The length, breadth and height of a cuboid are in the ratio 27 : 8 : 1. The cuboid is melted and recast into a cube. If p is the surface area of the cuboid and q is the surface area of the cube, then what is p/q equal to?

Question

Prepp Team · Accepted Answer

Understanding the Cuboid and Cube Problem

This problem involves a cuboid that is melted and reshaped into a cube. The key principle here is that when a solid is melted and recast into another shape, its volume remains constant. We are given the ratio of the length, breadth, and height of the cuboid and asked to find the ratio of the surface area of the cuboid to the surface area of the resulting cube.

Setting up the Dimensions and Volume of the Cuboid

The length, breadth, and height of the cuboid are given in the ratio $27 : 8 : 1$. Let's introduce a constant factor, $x$, to represent the actual dimensions.

Length ($l$) = $27x$
Breadth ($b$) = $8x$
Height ($h$) = $x$

The volume of the cuboid ($V_{cuboid}$) is calculated by multiplying its length, breadth, and height.

$V_{cuboid} = l \times b \times h = (27x)(8x)(x)$

$V_{cuboid} = 27 \times 8 \times x^3$

$V_{cuboid} = 216x^3$

Volume Conservation and Cube Side Length

When the cuboid is melted and recast into a cube, the volume of the material remains the same. Therefore, the volume of the cube ($V_{cube}$) is equal to the volume of the cuboid.

$V_{cube} = V_{cuboid} = 216x^3$

Let the side length of the cube be $a$. The volume of a cube is given by $a^3$.

$a^3 = 216x^3$

To find the side length $a$, we take the cube root of both sides.

$a = \sqrt[3]{216x^3}$

$a = \sqrt[3]{216} \times \sqrt[3]{x^3}$

Since $6^3 = 216$, we have $\sqrt[3]{216} = 6$.

$a = 6x$

So, the side length of the cube is $6x$.

Calculating Surface Areas

We need to find the surface area of the cuboid ($p$) and the surface area of the cube ($q$).

The surface area of a cuboid with dimensions $l, b, h$ is given by the formula $2(lb + bh + hl)$.

$p = 2((27x)(8x) + (8x)(x) + (x)(27x))$

$p = 2(216x^2 + 8x^2 + 27x^2)$

$p = 2((216 + 8 + 27)x^2)$

$p = 2(251x^2)$

$p = 502x^2$

The surface area of a cube with side length $a$ is given by the formula $6a^2$.

$q = 6a^2$

Since $a = 6x$, we substitute this value into the formula.

$q = 6(6x)^2$

$q = 6(36x^2)$

$q = 216x^2$

Finding the Ratio p/q

Now, we need to find the ratio of the surface area of the cuboid ($p$) to the surface area of the cube ($q$), which is $p/q$.

$\frac{{p}}{{q}} = \frac{{502x^2}}{{216x^2}}$

The term $x^2$ cancels out from the numerator and denominator.

$\frac{{p}}{{q}} = \frac{{502}}{{216}}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

$\frac{{502 \div 2}}{{216 \div 2}} = \frac{{251}}{{108}}$

The fraction $\frac{{251}}{{108}}$ cannot be simplified further as 251 is a prime number and 108 is not divisible by 251.

Conclusion

The ratio of the surface area of the cuboid to the surface area of the cube is $\frac{{251}}{{108}}$.

Revision Table: Cuboid vs Cube Properties

Property	Cuboid	Cube
Dimensions	Length (l), Breadth (b), Height (h)	Side (a)
Volume Formula	$l \times b \times h$	$a^3$
Surface Area Formula	$2(lb + bh + hl)$	$6a^2$

Additional Information: Melting and Recasting Concepts

The process of melting and recasting is a common theme in geometry problems. It highlights the principle of conservation of volume. When a solid substance changes its shape or form (e.g., from solid to liquid and back to solid in a new shape) without losing any material, its total volume remains unchanged.

Volume Conservation: This principle is crucial. The amount of 'stuff' (volume) stays the same, even if the shape changes.
Surface Area Change: While volume is conserved, the surface area almost always changes when a shape is transformed. This is because surface area depends on how the volume is distributed in space. For a fixed volume, a sphere has the minimum surface area, and shapes like thin sheets or wires have very large surface areas.
Applications: This concept applies to various scenarios, such as melting ice into water and then refreezing it in a different mold, reshaping clay, or manufacturing metal objects by casting.

Understanding the difference between volume (the space occupied) and surface area (the total area of the boundaries) is key to solving such problems effectively.

Prepp Team · Answer

$\frac{{247}}{{108}}$

Prepp Team · Answer

$\frac{{503}}{{216}}$

Prepp Team · Answer

$\frac{{505}}{{216}}$

Understanding the Cuboid and Cube Problem

Setting up the Dimensions and Volume of the Cuboid

Volume Conservation and Cube Side Length

Calculating Surface Areas

Finding the Ratio p/q

Conclusion

Revision Table: Cuboid vs Cube Properties

Additional Information: Melting and Recasting Concepts

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