Answer
a. 310 hours;
b. $\sigma \approx 267$;
c. $P(\mu + 1\sigma \leq t)\approx0.206$
d. $m\approx240$
Work Step by Step
We are given $f(x)=\frac{1}{58\sqrt t}$
for t in $[1,900]$
a. $\mu=\int^{900}_{1} t(\frac{1}{58\sqrt t})dt$
$=\int^{900}_{1}( \frac{1}{58}\sqrt t)dt$
$=\frac{1}{87}t^\frac{3}{2}|^{900}_{1}$
$=\frac{931}{3} \approx 310$
b. $Var(X) =\int^{900}_{1} t^{2}(\frac{1}{58\sqrt t})dt - (310)^{2}$
$=\int^{900}_{1}( \frac{1}{58}t\sqrt t)dt-96306$
$=\frac{1}{145}.t^\frac{5}{2}|^{900}_{1} -96306$
$\approx 71280$
$\rightarrow \sigma= \sqrt 71280 \approx 267$
c. $P(\mu + 1\sigma \leq t)=\int^{900}_{\mu + 1\sigma}(\frac{1}{58\sqrt t})dt$
$=\frac{1}{29}t^\frac{1}{2}|^{900}_{577} \approx0.206$
d. $\int^{m}_{1}(\frac{1}{58\sqrt t})dt=\frac{1}{2}$
$\frac{1}{29}(m^{\frac{1}{2}}-1)=\frac{1}{2}$
$m^{\frac{1}{2}}=\frac{31}{2}$
$m\approx240$