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A sector of a circle of radius 4cm is bent to form a cone. Calculate the total surface area of the object by whose base radius is 7cm.

Mathematics

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Answer

748 cm2\text{748 cm}^2

Step-by-step solution

6 steps

  1. Identify the given dimensions of the cylinder.

    The radius of the cylinder is r=7 cmr = 7 \text{ cm}.
    The height of the cylinder is h=10 cmh = 10 \text{ cm}.

  2. Write down the formula for the total surface area of a cylinder.

    The total surface area (AA) of a cylinder is given by the formula:

    A=2πr2+2πrhA = 2\pi r^2 + 2\pi rh

    where 2πr22\pi r^2 is the area of the two circular bases and 2πrh2\pi rh is the lateral surface area.

  3. Substitute the given values into the formula.

    Using π=227\pi = \frac{22}{7}:

    A=2×227×(7 cm)2+2×227×7 cm×10 cmA = 2 \times \frac{22}{7} \times (7 \text{ cm})^2 + 2 \times \frac{22}{7} \times 7 \text{ cm} \times 10 \text{ cm}

  4. Calculate the area of the bases.

    2πr2=2×227×49 cm22\pi r^2 = 2 \times \frac{22}{7} \times 49 \text{ cm}^2 2πr2=2×22×7 cm22\pi r^2 = 2 \times 22 \times 7 \text{ cm}^2 2πr2=308 cm22\pi r^2 = 308 \text{ cm}^2

  5. Calculate the lateral surface area.

    2πrh=2×227×7 cm×10 cm2\pi rh = 2 \times \frac{22}{7} \times 7 \text{ cm} \times 10 \text{ cm} 2πrh=2×22×10 cm22\pi rh = 2 \times 22 \times 10 \text{ cm}^2 2πrh=440 cm22\pi rh = 440 \text{ cm}^2

  6. Add the areas to find the total surface area.

    A=308 cm2+440 cm2A = 308 \text{ cm}^2 + 440 \text{ cm}^2 A=748 cm2A = 748 \text{ cm}^2

    The total surface area of the object is 748 cm2\text{748 cm}^2.

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