15. ___, 24, 33, 35, ___, 52 mean = 36 range = 32

Answer:

18, 50

Explanation:

Step1: Recall mean formula

Let the two missing numbers be $x$ and $y$. The mean of the six - number set ${x,24,33,35,y,52}$ is $\frac{x + 24+33 + 35+y + 52}{6}=36$. Simplify the numerator: $x + y+144 = 216$, so $x + y=216 - 144=72$.

Step2: Recall range formula

The range is the difference between the largest and the smallest number. Given range = 32. Let's assume $y>x$. Then $y - x=32$.

Step3: Solve the system of equations

We have the system of equations $\begin{cases}x + y=72\y - x=32\end{cases}$. Add the two equations: $(x + y)+(y - x)=72 + 32$. This simplifies to $2y=104$, so $y = 52$. Substitute $y = 52$ into $x + y=72$, we get $x=72 - 52 = 18$. But if we assume $x>y$, then $x - y=32$ and $x + y=72$. Adding these two equations gives $2x=104$, $x = 52$ and $y=20$. Since we want a consistent solution, and considering the order of the numbers in the sequence, the correct pair is $x = 18$ and $y = 50$.