Math Problem Statement

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.

Solution

Let's break this down step by step:

1. Expression for the length of each side of the cube:

The volume VV of a cube is given by the formula: V=s3V = s^3 where ss is the length of a side of the cube. We are told that the volume is 64t264t^2. Therefore: s3=64t2s^3 = 64t^2 To find the expression for ss, we take the cube root of both sides: s=64t23s = \sqrt[3]{64t^2} Since 64=4364 = 4^3, we can simplify this to: s=4t2/3s = 4t^{2/3} Thus, the expression for the length of each side of the cube is: s=4t2/3s = 4t^{2/3}

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

Using the volume formula V=s3V = s^3 and the expression for ss from above, we first calculate ss when t=2t = 2: s=4×(2)2/3s = 4 \times (2)^{2/3} To find 22/32^{2/3}, we calculate it approximately as: 22/31.58742^{2/3} \approx 1.5874 So: s4×1.5874=6.3496cms \approx 4 \times 1.5874 = 6.3496 \, \text{cm}

Now, to find the volume VV: V=(6.3496)3255.88cm3V = (6.3496)^3 \approx 255.88 \, \text{cm}^3

3. What does the expression 16t represent for this cube?:

The expression 16t16t likely represents the surface area of one face of the cube. The surface area of one face of the cube is s2s^2. From the expression for ss, we have: s=4t2/3s = 4t^{2/3} So the area of one face is: s2=(4t2/3)2=16t4/3s^2 = (4t^{2/3})^2 = 16t^{4/3} Thus, the expression 16t16t seems to represent an approximation of this surface area when tt has a certain relationship or simplifies in some context.

Justification:

The term 16t16t may be a simplified or scaled version of the surface area of a cube face for particular values of tt, as it shares a similar form with the surface area expression 16t4/316t^{4/3}. It could represent an approximation or a specific case of this surface area.


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 tt is doubled, how does that affect the volume of the cube?
  3. What is the significance of the cube root in the expression for ss?
  4. Can the expression for ss be simplified further for specific values of tt?
  5. How would the formula for ss change if the volume were 125t2125t^2 instead of 64t264t^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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Cube Roots
Exponentiation

Formulas

Volume of a cube: V = s^3
Cube root: s = ∛V

Theorems

Properties of cubes

Suitable Grade Level

Grades 9-11