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CLAT Exam  >  CLAT Questions  >  Sides of a triangle are 6, 10 and x for what ... Start Learning for Free
Sides of a triangle are 6, 10 and x for what value of x is the area of the △ the maximum?
  • a)
    8 cm2
  • b)
    9 cm2
  • c)
    12 cm2
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?
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Sides of a triangle are 6, 10 and x for what value of x is the area of...
Explanation:
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( s \) is the semi-perimeter of the triangle, and \( a, b, c \) are the lengths of the three sides.
Given that the sides of the triangle are 6, 10, and x, the semi-perimeter (\( s \)) is calculated as:
\[ s = \frac{6 + 10 + x}{2} = \frac{16 + x}{2} = 8 + \frac{x}{2} \]
Now, substituting the values of \( s \), \( a \), \( b \), and \( c \) into the area formula, we get:
\[ \text{Area} = \sqrt{(8 + \frac{x}{2})(8 - \frac{x}{2})(2)(4)(6)} \]
\[ = \sqrt{(64 - \frac{x^2}{4})(48)} \]
\[ = \sqrt{3072 - 12x^2} \]
To find the maximum area, we need to find the critical points of the function. Taking the derivative of the area with respect to \( x \) and setting it to zero, we get:
\[ \frac{d}{dx}(\sqrt{3072 - 12x^2}) = 0 \]
\[ -\frac{6x}{\sqrt{3072 - 12x^2}} = 0 \]
\[ x = 0 \]
However, since the side of a triangle cannot be zero, the maximum area does not exist for this triangle with sides 6, 10, and x. Hence, the correct answer is option None of these (D).
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Sides of a triangle are 6, 10 and x for what value of x is the area of the the maximum?a)8 cm2b)9 cm2c)12 cm2d)None of theseCorrect answer is option 'D'. Can you explain this answer?
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