Answer:
Part a)
Given that
[tex]f(X)=\int_{0}^{X}\frac{2}{(x+2)^{2}}dx[/tex]
Thus the probability that a randomly selected clock lasts for at most 6 years is
[tex]f(6)=\int_{0}^{6}\frac{2}{(x+2)^{2}}dx[/tex]
[tex]f(6)=[\frac{2}{(x+2)}]_{0}^{6}\\\\f(6)=0.75[/tex]
Part b)
The probability that a clock lasts between 6 to 9 years equals
f(9)-f(6)
[tex]f(9)=\int_{0}^{9}\frac{2}{(x+2)^{2}}dx\\\\f(9)=0.818[/tex]
Thus probability becomes 0.818-0.75 = 0.0681
part c )
See attached figure
For shaded region of part a see attached figure
