Calculating Canvas for a Conical Tent

The question asks us to find the amount of canvas needed to make a conical tent with a given radius and height. The canvas forms the curved surface of the tent, which is the lateral surface area of the cone.

To find the lateral surface area of a cone, we use the formula:

\[ \text{Lateral Surface Area} = \pi \cdot r \cdot l \]

where \(r\) is the radius of the base and \(l\) is the slant height of the cone.

We are given the radius \(r = 6\) units and the height \(h = 8\) units. However, the formula requires the slant height \(l\). The radius, height, and slant height of a cone form a right-angled triangle, with the slant height as the hypotenuse. We can use the Pythagorean theorem to find the slant height.

Finding the Slant Height of the Cone

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (slant height, \(l\)) is equal to the sum of the squares of the other two sides (radius, \(r\), and height, \(h\)).

\[ l^2 = r^2 + h^2 \]

Given \(r = 6\) and \(h = 8\):

\[ l^2 = 6^2 + 8^2 \]

\[ l^2 = 36 + 64 \]

\[ l^2 = 100 \]

To find \(l\), we take the square root of 100:

\[ l = \sqrt{100} \]

\[ l = 10 \text{ units} \]

So, the slant height of the conical tent is 10 units.

Calculating the Amount of Canvas (Lateral Surface Area)

Now that we have the radius \(r = 6\) units and the slant height \(l = 10\) units, we can calculate the lateral surface area using the formula:

\[ \text{Lateral Surface Area} = \pi \cdot r \cdot l \]

\[ \text{Lateral Surface Area} = \pi \cdot 6 \cdot 10 \]

\[ \text{Lateral Surface Area} = 60\pi \text{ square units} \]

To get a numerical value, we use an approximate value for \(\pi\). Using the approximation \(\pi \approx \frac{22}{7}\):

\[ \text{Lateral Surface Area} \approx 60 \cdot \frac{22}{7} \]

\[ \text{Lateral Surface Area} \approx \frac{1320}{7} \]

Now, we perform the division:

\[ \frac{1320}{7} \approx 188.571428... \]

The question asks us to round the answer off to two places of decimals. Looking at the third decimal place (1), we round down.

\[ \text{Lateral Surface Area} \approx 188.57 \text{ square units} \]

Therefore, approximately 188.57 square units of canvas are needed to make the tent.

Summary of Conical Tent Canvas Calculation

Given:

• Radius \(r = 6\) units
• Height \(h = 8\) units

Steps:

1. Calculate the slant height \(l\) using the Pythagorean theorem: \(l = \sqrt{r^2 + h^2}\).
2. Calculate the lateral surface area using the formula: \(\text{LSA} = \pi \cdot r \cdot l\).
3. Round the result to two decimal places.

Calculation:

\[ l = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ units} \]

\[ \text{LSA} = \pi \cdot 6 \cdot 10 = 60\pi \]

Using \(\pi \approx \frac{22}{7}\):

\[ \text{LSA} \approx 60 \cdot \frac{22}{7} \approx 188.5714... \]

Rounded to two decimal places: 188.57 square units.

Revision Table: Conical Tent Calculations

Additional Information: Surface Area of Cones

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

• Lateral Surface Area: This is the area of the curved surface connecting the base to the apex. It's the part that would be made of canvas for a tent. The formula is \(\text{LSA} = \pi r l\).
• Base Area: For a cone with a circular base, the area of the base is \(\text{Base Area} = \pi r^2\).
• Total Surface Area: This is the sum of the lateral surface area and the base area. For a closed cone, the total surface area is \(\text{TSA} = \text{LSA} + \text{Base Area} = \pi r l + \pi r^2 = \pi r (l + r)\). For an open tent like the one in the question, only the lateral surface area is relevant for the canvas needed.
• Volume of a Cone: The volume of a cone is given by \(V = \frac{1}{3} \pi r^2 h\).

Understanding these formulas is crucial for solving problems involving cones.