SOLUTION: Find the # of square units contained in the region bordered by the graphs of the functions y=3x, x=3y, 3x+y=30?

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Question 305820: Find the # of square units contained in the region bordered by the graphs of the functions y=3x, x=3y, 3x+y=30?
Found 2 solutions by Fombitz, CharlesG2:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!

Find the area of triangle OAC and subtract from it the area of triangle OAB.
Find the intersection points A and C.
+3x=30-3x+
+6x=30
+x=5
+y=3x=15
(5,15) is point C.
+x%2F3=30-3x
+x=90-9x
+10x=90
+x=9
+y=x%2F3=3
(9,3) is point B.
The area of a triangle
A=%281%2F2%29bh
h for each is the y-coordinate of points B and C.
.
.
.
A%28OAC%29=%281%2F2%29%28OA%29%2815%29=%281%2F2%29%2810%29%2815%29=75
A%28OAB%29=%281%2F2%29%28OA%29%283%29=%281%2F2%29%2810%29%283%29=15
A=A%28AOC%29-A%28OAB%29=75-15=60

Answer by CharlesG2(834) About Me  (Show Source):
You can put this solution on YOUR website!
Find the # of square units contained in the region bordered by the graphs of the functions y=3x, x=3y, 3x+y=30?
put all 3 in y = mx + b form where m is the slope and b is the y-intercept (vertical intercept or point (0,b)):
1. y = 3x or y = 3x + 0
2. x = 3y --> x/3 = y or y = (1/3)x + 0
3. 3x + y = 30 --> y = -3x + 30
1/3 = -1 * 1/(-3) = -1 * -1/3 = 1/3 so equation 2 and 3 are perpendicular (the slopes are negative reciprocals)
+graph%28+300%2C+300%2C+-5%2C+20%2C+-5%2C+20%2C+3%2Ax%2C+x%2F3%2C+-3%2Ax+%2B+30%29+
this is a right triangle
equation 1 and 2 intersect at (0,0)
equation 2 and 3 intersect at:
(1/3)x = -3x + 30
x = -9x + 90
10x = 90
x = 9, eq.1 y = 3, eq. 2 y = 3 (9,3)
equation 1 and 3 intersect at:
3x = -3x + 30
6x = 30
x = 5, eq.1 y = 15, eq. 2 y = 15 (5,15)
from (0,0) to (9,3) is the base length is:
base = sqrt((9-0)^2 + (3-0)^2) = sqrt(9^2 + 3^2)
base = sqrt(81 + 9) = sqrt(90) = 3sqrt(10)
from (9,3) to (5,15) is the height length is:
height = sqrt((5-9)^2 + (15-3)^2) = sqrt((-4)^2 + 12^2)
height = sqrt(16 + 144) = sqrt(160) = 4sqrt(10)
area = 1/2 * base * height = 1/2 * 3sqrt(10) * 4sqrt(10)
area = 2 * 3 * 10 = 6 * 10 = 60 square units