The shopkeeper says, 'when the area of the base is the same as the area of the four sides the volume will be at a maximum'. Investigate the shopkeepers claim and see if it is correct.

Authors Avatar

Trays

A shopkeeper asks a company to make some trays. The shopkeeper says, ‘when the area of the base is the same as the area of the four sides the volume will be at a maximum’.

I will use different mathematical means to investigate the shopkeepers claim and see if it is correct.

Formula

V= Volume                        HLW

W= Area of walls        4(L X H)

B= Area of base                LW

Investigation 1

As you can see the above shows that when the height of the sides equals 3. the trays volume is at a maximum. This shows the shopkeepers claim to be correct so far as the area of all four sides equals the area of the base.

I will now use a formula to further test the shopkeeper’s theory:

(L-2x)²        =        4x(L-2x)

÷(L-2x)                ÷(L-2x)

L-2x                =        4x

L                =        6x

X                =        1/6L

This shows the height in terms of length.

In terms of L, x is 1/6 of L.

When:

x1/6L

V= 1/6L(L-2/6L)²

V= 1/6L(2/3L)²

V= 1/6L X 2/3L X 1/6L

V= 2/27

I will now substitute L=18cm into that equation:

Join now!

2/6L=2/27 X 18³ = 432cm³

This confirms the first result. Now I will use trial and error to again confirm the maximum value:

Results in the table so far agree that 432cm³ is the maximum volume.

Investigation 2

I will now use a tray using Length=15cm.

This can also be used to prove the shopkeeper is correct.

(15-2x)²        =        4(15-2x)

÷(15-2x)                ÷(L-2x)

15-2x        =        4x

15                =        6x

X                =        2.5

V                =        (15-2x)(15-2x)

V                =        (225-30x+4x²)x

V                =        225x-60x²+4x³

V                =        250cm³

This formula shows that when the height of the sides equal 2.5, the trays volume will be at its ...

This is a preview of the whole essay