6. Sketch the region enclosed by the graphs of x = 0, 6y-5x=0 and x+3y=21. Find the area

7. Find the volume of the solid formed when the region is revolved around the y axis

X = 0

6y - 5x = 0 so it intersects x=0 at (0,0)

x + 3y = 21

so it intersect x=0 at (0,7)

multiply by 2

2x + 6y = 42

subtracting 6y - 5x = 0

7x = 42

x = 6

so the third line intersects w second line at (6,5)

6. area of triangle = 1/2*base*height

= 1/2*7*6

=21

7. volume = volume of 2 cones

cone volume = 1/3*pi*radius^2*height

lower cone: radius=6, height=5

upper cone: radius=6, height=7-5=2

volume = 1/3*pi*6^2 * (5+2)

=84pi

=263.89

Given three equations, x=0

6y-5x=0

x+3y=21

The lines cross at:

x=0, y=0

x=0, y=7

x=6, y=5

so base of triangle is the y-axis between 0 and 7

height is x=6

area=6*7/2=20.5

volume=1/3 * (base area) * height

=1/3 * (pi*6^2) * 7

=84pi

=263.9