To design a conical tent with a capacity of 5000 to 5200 liters, we can approach it in the following steps:

Step 1: Volume Calculation

The volume $V$ of a cone is given by: $V = \frac{1}{3} \pi r^2 h$ where:

• $r$ is the radius of the base of the cone,
• $h$ is the height of the cone.

Since the tent needs a capacity of 5000 to 5200 liters, we convert liters to cubic meters (as 1 cubic meter = 1000 liters). Thus, the volume should be: $V = 5 \text{ to } 5.2 \text{ cubic meters}$

Step 2: Solve for Dimensions

To find the tent’s dimensions, let's work with the midpoint of the range, 5.1 cubic meters. Assuming the tent’s height $h$ and base radius $r$ need to be practical for a tent, we can test reasonable values. Let’s rearrange for $r$ in terms of $h$: $r^2 = \frac{3V}{\pi h}$

Let's assume a height (say, 2.5 meters) and calculate $r$ based on it.

1. If $h = 2.5$ meters: $r^2 = \frac{3 \times 5.1}{\pi \times 2.5} \approx 1.948$ $r \approx \sqrt{1.948} \approx 1.395 \text{ meters}$

2. Resulting Dimensions:

   • Radius $r \approx 1.395$ meters
   • Height $h = 2.5$ meters

Step 3: Surface Area Calculation

The surface area $A$ of a cone (excluding the base) is given by: $A = \pi r l$ where $l$ is the slant height, calculated as: $l = \sqrt{r^2 + h^2}$ Substitute the values: $l = \sqrt{1.395^2 + 2.5^2} \approx 2.846 \text{ meters}$ Then: $A = \pi \times 1.395 \times 2.846 \approx 12.48 \text{ square meters}$

Step 4: Cost Calculation

The cost of materials is $6.85$ per square meter, so: $\text{Cost} = 12.48 \times 6.85 \approx 85.48$

Summary

To build this conical tent:

• Dimensions: Radius = 1.395 meters, Height = 2.5 meters
• Material Area: 12.48 square meters
• Material Cost: $85.48

Would you like further details or have any questions?

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1. How can the height and radius be adjusted to maintain the volume?
2. What impact would adding a floor have on the total material area and cost?
3. How is the cost affected if we increase the tent’s height slightly?
4. What if the cone’s volume requirement was increased to 6000 liters?
5. Can we include insulation in the material cost estimate?

Tip: When designing, rounding dimensions slightly may ease construction without major cost increase.