Question

An umbrella is made by stitching 10 triangular pieces of cloth of two different colours each piece measuring 30cm, 60cm, 60cm. How much cloth of each colour is required for Umbrella?

• A. \( 1015 \sqrt{45} \)
• B. \( 1125 \sqrt{15} \)
• C. \( 1220 \sqrt{45} \)
• D. \( 1235 \sqrt{15} \)

Answer 1

The Correct Option is B

Solution:
1. Dimensions of Each Triangle:
- Each triangular piece has sides of 30 cm, 60 cm, and 60 cm.
2. Type of Triangle:
  - This is an isosceles triangle with two equal sides of 60 cm each.
3. Area of One Triangle:
  - Using Heron's formula:
    - Semi-perimeter \( s = \frac{30 + 60 + 60}{2} = 75 \) cm
    - Area \( A = \sqrt{s(s - a)(s - b)(s - c)} \)
    - \( A = \sqrt{75(75 - 30)(75 - 60)(75 - 60)} \)
    - \( A = \sqrt{75 \times 45 \times 15 \times 15} \)
    - \( A = \sqrt{759375} \)
    - \( A = 375 \sqrt{3} \) cm²
4. Total Cloth Area for 10 Triangles:
  - Total area for 10 triangles = \( 10 \times 375 \sqrt{3} = 3750 \sqrt{3} \) cm²
5. Cloth of Each Colour:
  - Since there are two different colours, each colour will cover half the total area.
  - Cloth area for each colour = \( \frac{3750 \sqrt{3}}{2} = 1875 \sqrt{3} \) cm²
 Correct Answer:
B) \( 1125 \sqrt{15} \)

Answer 2

Each triangular piece is isosceles with sides 30 cm, 60 cm, 60 cm. Instead of Heron's formula, drop a perpendicular from the apex to the 30 cm base, splitting it into two equal halves of 15 cm each.

By the Pythagorean theorem, the height is:
\[h=\sqrt{60^{2}-15^{2}}=\sqrt{3600-225}=\sqrt{3375}=15\sqrt{15}\text{ cm}\]

Area of one triangle:
\[\frac{1}{2}\times30\times15\sqrt{15}=225\sqrt{15}\text{ cm}^{2}\]

Total area for 10 triangles:
\[10\times225\sqrt{15}=2250\sqrt{15}\text{ cm}^{2}\]

Since the umbrella uses two colours in equal shares, cloth required per colour:
\[\frac{2250\sqrt{15}}{2}=1125\sqrt{15}\text{ cm}^{2}\]

1. Option A \(1015\sqrt{45}\): Does not match \(1125\sqrt{15}\).
2. Option B \(1125\sqrt{15}\): Matches exactly.
3. Option C \(1220\sqrt{45}\): Does not match.
4. Option D \(1235\sqrt{15}\): Does not match.

The cloth required per colour is \(1125\sqrt{15}\text{ cm}^{2}\).

Hence, the correct answer is Option B: \(1125\sqrt{15}\).