Find the volume of the solid whose base of a solid is the region bounded bythegraphsofy=3x,y=6,andx=0. Thecross␣sections perpendicular to the x ␣ axis are rectangles of perimeter 20.
To find the volume of the solid, we need to integrate the cross-sectional areas perpendicular to the x-axis along the given boundaries.
First, let's analyze the information provided. The base of the solid is the region bounded by the graphs of y = 3x, y = 6, and x = 0. We need to find the limits of integration for x in order to calculate the total volume.
The graph of y = 3x represents a straight line passing through the origin with a slope of 3. The graph of y = 6 is a horizontal line parallel to the x-axis. The line x = 0 is the y-axis.
We can see that the region bounded by these three graphs is a triangular-shaped area. To find the limits of integration for x, we need to determine where the triangular region begins and ends.
First, let's find the x-coordinate where the line y = 3x intersects the line y = 6:
3x = 6
x = 2
So, the region starts at x = 0 and ends at x = 2.
Now, let's focus on the cross-sections perpendicular to the x-axis. It is stated that these cross-sections are rectangles with a perimeter of 20.
The perimeter of a rectangle with length L and width W is given by: P = 2L + 2W.
In this case, since the cross-sections are rectangles with a perimeter of 20, we can express the relationship between the length and width as follows: 2L + 2W = 20.
To simplify the problem, let's express the width W in terms of the length L: W = (20 - 2L) / 2.
Now, we can calculate the cross-sectional area A of each rectangle by multiplying the length L and width W: A = L * W.
Substituting the width expression into the area formula, we get: A = L * ((20 - 2L) / 2).
To determine the total volume, we need to integrate the cross-sectional areas with respect to x: V = ∫[0 to 2] A dx.
Plugging in the expression for A, we have: V = ∫[0 to 2] (L * ((20 - 2L) / 2)) dx.
Evaluating this integral will give us the volume of the solid.