Let $\displaystyle {p}{\left({t}\right)}$ be the probability density function given by $\displaystyle {p}{\left({t}\right)}=\alpha\cdot{f{{\left({t}\right)}}}$ where $\displaystyle {f{{\left({t}\right)}}}={\left\lbrace\begin{array}{ccc} {10}{x}+{9}&\text{if}&{0}\leq{t}\leq{10}\\-{5}{x}+{159}&\text{if}&{10}<{t}<{30}\\{0}&&\text{otherwise}\end{array}\right.}$

Find the value of $\displaystyle \alpha$ that makes $\displaystyle {p}{\left({t}\right)}$ a probability density function:
$\displaystyle \alpha$ = Preview Question 6 Part 1 of 3  

Find the median value of $\displaystyle {p}{\left({t}\right)}$:
median = Preview Question 6 Part 2 of 3  

Find the following probability:
$\displaystyle {P}{\left({5.9}<{X}<{23.5}\right)}$ = Preview Question 6 Part 3 of 3  

 
 
 
Enter your answer as a number (like 5, -3, 2.2) or as a calculation (like 5/3, 2^3, 5+4). If you enter your answer as a decimal, your answer should be accurate to at least 6 decimal places.