Final answer:
To find the volume of the given solid, we need to calculate the volume of a triangular prism. We can do this by finding the area of the triangle as the base and the height as the difference in z-values between the surface and the triangle's plane. Then, we multiply the base area by the height to find the volume of the triangular prism.
Step-by-step explanation:
To find the volume of the given solid, we first need to understand the shape of the solid. The solid is under the surface z = 5xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2). Visualizing this shape, we see that it is a triangular prism.
The volume of a triangular prism can be calculated by multiplying the base area by the height. In this case, the base area is the area of the triangle, which can be found using the formula for the area of a triangle. The height of the prism is the difference in z-values between the surface and the triangle's plane.
So, let's calculate:
- Find the area of the triangle using the formula: Area = 1/2 * base * height. In this case, the base length is the distance between (1, 1) and (4, 1), which is 4 - 1 = 3. The height is the distance between (1, 1) and (1, 2), which is 2 - 1 = 1. Therefore, the area of the triangle is 1/2 * 3 * 1 = 1.5 square units.
- Find the height of the prism, which is the difference in z-values between the surface and the triangle's plane. The surface is given by z = 5xy. At the vertices of the triangle, z = 5*1*1 = 5. The z-value on the plane of the triangle is 0. Therefore, the height of the prism is 5 - 0 = 5 units.
- Multiply the base area by the height to find the volume of the triangular prism: Volume = Area * Height = 1.5 * 5 = 7.5 cubic units.