Question

type the correct answer in the box.
the path of a swooping bird is modeled by the hyperbola ( 4 y ^ { 2 } - 1,225 x ^ { 2 } = 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.
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Explanation:

Step1: Analyze the equation for minimum \(y\)

We have the equation \(4y^{2}-1225x^{2} = 4900\). Rearrange it to solve for \(y^{2}\): \(y^{2}=\frac{4900 + 1225x^{2}}{4}\). Since \(x^{2}\geq0\) (because the square of a real - number is non - negative), the minimum value of \(y^{2}\) occurs when \(x = 0\).

Step2: Substitute \(x = 0\) into the equation for \(y\)

When \(x = 0\), we have \(4y^{2}=4900\). Then \(y^{2}=\frac{4900}{4}=1225\). Taking the square root of both sides (and considering \(y\geq0\) since \(y\) represents height), we get \(y=\sqrt{1225}\).

Answer:

\(35\)