Question

A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function

$f\left(x\right)=\left\{\begin{array}{ll}5{(1-x)}^4& 0<x<1\\ 0& otherwise\end{array}\right.$

what must the capacity of the tank be so that the probability of the supply being exhausted in a given week is role="math" localid="1646634562935" $.01?$

Short answer

the capacity of the tank$c=0.602$

Step-by-step solution

 Given Information.

$f\left(x\right)=\left\{\begin{array}{ll}5{(1-x)}^4& 0<x<1\\ 0& otherwise\end{array}\right.$

Explanation.

$c\Rightarrow p(x>c)=0.07$

role="math" localid="1646635699895" ${\int }_c^{1}5{(1-x)}^{4}dx=0.01$

${{\left[\frac{-5{(1-x)}^5}{5}\right]}_c}^1=0.07$

Explanation.

$1-c={(0.01)}^{\frac{1}{5}}$

$c={(1-0.01)}^{\frac{1}{5}}$

$=0.602$