A joker’s cap is in the form of a right circular cone of base radius 7cm and height 27cm. Find the area of the sheet required to make 10 such caps.

To find the area of the sheet required to make 10 joker's caps, we first need to calculate the surface area of one cap, which is a right circular cone. The surface area of a cone consists of the base area and the lateral surface area.

Step 1: Calculate the Base Area

The base area (A_base) of the cone can be calculated using the formula:

A_base = πr²

Where:

• r = radius of the base = 7 cm

So, substituting the values:

A_base = π × (7 cm)² = π × 49 cm² ≈ 153.94 cm²

Step 2: Calculate the Slant Height

Next, we need the slant height (l) of the cone, which can be found using the Pythagorean theorem:

l = √(r² + h²)

Where:

• h = height of the cone = 27 cm

Calculating the slant height:

l = √((7 cm)² + (27 cm)²) = √(49 cm² + 729 cm²) = √778 cm² ≈ 27.86 cm

Step 3: Calculate the Lateral Surface Area

The lateral surface area (A_lateral) of the cone is given by:

A_lateral = πrl

Substituting the values:

A_lateral = π × 7 cm × 27.86 cm ≈ 615.75 cm²

Step 4: Total Surface Area of One Cap

The total surface area (A_total) of one cap is the sum of the base area and the lateral surface area:

A_total = A_base + A_lateral

A_total ≈ 153.94 cm² + 615.75 cm² ≈ 769.69 cm²

Step 5: Area for 10 Caps

To find the area required for 10 caps, simply multiply the total surface area of one cap by 10:

Area for 10 caps = 10 × A_total ≈ 10 × 769.69 cm² ≈ 7696.9 cm²

Final Answer

The area of the sheet required to make 10 joker's caps is approximately 7696.9 cm².