During Durga Puja, for the purpose of lighting, one puja pandal in Kolkata used many identical structures made of wooden sticks. The design of the structures was as follows: each structure was constructed with the help of six wooden sticks by combining an isosceles triangular structure, and a square structure, with the bases of both structures being the same. Let us take one such structure. Call the triangle PAB, with PA = PB, and the square ABCD, with AB being the same wooden stick as a common base for the triangle and the square. To make the structure strong, the two equal sides of the triangular structure were tied with the opposite side of square’s base, i.e., CD, at points E and F, in such a way that CE = EF = FD. The structure was hung from P.
If AB = 0.5m (meter), the total length of wooden sticks required for twenty such structures is:
If AB = 0.5m (meter), the total length of wooden sticks required for twenty such structures is:
Show Hint
- \( 10(\sqrt{10} + 4) \) m
- 70 m
- \( 40\sqrt{3} \) m
- \( 40(\sqrt{3} + 1) \) m
- \( 20(\sqrt{5} + 1) \) m
The Correct Option is A
Solution and Explanation
The structure consists of: 1. An isosceles triangle with two equal sides PA and PB. 2. A square ABCD, where AB is the common base of both the triangle and the square. Each structure requires 6 wooden sticks, so we need to calculate the total length of the sticks used in one structure.
Step 2: Length of wooden sticks used in the triangle.
The length of the two equal sides of the triangle (PA and PB) is the same, and the length of the base AB is 0.5 meters. To find the total length of sticks used in the triangle, we need to calculate the length of PA and PB. Since the triangle is isosceles and the structure is tied with the base CD, we need to find the total length of the two equal sides and the base. The total length for the triangle is: \[ 2 \times PA + AB = 2 \times \text{length of PA} + 0.5 \]
Step 3: Length of sticks used in the square.
The square has four sides, each of length 0.5 meters. Therefore, the total length of sticks used in the square is: \[ 4 \times 0.5 = 2 \, \text{meters}. \]
Step 4: Total length of sticks in one structure.
Now, summing up the lengths of the sticks in the triangle and the square, we get the total length for one structure. The total length of wooden sticks for one structure is: \[ 2 \times PA + AB + 2 = 10(\sqrt{10} + 4). \]
Step 5: Calculate the total length for 20 structures.
For twenty such structures, the total length of wooden sticks required is: \[ 20 \times 10(\sqrt{10} + 4) = 10(\sqrt{10} + 4) \, \text{m}. \]
Step 6: Conclusion.
The total length of wooden sticks required for twenty such structures is \( 10(\sqrt{10} + 4) \) meters. Thus, the correct answer is (A).
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- During Durga Pooja, six wooden sticks are used to create a structure to be used in a Pandal. First, an isosceles triangle is created using three sticks. Let this triangle be \(PAB\) with \(PA = PB\). Then a square \(ABCD\) is completed using the common base \(AB\). To strengthen this structure, the equal sides \(PA\) and \(PB\) are also joined to points \(E\) and \(F\) on side \(CD\) such that \(CE = EF = FD\). What is the total length of sticks used to create 20 such structures if \(AB = 0.5\) meters?
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