Question

A rectangular tank with a square base, an open top, and a volume of 2916 ft3 is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

The dimensions of the tank with minimum surface area are ____?? ft.

Answer 1

Answer:

The dimensions are x=18, y=9

Step-by-step explanation:

We can solve this by having a front face with a length of x, a side face of length and the height of y. We can write the volume of the tank as:

$V=x^2y=2916$

We can write the surface area by adding the area of all sides of the tank:

$S=x^2+4xy$

We can write the volume equation in terms of x:

$y=2916/(x^2)$

and substitute into the Area equation:

$S=x^2+4x(2916/x^2)$

To find the minimum surface area we must use the first derivative:

$S'=2x-11664/x^2$

We must see where the equation is zero or undefined:

$S'=2x-11664/x^2=0$

x can't be zero, we solve the equation:

$2x^3-11664=0$

$x=18$

We must solve for y:

$y=2916/18^2=2916/324=9$

The dimensions are x=18, y=9

We can test this using the second derivative test:

$S''=2+23328/x^3$

substitute x=18, if the expression S''(18)>0 then it proves that the minimum surface area will be obtained with the value x=18. This is true

Answer 2

Answer:

The dimensions of the tank that has the minimum surface area are:

$Sides(x)=18 ft.\\Height(y)=9 ft.$

Step-by-step explanation:

Given information

Volume (v) of the tank = 2916 cubic ft.

Let,

The sides of rectangle = $x$

And, height of rectangle = $y$

So, Volume $(v)=x^2*y$

And surface area $(S)=x^2+4xy$

Now, substitute the value of $y$ in surface area equation from the volume equation:

So,

$S=x^2+4x(2916/x^2)$

To find minimum surface area, take first derivative of the above equation$ds/dx=2x-(11664/x^2)$

solving the above equation by equating with zero to check if it is undefined

we get,

$2x-(11664/x^2)=0$

$2x^3-11664=0\\x=18$

on solving the volume equation for the value of $y$:

$x^2y=2916\\y=(2916/18^2)\\y=9$

So,

The dimensions of the tank that has the minimum surface area are:

$Sides(x)=18 ft.\\Height(y)=9 ft.$

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