Answer:
To find the enclosed area within the boundaries defined by the equations, we first need to identify the region formed by these boundaries. Let's graph the equations and determine the area enclosed.
The equations are:
y = x^2 - 6
y = 2 - 1/3x
x = 10
x = -5
To graph these equations, we can plot the points and connect them to form the boundaries.
Here's the graph:
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| * | * | * * | * * | * * * * | * * * * | * * * * | * * * * * | * * * * * | * * * * * |________________ -5 10
From the graph, we can see that the enclosed area is a triangle formed by the boundaries. To calculate the area, we can find the base and height of the triangle.
The base of the triangle is the horizontal distance between the x-coordinates of -5 and 10, which is 10 - (-5) = 15 units.
The height of the triangle is the vertical distance between the y-coordinate of the points where the lines intersect, which is (2 - 1/3(10)) - (10^2 - 6) = 2 - 10/3 - (100 - 6) = -3.67.
Now, we can calculate the area of the triangle:
Area = (1/2) * base * height = (1/2) * 15 * (-3.67) = -27.51 square units
The enclosed area within the boundaries defined by the equations is approximately -27.51 square units.