Let's solve the equation step by step to determine the height of the bird when it is closest to the ground.
The given equation of the hyperbola is:
[tex]\[ 4y^2 - 1,225x^2 = 4,900 \][/tex]
We start by converting this equation into the standard form of a hyperbola:
[tex]\[ \frac{4y^2}{4,900} - \frac{1,225x^2}{4,900} = 1 \][/tex]
Simplifying the coefficients, we get:
[tex]\[ \frac{y^2}{1,225} - \frac{x^2}{4} = 1 \][/tex]
Here, the standard form of a hyperbola with vertical transverse axis is:
[tex]\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \][/tex]
where [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] are the denominators of [tex]\( y^2 \)[/tex] and [tex]\( x^2 \)[/tex] terms, respectively.
Comparing the two equations, we identify that:
[tex]\[ a^2 = 1,225 \][/tex]
[tex]\[ b^2 = 4 \][/tex]
To find [tex]\( a \)[/tex], we take the square root of [tex]\( a^2 \)[/tex]:
[tex]\[ a = \sqrt{1,225} = 35 \][/tex]
Since [tex]\( a \)[/tex] represents the distance from the center to a vertex along the y-axis and since the origin lies at ground level, the vertex of the hyperbola gives the height of the bird when it is closest to the ground.
Thus, the height of the bird from the ground when it is closest is:
[tex]\[ 35 \text{ meters} \][/tex]
So, the bird is closest to the ground when its height is [tex]\( 35 \)[/tex] meters from the ground.
