A triangle is bounded by the x-and y-axes and must pass through P(2,2)
a.) find the area of the triangle as a function of x (Hint: let C be the point (0,2) and D be the point (2,0). Use the fact that ACP and PDB are similar triangles.)
b.) What is the domain of the function you found in a.?
To find the area of the triangle as a function of x, we can use the fact that triangles ACP and PDB are similar. This implies that the ratios of corresponding side lengths are equal.
Let's consider triangle ACP with vertices A(0,0), C(0,2), and P(2,2). The base of this triangle is AC, which has a length of 2. The height is the y-coordinate of point P, which is also 2.
Now, let's consider triangle PDB with vertices P(2,2), D(2,0), and B(0,0). The base of this triangle is BD, which has a length of 2. The height is the x-coordinate of point P, which is also 2.
Since the corresponding side lengths of triangles ACP and PDB are equal, we can use these side lengths to find the area of the triangle.
The area of triangle ACP is (1/2) * base * height = (1/2) * 2 * 2 = 2.
The area of triangle PDB is also 2.
Therefore, the area of the whole triangle bounded by the x- and y-axes and passing through P(2,2) is the sum of the areas of triangles ACP and PDB, which is 2 + 2 = 4.
To find the domain of the function we found in part a., we need to consider the possible values of x. Since the triangle is bounded by the x-axis, the x-coordinate of point P(2,2) is within the domain. So, the domain of this function is all real numbers.