Question

The path of a swooping bird is modeled by the hyperbola 4y2 − 1,225x2 = 4,900, where x is the horizontal distance measured from the point where the bird is closest to the ground and y represents the height of the bird from the ground. Hint: Assume that the origin lies at ground level.

The bird is closest to the ground when its height is__ meters from the ground.

Answer 1

The graph of the path of the swooping bird is shown in figure below. The general form of the hyperbola equation is given by:


`(y^(2))/(a^(2))-(x^(2))/(b^(2))=1`

So we can order the equation of the problem by multiplying it by the following term:


`(1)/(49000)`

Therefore:


`(y^(2))/(35^(2))-(x^(2))/(2^(2))=1`

∴
`(y^(2))/(1225)-(x^(2))/(4)=1`

Given that the origin lies at ground level, the bird is closest to the ground at the vertices of the parable, that is, when x = 0 (this will give us two solutions, but we will take the positive value because the bird flight over the air)


`(y^(2))/(1225)-(0^(2))/(4)=1 \rightarrow y=√(1225) \rightarrow \boxed{height=35m}`

Answer 2

Hello,
Please, see the attached file.
Thanks.