Julia plans to build a storage shed with a floor area of 60m2 . If she wants to use the least amount of materials so that she optimizes her cost. Find the dimensions and the minimum perimeter.

To optimize the cost of building a storage shed, Julia should aim to minimize the perimeter while maintaining a fixed floor area of 60 m². To do this, we need to find the dimensions of the shed that result in the least perimeter.

For a given area, a square will have the smallest perimeter. Therefore, we should consider the storage shed to be a rectangle (including a square as a special case) and minimize the perimeter.

Let's denote the length of the shed by \( l \) and the width by \( w \).

### Step 1: Establish the relationship between length and width
Given the floor area:
\[
l \times w = 60
\]

### Step 2: Express the perimeter in terms of one variable
The perimeter \( P \) of the rectangle is given by:
\[
P = 2l + 2w
\]

Using the area relationship \( w = \frac{60}{l} \), we can express the perimeter in terms of \( l \):
\[
P = 2l + 2 \left( \frac{60}{l} \right)
\]
\[
P = 2l + \frac{120}{l}
\]

### Step 3: Find the critical points
To minimize the perimeter, we take the derivative of \( P \) with respect to \( l \) and set it to zero:
\[
\frac{dP}{dl} = 2 - \frac{120}{l^2}
\]
\[
0 = 2 - \frac{120}{l^2}
\]
\[
\frac{120}{l^2} = 2
\]
\[
l^2 = 60
\]
\[
l = \sqrt{60} \approx 7.75 \, \text{m}
\]

### Step 4: Determine the corresponding width
\[
w = \frac{60}{l} = \frac{60}{\sqrt{60}} = \sqrt{60} \approx 7.75 \, \text{m}
\]

### Step 5: Calculate the minimum perimeter
Since the shed is optimized to be a square:
\[
P = 2l + 2w = 2 \times 7.75 + 2 \times 7.75 = 4 \times 7.75 \approx 31 \, \text{m}
\]

### Conclusion
The dimensions of the storage shed that use the least amount of materials are approximately \( 7.75 \, \text{m} \times 7.75 \, \text{m} \). The minimum perimeter is approximately 31 meters.

Would you like further details or have any questions?

Here are 5 numbered questions you might consider asking:
1. How do you find the surface area of a rectangular prism?
2. What is the formula for the volume of a cylinder?
3. How do you calculate the perimeter of a triangle?
4. What is the method to find the area of a trapezoid?
5. How do you determine the diagonal length of a rectangle?

**Tip:** For optimization problems involving dimensions and minimizing material use, always consider geometric properties such as symmetry and area-to-perimeter ratios.

Discover how to optimize the cost of building a storage shed by minimizing its perimeter while maintaining a floor area of 60 square meters. Learn the step-by-step solution and calculations involving geometric optimization principles. Suitable for high school students and beyond.

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