Question

An aquarium with rectangular sides and bottom (and no top) is to hold 5gel. Find its proportions so that it will use the least amount of material.

Short answer

The proportion are I=W=2/3h so that it will use the least amount of material.

Step-by-step solution

Given data

The bottom thickness of an open box is given to be three times that of the sides and the volume is 5 gallons.

Concept of Partial Differentiation

The process of finding the partial derivative of a function is called partial differentiation. In this process, the partial derivative of a function with respect to one variable is found by keeping the other variable constant.

Partial derivative formula:

Draw the diagram of the box

Consider an open-top box of length l, width w and constructed of a material of thickness t, as shown in the figure given below.

Solve for the surface area of the bottom of the box

The volume V of the open box is given as follows:

V=lwh

Equate the volume to the given date to get the thickness

V=lwh

The bottom thickness of the box is given to be three times that of the sides.

This means that using the same material throughout,the bottom needs three layers to make it three times thicker than the side.

This implies that the surface area of the bottom of the box needs to be multiplied by 3.

Thus the surface area of the box is given as follows:

S = lw . 3+2[lh + wh]

S = 3lw + 2(l + w)h

Now S is a function of two variables l and w so,to minimize S, find ∂S/∂l,∂S/∂w and set them equal to 0 to get:

Solve the equations for length, width and height

Solve each of the above equation:

Multiply the equations as shown:

Use this value of lw in the above equations to obtain

Therefore, the height is calculated as follows

Therefore, the proportional dimensions of the box which can be constructed with the use of minimum material are .