NCERT Class X Chapter 12: Surface Areas And Volumes Example 4

NCERT Class X Chapter 12: Surface Areas And Volumes Example 4

Question:

Mayank made a bird-bath for his garden in the shape of a cylinder with a hemispherical depression at one end. The height of the cylinder is 1.45 m and its radius is 30 cm. Find the total surface area of the bird-bath. (Use \( \pi = \frac{22}{7} \))

Given:

  • Height of the cylinder, \( h = 1.45\,\text{m} = 145\,\text{cm} \)
  • Radius of the cylinder, \( r = 30\,\text{cm} \)
  • \( \pi = \frac{22}{7} \)

To Find:

Total surface area of the bird-bath.

Formula:

  • Total surface area = Curved surface area of cylinder + Area of base + Curved surface area of hemisphere
  • Curved surface area of cylinder: \( 2\pi r h \)
  • Area of base: \( \pi r^2 \)
  • Curved surface area of hemisphere: \( 2\pi r^2 \)

Solution:

Step 1: Write the expression for total surface area.

$$ \text{Total surface area} = 2\pi r h + \pi r^2 + 2\pi r^2 = 2\pi r h + 3\pi r^2 $$

Step 2: Substitute the given values (\( r = 30\,\text{cm} \), \( h = 145\,\text{cm} \), \( \pi = \frac{22}{7} \)).

$$ \text{Total surface area} = 2 \times \frac{22}{7} \times 30 \times 145 + 3 \times \frac{22}{7} \times (30)^2 $$

Step 3: Calculate each term separately.

$$ 2 \times \frac{22}{7} \times 30 \times 145 = \frac{2 \times 22 \times 30 \times 145}{7} = \frac{191400}{7} = 27342.86 $$ $$ 3 \times \frac{22}{7} \times 30^2 = \frac{3 \times 22 \times 900}{7} = \frac{59400}{7} = 8485.71 $$

Step 4: Add the results to get the total surface area in cm².

$$ \text{Total surface area} = 27342.86 + 8485.71 = 35828.57\,\text{cm}^2 $$

Step 5: Convert the area from cm² to m².

$$ 1\,\text{m}^2 = 10{,}000\,\text{cm}^2 $$ $$ \text{Total surface area} = \frac{35828.57}{10000} = 3.582857\,\text{m}^2 $$

Step 6: Round off to two decimal places.

$$ \text{Total surface area} \approx 3.58\,\text{m}^2 $$

Result:

The total surface area of the bird-bath is approximately \( 3.58\,\text{m}^2 \).

Next question solution:

Exercise 12.1 Question 1
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