Mathematics: Acompany needs to make a cylindrical can that can hold precisely 2.3 liters of liquid. if the entire can is to be made out of the same material, find the dimensions of the can that will minimize the cost. round your answer to the nearest two decimal places.recall that the volume of a cylinder is πr^2h and the surface area is 2πrh+2πr^2 where r is the radius and h is the height. also note that 1 liter is equal to 1000 cm^3.radius= =

Acompany needs to make a cylindrical can that can

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09.01.2022

Dimensions of the can should be r = 7.15 cm and height = 14.33 cm.

Step-by-step explanation:

A cylindrical can can hold the amount of water = 2.3 liters

Since 1 liter = 1000 cm³

Therefore, 2.3 liters = 2.3×1000 = 2300 cm³

Volume of a cylinder 'V' = πr²h

2300 = πr²h

$h=\frac{2300}{\pi r^{2} }$ ---------(1)

We have to calculate the dimensions of the can that will minimize the cost.

Total surface area A = 2πrh + 2πr²

To minimize the cost of the material used we will find the derivative of the total surface area and equate the derivative to zero.

A = $2\pi r(\frac{2300}{\pi r^{2}})+2\pi r^{2}$

A = $\frac{4600}{r}+2\pi r^{2}$

A' = $-\frac{4600}{r^{2}}\frac{dh}{dr}+4\pi r$

A' = 0

$-\frac{4600}{r^{2}}\frac{dh}{dr}+4\pi r$ = 0

$-\frac{4600}{r^{2}}+4\pi r=0$

$4\pi r=\frac{4600}{r^{2} }$

$r^{3}=\frac{4600}{4\times \pi }$

$r=\sqrt[3]{\frac{4600}{4\times 3.14} }$

$r=\sqrt[3]{366.242}$

r = 7.15 cm

From equation (1)

$h=\frac{2300}{\pi r^{2}}$

= $\frac{2300}{\pi \times (7.15)^{2} }$

= 14.33 cm

Therefore, can having the radius = 7.15 cm and h = 14.33 cm will have the minimum cost.

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