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:

$\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$

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

$\frac{1}{49000}$

Therefore:

$\frac{y^{2}}{35^{2}}-\frac{x^{2}}{2^{2}}=1$

∴ $\frac{y^{2}}{1225}-\frac{x^{2}}{4}=1$

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)

$\frac{y^{2}}{1225}-\frac{0^{2}}{4}=1 \rightarrow y=\sqrt{1225} \rightarrow \boxed{height=35m}$

Answer 2

Hello,
Please, see the attached file.
Thanks.