Math / Geometry

QuestionIn the diagrams below, the triangular prism and the cube are equivalent
Determine the total surface area of the cube.

Studdy Solution

STEP 1

What is this asking? We need to find the surface area of a cube, given that it has the same volume as a triangular prism. Watch out! Don't mix up surface area and volume!
We're dealing with two different 3D shapes here, so keep the formulas straight.

STEP 2

1. Find the volume of the triangular prism.
2. Find the side length of the cube.
3. Calculate the surface area of the cube.

STEP 3

The volume of a triangular prism is given by $V = (\text{Area of the triangular base}) \cdot \text{height}$ .
The area of a triangle is $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .

STEP 4

The triangle has a base of $9$ dm and a height of $6$ dm.
So, the area is $\frac{1}{2} \cdot 9 \text{ dm} \cdot 6 \text{ dm} = 27 \text{ dm}^2$ .

STEP 5

The height of the prism is 27 dm.
Multiplying the area of the base by the height of the prism, we get $27 \text{ dm}^2 \cdot 27 \text{ dm} = 729 \text{ dm}^3$ .

STEP 6

The volume of a cube is given by $V = \text{side}^3$ .

STEP 7

Since the cube and the triangular prism have the same volume, we can write $x^3 = 729 \text{ dm}^3$ , where $x$ is the side length of the cube.

STEP 8

Taking the cube root of both sides, we get $x = \sqrt[3]{729} = 9 \text{ dm}$ .
So, the side length of the cube is 9 dm!

STEP 9

The surface area of a cube is given by $SA = 6 \cdot \text{side}^2$ .

STEP 10

We found that the side length of the cube is $9$ dm.
Therefore, the surface area is $6 \cdot (9 \text{ dm})^2 = 6 \cdot 81 \text{ dm}^2 = 486 \text{ dm}^2$ .

STEP 11

The total surface area of the cube is $486 \text{ dm}^2$ .

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