1. Interpret the region

The point $A(x,y)$ is uniformly chosen from the region bounded by:

• the $y$-axis: $x=0$
• the line $2y+x=6 \Rightarrow x=6-2y$
• the line $5x-6y=30 \Rightarrow x=6+\frac{6y}{5}$

We first find the vertices of the bounded region.

2. Find intersection points

(i) Intersection of $x=0$ with $2y+x=6$

Putting $x=0$: $2y=6 \Rightarrow y=3$ So one point is: $(0,3)$

(ii) Intersection of $x=0$ with $5x-6y=30$

Putting $x=0$: $-6y=30 \Rightarrow y=-5$ So another point is: $(0,-5)$

(iii) Intersection of the two lines

Solve $x+2y=6$ $5x-6y=30$ From the first equation, $x=6-2y$ Substitute into the second: $5(6-2y)-6y=30$ $30-10y-6y=30$ $-16y=0 \Rightarrow y=0$ Then $x=6$ So the third point is: $(6,0)$

Thus the bounded region is the triangle with vertices: $(0,3),\ (0,-5),\ (6,0)$

3. Total area of the triangle

Take the side along the $y$-axis from $(0,-5)$ to $(0,3)$ as base. Its length is: $3-(-5)=8$ The perpendicular distance of $(6,0)$ from the line $x=0$ is $6$.

Hence, $\text{Total area}=\frac{1}{2}\times 8 \times 6=24$

4. Find the part where $y<1$

The line $y=1$ cuts the triangle. We need the area of the portion below this line.

Find where $y=1$ meets the two slanted sides.

On $x+2y=6$

$x+2(1)=6 \Rightarrow x=4$ So intersection point is $(4,1)$.

On $5x-6y=30$

$5x-6(1)=30 \Rightarrow 5x=36 \Rightarrow x=\frac{36}{5}$ But this point is not on the bounded segment of this line inside the triangle, because the segment in the triangle runs from $(0,-5)$ to $(6,0)$, where $y$ ranges only from $-5$ to $0$. So $y=1$ does not meet that side of the triangle.

Thus, the region with $y>1$ is the small triangle with vertices: $(0,3),\ (0,1),\ (4,1)$

Its area is: $\frac{1}{2}\times 2 \times 4=4$

Therefore, area where $y<1$ is: $24-4=20$

5. Probability

Since the point is uniformly distributed in the triangular region, $P(y<1)=\frac{\text{Area where } y<1}{\text{Total area}}=\frac{20}{24}=\frac{5}{6}$

6. Option check

$\frac{5}{6}$ corresponds to Option B.

7. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

So, the derived answer agrees with the stored answer.