Question

It would be really helpful if anyone solved this for me !!!! It's differentiation.

Answer 1

Determine point A:

$y(x)=\frac{10}{2x+1}-2\\\frac{10}{2x+1}-2=0\implies x=2$

so A(2,0)

Find the function for the tangent:

$\frac{dy}{dx}=-\frac{-20}{(2x+1)^2}$

the tangent function is a line

$y=ax + b = \frac{dy}{dx}|_A\cdot x + b$

(with slope being the derivative evaluated at point A(2,0))

$\frac{dy}{dx}|_A=-\frac{4}{5}$ so a=-4/5.

Determine b by using what we know about A:

$y=-\frac{4}{5}x+b\\0=-\frac{4}{5}\cdot 2+b\implies b=\frac{8}{5}$

so the function is

$y = -\frac{4}{5}x+\frac{8}{5}\\5y + 4x = 8$

(i) this is demonstrated above

(ii) distance AC: we know A(2,0) and C(0,5/8), the distance is:

$\sqrt{(0-2)^2+(\frac{5}{8}-0)^2}=\sqrt{281}/8\approx 2.1$