Unusual Shape For A Matchbox

[kuiperbelt]

A matchbox consists of a tray inside an open ended cover. The ends are of square cross section and the volume is 250 square cms. If the depth of the tray is y cm, neglecting the thickness of the material show that the minimum area of material required to construct the matchbox is 346.10^-4 square metres.

[Etch]

The matchbox is a cuboid having dimensions
y cm × y cm × 250/y² cm

So the area of material required to construct the matchbox consists of 2 end panels for the tray, each having area
y × y = y² square cm
and 7 top/side/bottom panels (4 for the top, sides and bottom of the cover, and 3 for the sides and bottom of the tray), each having area
y × 250/y² = 250/y square cm

The total area of material required is therefore
A = 2 × y² + 7 × 250/y square cm
A = 2y² + 1750/y ....... (1)

dA/dy = 4y - 1750/y²
A is a minimum when dA/dy = 0 so when 4y - 1750/y² = 0
thus y³ = 1750/4
and y = 437.5^(1/3) ≈ 7.59

Using this value of y in (1) gives
Amin ≈ 346 square cm
Divide by 100 × 100 = 10^4 to convert to square metres
Amin ≈ 346×10^-4 square metres

[THECORBYLOON]

How do we know the matchbox is a cuboid and how is the volume measured in cm²?

[ZebuSanctuary]

Putting aside the obvious anomaly, the following account presupposes the volume = 250 cm³

Total SA = 2y² + 1750/y ---> This can be shown in the link below.

https://ibb.co/wMCjKhR

Now differentiate Total SA wrt y

dTSA/dy = 4y - 1750/y²

dTSA/dy = 0 at maxima or minima thus;

0 = 4y - 1750/y²

y³ = 1750/4

y = 7.59

Substituting y value into TSA equation;

TSA = 2.(7.59)² + 1750/7.59

TSA = 345.8 cm² rounding up and converting to m² (divide by 10,000)

Total SA = 346.10^-4 m²

Proof;

d²TSA/dy² = 4 + 1750/y³ ---> Clearly 'Positive' when y = 7.59 hence a minima.

[THECORBYLOON]

How do we know the matchbox is a cuboid?

[ZebuSanctuary]

// How do we know the matchbox is a cuboid? //

Definition of Cube and Cuboid (Courtesy of the internet);

Cube: A three-dimensional shape which has six square-shaped faces of equal size and has an angle of 90 degrees between them is called a cube. It has 6 faces, 12 edges and 8 vertices. Opposite edges are equal and parallel. Each vertex meets three faces and three edges.

Cuboid: A three-dimensional figure with three pairs of rectangular faces attached opposite each other. These opposite faces are the same. Out of these six faces, two can be squares. The other names for cuboids are rectangular boxes, rectangular parallelepipeds, and right prisms.

The dimension X in this question is around 4.3 cm.

Hope this helps.

[THECORBYLOON]

Ahhhhhhhhh...cheers for enlightening me. I assumed wrongly that cuboid and cube were the same.

[Atheist]

Zebu; A cuboid is a stretched cube?

[ZebuSanctuary]

// A cuboid is a stretched cube? //

On the proviso, when stretching, the middle of the cuboid doesn't begin to taper, your analogy describes aptly one of the two types of cuboid, as defined above in >21.07.