Question

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

Answer 1

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

Answer 2

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