Question

Find the area of the region bounded by the $y$-axis, the line $y=6$, and the line $y = \frac{1}{2}x$.

Answer 1

To find the area of the region bounded by the y-axis, the line y=6, and the line y=1/2x, which forms a right triangle, we calculate it as Area = 1/2 × base × height = 1/2 × 12 × 6 = 36 square units.

Step-by-step explanation:

To find the area of the region bounded by the y-axis, the line y=6, and the line y =

`(1)/(2)x`

, we first need to identify the shape of this region. It forms a right triangle where the base is on the x-axis, and the height is along the y-axis.

First, we set y =

`(1)/(2)x`

equal to y = 6 to find the x-coordinate of the point where the two lines intersect:

y =
`(1)/(2)x` = 6x = 12

Now we know the base of the triangle extends from x = 0 to x = 12 along the x-axis, and the height is y = 6.

The area of a triangle is given by:

Area =
`(1)/(2)` × base × height

So the area of our region is:

Area =
`(1)/(2)` × 12 × 6Area =
`(1)/(2)` × 72Area = 36

Therefore, the area of the region bounded by the given lines is 36 square units.

Answer 2

For a better understanding of the solution provided here please go through the diagram in the file that has been attached.

As can be clearly seen from the question and the graph, the area is bounded by a straight line
`y=(1)/(2)x` which passes through the origin, the y axis and the horizontal line
`y=6`.

As we can see from the graph, this is a right triangle. To find the point of intersection of the horizontal line y=6 and the line
`y=(x\1)/(2)x`, we will have to equate the two as:

`(1)/(2)x=6`

Therefore,
`x=12`

Thus,
`y=(1)/(2)x= (1)/(2)* 12=6`

Hence the point of intersection is (12,6) which is depicted on the graph too.

As this is a case of a simple right triangle, the area of the bounded region will thus be:

`Area=(1)/(2)* base* height`

Let the base be the y=6 line whose length is 12 and let height be the y axis whose length is 6, thus, the area will be:

`Area=(1)/(2)* 6* 12=36` squared units.