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The problem involves finding the dimensions of a rectangular prism with a fixed volume of \(170 \, \text{m}^3\) that has the minimum possible surface area.

To tackle this:

1. **Volume Constraint:** For a rectangular prism with dimensions \(x\), \(y\), and \(z\), the volume \(V\) is given by:
   \[
   V = x \cdot y \cdot z = 170 \, \text{m}^3
   \]

2. **Surface Area Minimization:** The surface area \(S\) of the prism is:
   \[
   S = 2(xy + xz + yz)
   \]
   To minimize \(S\) with a constant volume, a cube-like shape often provides the smallest surface area for a given volume. However, since 170 is not a perfect cube, we need to find the dimensions \(x\), \(y\), and \(z\) that satisfy the volume requirement and yield the minimum surface area.

### Solution Steps

#### Part (a): Finding the Dimensions
1. Assume the prism's base is square, so \(x = y\), and solve for \(z\) in terms of \(x\):
   \[
   x^2 \cdot z = 170 \Rightarrow z = \frac{170}{x^2}
   \]

2. Substitute \(z\) into the surface area formula:
   \[
   S = 2(x^2 + x \cdot z + x \cdot z) = 2(x^2 + 2x \cdot \frac{170}{x^2})
   \]
   Simplify and differentiate to find the minimum.

#### Part (b): Surface Area Calculation
Once \(x\), \(y\), and \(z\) are found, substitute back into the surface area formula to find the least possible surface area.

Would you like a detailed step-by-step solution for these calculations?

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### Further Questions
1. How do we apply calculus to find minimum values in constrained optimization problems?
2. Why is a cube the shape with the smallest surface area for a given volume?
3. What happens to surface area when dimensions are kept proportional but volume changes?
4. Can this approach work with other 3D shapes (like cylinders) for volume constraints?
5. How does changing the units (e.g., meters to centimeters) affect surface area and volume calculations?

**Tip:** Minimizing surface area with volume constraints is a classic application of optimization techniques in calculus.

Yasmin is constructing a rectangular prism with volume exactly 170 m³. It will have the least possible surface area.

a) Describe the prism. What will be its dimensions?

b) What is the surface area of this rectangle?

Learn how to find the dimensions of a rectangular prism with a fixed volume of 170 m³ that minimizes surface area. This problem involves applying optimization techniques and geometry principles to determine dimensions and calculate surface area, suitable for high school students in Grades 10-12.

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