A cube has a volume of 64t$$ ^{\placeholder{}}^2 $$

﻿﻿﻿Write an expression for the length of each side of the cube.

﻿﻿﻿Calculate the volume of the cube when t =2 cm.

﻿﻿﻿What does the expression 16t? represent for this cube?
Justify your answer.

Let's break this down step by step:

### 1. **Expression for the length of each side of the cube**:
The volume $ V $ of a cube is given by the formula:
\[
V = s^3
\]
where $ s $ is the length of a side of the cube. We are told that the volume is $ 64t^2 $. Therefore:
\[
s^3 = 64t^2
\]
To find the expression for $ s $, we take the cube root of both sides:
\[
s = \sqrt[3]{64t^2}
\]
Since $ 64 = 4^3 $, we can simplify this to:
\[
s = 4t^{2/3}
\]
Thus, the expression for the length of each side of the cube is:
\[
s = 4t^{2/3}
\]

### 2. **Calculate the volume of the cube when $ t = 2 $ cm**:
Using the volume formula $ V = s^3 $ and the expression for $ s $ from above, we first calculate $ s $ when $ t = 2 $:
\[
s = 4 \times (2)^{2/3}
\]
To find $ 2^{2/3} $, we calculate it approximately as:
\[
2^{2/3} \approx 1.5874
\]
So:
\[
s \approx 4 \times 1.5874 = 6.3496 \, \text{cm}
\]

Now, to find the volume $ V $:
\[
V = (6.3496)^3 \approx 255.88 \, \text{cm}^3
\]

### 3. **What does the expression 16t represent for this cube?**:
The expression $ 16t $ likely represents the surface area of one face of the cube. The surface area of one face of the cube is $ s^2 $. From the expression for $ s $, we have:
\[
s = 4t^{2/3}
\]
So the area of one face is:
\[
s^2 = (4t^{2/3})^2 = 16t^{4/3}
\]
Thus, the expression $ 16t $ seems to represent an approximation of this surface area when $ t $ has a certain relationship or simplifies in some context.

### Justification:
The term $ 16t $ may be a simplified or scaled version of the surface area of a cube face for particular values of $ t $, as it shares a similar form with the surface area expression $ 16t^{4/3} $. It could represent an approximation or a specific case of this surface area.

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Let me know if you need further details!

Here are 5 questions to expand on this concept:
1. How would you calculate the total surface area of the cube?
2. If $ t $ is doubled, how does that affect the volume of the cube?
3. What is the significance of the cube root in the expression for $ s $?
4. Can the expression for $ s $ be simplified further for specific values of $ t $?
5. How would the formula for $ s $ change if the volume were $ 125t^2 $ instead of $ 64t^2 $?

**Tip**: When solving cube root problems, alwayssimplify the constants first to make calculations easier, then focus on handling the exponents separately. This reduces the complexity of the overall expression.

A cube has a volume of 64t^2. Write an expression for the length of each side of the cube. Calculate the volume of the cube when t = 2 cm. What does the expression 16t represent for this cube? Justify your answer.

Explore how to find the side length of a cube with a volume of 64t^2 and calculate the volume when t = 2 cm. This step-by-step guide also explains the meaning of the expression 16t in relation to cube surface area. Suitable for students in Grades 9-11.

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