consider the given density curve. what is the value of the median? 3 4 4.5 6

Answer:

4.5

Explanation:

Step1: Recall median property

For a density - curve, the median divides the area under the curve into two equal parts.

Step2: Calculate area of rectangle

The density curve is a rectangle with height $h=\frac{1}{3}$ and base from $x = 3$ to $x = 6$. The area of the whole rectangle $A=h\times b=\frac{1}{3}\times(6 - 3)=1$.

Step3: Let median be $m$

We want to find $m$ such that $\int_{3}^{m}\frac{1}{3}dx = 0.5$.

Step4: Evaluate integral

Integrating $\int_{3}^{m}\frac{1}{3}dx=\frac{1}{3}(m - 3)$.

Step5: Solve for $m$

Set $\frac{1}{3}(m - 3)=0.5$. Then $m-3 = 1.5$, so $m=4.5$.