Two sides of a square lie along the lines 2y = 20 - 3x and 3x+ 2y = 48 Find the area of the square
the first line contains the point (0,10)
The second line has equation 3x+2y-48 = 0
So the distance between the lines is
|3*0 + 2*10 - 48|/√(3^2+2^2) = 28/√13
That is the side of the square, so its area is 28^2/13 = 784/13
2y = 20 - 3x and 3x+ 2y = 48
y = -1.5 x + 10
and
y = - 1.5 x + 24
those lines are parallel, so the distance between them is the length of a side of your square
the slope of a line between them and perpendicular to them is
-1/-1.5 = 1/1.5
so form of our desired perpendicular is y = x/1.5 + b
a point on first line is (0,10)
so our line can be:
10 = 0 + b or b = 10
y = x/1.5 + 10
where does that hit line two?
x/1.5 + 10 = -1.5 x + 24
14 = x/1.5 + 1.5 x
21 = x + 2.25 x = 3.25 x
x = 6.46
so find y
y = - 1.5 x + 24 = -1.5(6.46) + 24 = 14.3
so our side starts at (0 , 10) and goes to (6.46 , 14.3)
length of side squared = Area = 6.46^2 + 4.3^2
2y = 20 - 3x ... y = -3/2 x + 10
3x+ 2y = 48 ... y = -3/2 x + 16
the slope of the distance between the lines is ... -[1 / (-3/2)] = 2/3
through the lower intercept ... y = 2/3 x + 10
intersection with upper line ... 2/3 x + 10 = -3/2 x + 16
... multiplying by 3/2 ... x + 15 = -9/4 x + 24 ... 13 x = 36 ... x = 36 / 13
... substitute back to find the y-coordinate of the intersection
the distance from the lower y-intercept , (10,0)
... to the intersection with the upper line
... is the length of the side of the square
oops ... the lower y-intercept is (0,10) , not (10,0)
To find the area of the square, we need to first find the length of one side of the square.
We have two equations representing the lines along which the sides of the square lie:
1) 2y = 20 - 3x
2) 3x + 2y = 48
We need to find the point of intersection of these two lines. We can do this by solving the system of equations.
To solve the system of equations:
Step 1: Rewrite the first equation in terms of x and substitute it into the second equation:
2y = 20 - 3x
Rewriting this equation in terms of x:
3x = 20 - 2y
Substituting this into the second equation:
3x + 2y = 48
(20 - 2y) + 2y = 48
20 = 48
Step 2: Simplify the equation:
20 = 48
Step 3: Since the equation is not true, this means that the two lines do not intersect. Therefore, there is no square that satisfies both equations.
Hence, we cannot find the area of the square as there is no square defined by the given lines.