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'll begin by visualizing the solid and understanding its properties.
The given region bounded by the graphs of y = 3x, y = 6, and x = 0 forms a triangular base for the solid. The three vertices of the triangle are (0, 0), (2, 6), and (2, 3).
The cross-sections of the solid are rectangles perpendicular to the x-axis, and their perimeter is given as 20 units.
To determine the height and width of each rectangle, we need to find the equation of a cross-section at a general x-coordinate, let's call it x.
Since the base of the solid is a triangle, the height of each rectangle will be the difference between the y-values of the upper and lower boundaries of the solid at the specific x-coordinate.
The lower boundary is the line y = 3x, and the upper boundary is the line y = 6.
So, the height of each rectangle is 6 - 3x.
Now, let's consider the width of each rectangle. The problem states that the perimeter of each rectangle is 20 units. Since a rectangle has two equal widths, the width of each rectangle will be 20 / 2 = 10 units.
To calculate the volume of the solid, we'll integrate the area of each cross-section from the starting x-value to the ending x-value.
The starting x-value is 0 (where the base triangle touches the y-axis), and the ending x-value can be found by solving the equation y = 3x for y = 6 (the point where the upper boundary intersects the triangle).
Substituting y = 6 into the equation y = 3x, we get:
6 = 3x
x = 2
So, the limits of integration are from x = 0 to x = 2.
The volume, V, is given by the integral of the area of each cross-section, which is equal to the height times the width, integrated with respect to x:
V = ∫(0 to 2) (6 - 3x) * 10 dx
Evaluating this integral will give us the volume of the solid.