Question:
In the graph given:
The solid curve is the graph of $ = 2.cosh (\frac{}{2}$)
and the dotted line is $ = 8x$
a) Calculate the value of when the two graphs intersect, correct to 4 .significant figures.
b) Hence calculate the area of the shaded region
So i'm to solve simultanious equations to find the two intersects on the graph.
Can anyone show me how to solve these please?
$2.cosh (\frac{}{2}) = 8x$
$$\cosh \frac {x}{2} = 4$$ $$\implies \frac {e^\frac {x}{2} + e^{-\frac {x}{2}}}{2} = 4$$ $$\frac {e^x +1}{e^\frac {x}{2}} = 8$$ $$e^x -8e^\frac {x}{2} + 1 = 0$$ Let $e^\frac {x}{2} = u $. Then our equation becomes $$u^2-8u + 1 = 0$$
Can you solve the quadratic equation and simplify?