In this problem we are asked to find the coefficients a and b and i got stuck on

Question

In this problem we are asked to find the coefficients a and b and i got stuck on cramster's solution.

In the solution they went from A = (.5)a(10^4) - (.25)b(10^8) to

A = (.25)a(10^4) and i cannot figure out how they did it.

the question has a picture of a dam which is parabola shaped. this dam starts at the xy origin (0,0), goes up (function: y=ax-bx^3) and out 100m. when x is 100 m, y=0. first they said when x=100, y=0 and solved for b. then when solving for the area they seemed to just have b as b instead of the value they obtained and i got confused.
A= integral from 0 to 100(ax-bx^3)dx
A=integral from 0 to 100 (ax- (10^-4)x^3)dx... (10^-4 is the value they found for b when setting x = 100)
next
A=(.5)a[(x^2)-(.5)(10^-4)(x^4)] from 0 to 100

A= (.5)a(10^4)-(.25)b(10^8)
finally
A= (.25)a(10^4)

I cannot figure out what they did here so if someone could help that would be great.

Explanation / Answer

when x = 100,

y = a * 100 - b * 100^3 = 0

b = a * 100/(100^3) = a * 10^-4

so,

b = a * 10^-4

A = Integrating from 0 to 100 (ax - bx^3) dx

A = integrating from 0 to 100 ( a * x^2/2 - b * x^4/4)

A = a * 100^2/2 - b * (100^4)/4 = 0.5 * a * 10^4 - b * 0.25 * 10^8

now substituting, b = a * 10^-4

A = 0.5 * a * 10^4 - 0.25 * a * 10^-4 * 10^8 = 0.5 * a * 10^4 - 0.25 * a * 10^4 = 0.25 * a * 10^4

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