Area = 9*pi
To find the area of the red cross section, we need to use the formula for the area of a circle, which is as follows:
Area = pi * r^2, where r is the radius of the circe. Here, we see that we are not directly given the radius of the red cross section, however we are given the radius of the base of the cone. If we look at the cone 2-dimensionally, we see that the base of the cone forms a right triangle with the vertex of the cone, and the red cross section makes a smaller right triangle with the vertex of the cone inside the bigger right triangle.
So, now we have 2 right triangles, one of which sits inside the other. We know that the entire height of the right triangle is 15 ft (or 9+6) and the height of the small right triangle is 9 ft. Since the two triangles share a vertex, angle, and are both right triangles, we know that these two triangles are similar triangles, therefore the sides are proportional.
We can use the proportionality to find the radius of the red cross section. We can set up the following proportion:
5 / 15 = x / 9
Solving for x, we get:
9 * 5 = 15 *x
45 = 15 * x
x = 3
Now, we have the radius of the red cross section, and we can plug into our area equation to find the area of the cross section as follows:
A = pi * 3^2
A = 9*pi