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 of a cube is given by the formula: where is the length of a side of the cube. We are told that the volume is . Therefore: To find the expression for , we take the cube root of both sides: Since , we can simplify this to: Thus, the expression for the length of each side of the cube is:
2. Calculate the volume of the cube when cm:
Using the volume formula and the expression for from above, we first calculate when : To find , we calculate it approximately as: So:
Now, to find the volume :
3. What does the expression 16t represent for this cube?:
The expression likely represents the surface area of one face of the cube. The surface area of one face of the cube is . From the expression for , we have: So the area of one face is: Thus, the expression seems to represent an approximation of this surface area when has a certain relationship or simplifies in some context.
Justification:
The term may be a simplified or scaled version of the surface area of a cube face for particular values of , as it shares a similar form with the surface area expression . 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:
- How would you calculate the total surface area of the cube?
- If is doubled, how does that affect the volume of the cube?
- What is the significance of the cube root in the expression for ?
- Can the expression for be simplified further for specific values of ?
- How would the formula for change if the volume were instead of ?
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