Raymond B. answered • 07/16/22
Math, microeconomics or criminal justice
y= 6x -3x^2
is a parabola downward opening
take the derivative and set = 0
y = 6x -3x^2
y' =-6x +6 = 0
x = 1
y= 6 -3 = 3
vertex is (1,3) = maximum point on the parabola
axis of symmetry is x=1 which divides the parabola area into equal parts
area between the parabola and the x axis is 4.
half of 4 = 2
area between a line through the origin and the x axis, from x=0 to x=2 can be calculated as the sum of the area of a triangle with base = a and height = f(a) = 6a-3a^2
triangle area =af(a)/2 = 3a^2 -1.5a^3
plus the area between the parabola and the x axis from x=a to x=2
that area = 3x^2 - x^3 evaluated from a to 2
= 12-8 - (3a^2 - a^3) =4 -3a^2 + a^3
sum of the two areas = 4-0.5a^3 = 2
0.5a^3 = 2
a^3 =4 (this equation has 3 solutions but 2 are imaginary, ignore them)
a =(4)^(1/3)
f(a) = 6a -3a^2
f((4)^(1/3)) = 6(4)^(1/3) - 3(4)^(2/3)
line through (0,0) and (a, f(a)) is
y=mx with m=f(a)/a
y=(f(a)/a))x
y= [6(4)^(1/3)-3(4)^(2/3)]/4^(1/3)
y = (6-3(4)^(1/3))x
which is about
y = 1.2378x
the slope of the line through the origin, cutting the area between the x axis and the parabola = m = 6-3(4)^(1/3) = about 1.2378 = roughly 5/4
check the solution. Draw a rough graph of the parabola. Visually, the line seems to cut the area in about 1/2.
f(5/4) = 6(5/4) -3(5/4)^2 = 30/4-75/16 = (120-75)/16 = 45/16 = about 2.8
the point of intersection is about (5/4, 2.8)
calculate the area, it's roughly half the area under the parabola, even using rough approximations.
for m=3, the area gets divided into a 1 to 7 ratio
for m=2 the area gets divided in a 0.926 to 1.074 ratio
for m= 1 the division is 1.185 to 0.815
for m=0 zero area, nothing gets divided, it's a 4:0 ratio
for a 1:1 ratio, dividing the area in half,
m is somewhere between 1 and 2, roughly 5/4
