An isosceles triangle is formed with a rod of length l1 and coefficient of linear expansion -1 for the base and two thin rods each of length l2 and coefficient of linear expansion -2 for the two pieces- if the distance between the apex and the midpoint of the base remain unchanged as the temperatures varied- we get l1-l2-x- 2- 1- Find - x-

The length of the line from apex to the midpoint of base= √(l_2^2 l_1^24) .As the temperature increases, the new length remains the same as the old length. ...

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Standard XII

Mathematics

Sum of Coefficients of All Terms

An isosceles ...

Question

An isosceles triangle is formed with a rod of length $l_{1}$ and coefficient of linear expansion $α_{1}$ for the base and two thin rods each of length $l_{2}$ and coefficient of linear expansion $α_{2}$ for the two pieces, if the distance between the apex and the midpoint of the base remain unchanged as the temperatures varied, we get $\frac{l_{1}}{l_{2}} = x \sqrt{\frac{α_{2}}{α_{1}}}$ . Find ' $x$ '.

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Solution

The length of the line from apex to the midpoint of base= $\sqrt{l_{2}^{2} - \frac{l_{1}^{2}}{4}}$ .
As the temperature increases, the new length remains the same as the old length.
$⟹ \sqrt{l_{2}^{2} (1 + α_{2} t)^{2} - \frac{l_{1}^{2}}{4} (1 + α_{1} t)^{2}} = \sqrt{l_{2}^{2} - \frac{l_{2}^{2}}{4}}$
$⟹ \frac{l_{1}}{l_{2}} = 2 \sqrt{\frac{α_{2}}{α_{1}}}$

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The length of the line from apex to the midpoint of base=√l22−l214.As the temperature increases, the new length remains the same as the old length.⟹√l22(1+α2t)2−l214(1+α1t)2=√l22−l224⟹l1l2=2√α2α1