Question

There is a line through the origin that divides the region bounded by the parabola and the x-axis into two regions with equal area. What is the slope of that line?.

Answer 1

Using the slope line formula,

The slope of line y = mx is 1.0315.

We have given that,

parabola equation is y = 5x - 3x²

line passing through origin is in the form of y = mx

==> mx = 5x - 3x²

==> 3x²+ mx - 5x = 0

==> 3x² + x(m -5) = 0

==> x(3x + (m -5)) = 0

==> x = 0 , x = (5 -m)/3

The total area of the bounded region is then the integral of 5x - 3x² from x = 0 to x = (5-m)/3

Area under y = 5x - 3x2 and y = mx is (125/54)/2 = 125/108

==> [0 to (5 -m)/3] ∫(5x - 3x²-mx )dx = 125/108

==> [0 to (5 -m)/3]∫( (5 - m)x - 3x2 )dx = 125/108

==> [0 to (5 -m)/3] ((5 - m)x²/2 - x³ )= 125/108

==> (5 - m)[(5 -m)/3]²/2 - ((5 -m)/3)³ = 125/108

==> (5 -m)³/18 - (5 -m)³/27 = 125/108

==> (5 -m)³/54 = 125/108

==> (5 -m)³ = 125/2

==> (5 -m) = 3.9685

==> m = 5 - 3.9685

==> m = 1.0315

Hence slope of line = 1.0315

To learn more about slope of line , refer:

brainly.com/question/29044610

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