Calculating Cloth Required for a Conical Tent

This problem asks us to find the length of a piece of cloth needed to make a conical tent. We are given the dimensions of the tent (base diameter and slant height) and the width of the cloth. The key is to understand that the area of the cloth needed is equal to the surface area of the cone that forms the tent.

Understanding the Conical Tent and Cloth Material

A conical tent is shaped like a cone. The cloth used to make the tent forms the curved surface (lateral surface) of the cone. The base of the cone is usually the ground, which is not covered by the cloth in this type of tent. The cloth itself is provided as a long piece with a specific width.

The total area of the cloth is its length multiplied by its width. This area must be equal to the lateral surface area of the conical tent.

Calculating Key Dimensions

We are given the following information:

• Diameter of the base of the conical tent = 14 m
• Slant height of the conical tent = 9 m
• Width of the cloth = 2 m

First, we need to find the radius of the base of the cone from the given diameter.

Radius ($\text{r}$) = Diameter / 2

$\text{r} = \frac{14 \text{ m}}{2}$

$\text{r} = 7 \text{ m}$

The slant height ($\text{l}$) is given as 9 m.

Determining the Surface Area of the Tent

The cloth makes the lateral surface of the conical tent. The formula for the lateral surface area (LSA) of a cone is given by:

$\text{LSA} = \pi \times \text{radius} \times \text{slant height}$

$\text{LSA} = \pi \text{ r l}$

We usually use the value of $\pi$ as $\frac{22}{7}$ for calculations involving multiples of 7.

Substituting the values:

$\text{LSA} = \frac{22}{7} \times 7 \text{ m} \times 9 \text{ m}$

$\text{LSA} = 22 \times 9 \text{ m}^2$

$\text{LSA} = 198 \text{ m}^2$

So, the total area of the cloth required to make the tent is 198 square metres.

Finding the Length of Cloth Required

The cloth is a rectangle with a known width and an unknown length. The area of this rectangular cloth must be equal to the lateral surface area of the tent.

Area of cloth = Length of cloth $\times$ Width of cloth

We know the Area of cloth ($198 \text{ m}^2$) and the Width of cloth (2 m). We need to find the Length of cloth.

Let the length of the cloth be $\text{L}$ metres.

Area of cloth = $\text{L} \times 2 \text{ m}$

Setting the area equal to the LSA of the tent:

$\text{L} \times 2 \text{ m} = 198 \text{ m}^2$

Now, solve for $\text{L}$:

$\text{L} = \frac{198 \text{ m}^2}{2 \text{ m}}$

$\text{L} = 99 \text{ m}$

Therefore, 99 metres of the 2-m-wide cloth will be required to make the conical tent.

Step-by-Step Calculation Summary

The length of the cloth required is 99 m.

Revision Table: Conical Tent Calculations

Additional Information: Cone Geometry and Material Usage

When constructing a tent, the material used is typically a flat piece of fabric. For a conical tent, this fabric is cut from a larger piece. Imagine cutting a sector out of a circle; when you join the straight edges of the sector, it forms the curved surface of a cone. The radius of the original circle becomes the slant height of the cone, and the arc length of the sector becomes the circumference of the cone's base.

In this problem, we didn't need to worry about how the flat cloth is cut into a sector shape, only that its total area must match the required lateral surface area of the cone. The problem also explicitly states to "ignore wastage," which simplifies the calculation by assuming the entire area of the required length of cloth is used efficiently to form the tent surface.

The total surface area of a cone includes the base area ($\pi r^2$) plus the lateral surface area ($\pi r l$). However, for a tent, the base is usually open to the ground, so only the lateral surface area is relevant for the cloth needed.