\frac{246.4}{x^2}\\x^3=1.4667\\x=1.14\ cm$

Find the height of the container

$h=\frac{4}{x^2}$

substitute the value of x

$h=\frac{4}{1.14^2}=3.08\ cm$

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