Question

What is the average distance between the parabola yequals=77​x(1616minus−​x) and the​ x-axis on the interval left bracket 0 comma 16 right bracket[0,16]​?

Answer 1

To calculate the average distance between the given parabola and the x-axis

y = 77x(1616 - x)

x ∈ [0, 16]

avg distance =
`\int_(0)^(16)(77x(1616 - x)dx)/(\int_(0)^(16) x dx)`

=
`2\int_(0)^(16)(((124432x - 77x^(2))dx)/([x^(2)]_(0)^(16))`

=
`2\int_(0)^(16)((124432x^(2))/(2)- (77x^(3))/(3))/([x^(2)]_(0)^(16))`

=
`16^(2)[(62216 - 25.67* 16)/(16^(2))]`

avg distance = 61805 unit