ind thearea bounded by y= -0.5x+6, y= 1/4 x and the y-axis.
find the area* sorry
Make a sketch to see that you have a triangle
Find the intersection of the two non-vertical lines
1/4 x = - 1/2 x + 6
times 4
x = -2x + 24
3x = 24
x = 8
so we could consider the length along the y-axis as the base, and then the height would be 8
area = (1/2)(6)(8) = 24
To find the area bounded by the given lines and the y-axis, we need to determine the x-coordinate where these lines intersect. We can set the equations of the lines equal to each other and solve for x:
-0.5x + 6 = (1/4)x
To solve this equation, let's simplify it:
-0.5x + 6 = 0.25x
Add 0.5x to both sides:
6 = 0.75x
Divide both sides by 0.75:
x = 8
So, the lines intersect at x = 8.
Next, we need to determine the y-coordinates of the points of intersection. We can substitute this x-value (x = 8) into either of the original equations to get the y-coordinate. Let's use the equation y = -0.5x + 6:
y = -0.5(8) + 6
y = -4 + 6
y = 2
Hence, the two lines intersect at the point (8, 2).
To find the area bounded by these lines and the y-axis, we have a trapezoid. The height of the trapezoid is given by the difference between the y-coordinates of the points of intersection (2 - 0 = 2).
The formula for the area of a trapezoid is (1/2) × (sum of parallel sides) × height. In this case, one of the parallel sides is given by the equation y = -0.5x + 6, so its length is 6. The other parallel side is the y-axis, so its length is 0.
Plugging these values into the formula, we get:
Area = (1/2) × (6 + 0) × 2
Area = 6 × 1
Area = 6 square units
Therefore, the area bounded by the given lines and the y-axis is 6 square units.