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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 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...
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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).
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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