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Question

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fences are of same length x. The maximum area enclosed by the park is


A

x38

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B

x22

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C

πx2

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D

3x22

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Solution

The correct option is B

x22


Explanation for the correct option:

Step 1: Find the relation between angle and side:

Assume the two equal sides AB and AC have length x.

Also, assume ACB=θ

Draw a perpendicular line AD.

So, sinθ=ADx

And cosθ=DCx

Step 2: Find the area of the smaller triangle:

Use the area of triangle formula to find the area of ADC.

Ar=12×AD×DC=12×xsinθ×xcosθ=x22sinθcosθ arABC=12·bh

Step 3: Find the area of the whole triangle:

Use the obtained area to get the area of the bigger triangle.

ArABC=2ArADC

Find the area of ABC using the above relation.

ArABC=2·x22sinθcosθ=x22sin2θ

Step 4: Maximize the area of ABC:

Maximize the area of the triangle by maximizing sin2θ.

Maximum value of sin2θ=1

So, maximum value of ArABC=x22

Hence, option (B) is the correct answer.


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