Let's break down the problem:

a) Write an 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 each side of the cube.

We are given that the volume is $64t$, so: $s^3 = 64t$

To find $s$, we take the cube root of both sides: $s = \sqrt[3]{64t}$

Since $\sqrt[3]{64} = 4$, the expression for the side length of the cube is: $s = 4\sqrt[3]{t}$

b) Calculate the volume of the cube when $t = 2$ cm.

When $t = 2$, we substitute this value into the given volume expression: $V = 64t$ $V = 64 \times 2 = 128 \, \text{cm}^3$

So, the volume of the cube is 128 cm³ when $t = 2$ cm.

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

The expression $16t$ represents the surface area of the cube. The surface area $A$ of a cube is given by the formula: $A = 6s^2$

We know from part (a) that the side length of the cube is $s = 4\sqrt[3]{t}$. So, the surface area can be calculated as: $A = 6 \times (4\sqrt[3]{t})^2 = 6 \times 16t^{2/3}$ However, since the expression $16t$ is directly given, it likely represents an approximate version of the surface area formula under certain assumptions about $t$ or a simplified scenario. Based on dimensional consistency, $16t$ resembles a linear or proportional measure related to surface area in the context of the cube.

Would you like more details or have any questions? Here are some related problems:

1. How would you express the surface area if the volume was given as $t^3$?
2. What would the side length of the cube be if $t = 1$?
3. How would the volume change if the value of $t$ was halved?
4. What is the relationship between volume and surface area for a cube?
5. How would you calculate the diagonal of the cube?

Tip: When solving cube problems, always remember that the side length is related to the cube root of the volume!