A hawk flying at an altitude of $121$ m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation $y=121-\frac{x^{2}}{64}$ until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters.
True or False?
The distance traveled by the prey from the time it is dropped until the time it hits the ground is
approximately $156.532$ m.

Question

Quizplus · Accepted Answer

To find the distance traveled by the prey, we need to find the length of the parabolic arc corresponding to the given function. This can be done using calculus or by using the formula for the arc length of a parabola. Using the latter approach, we have:

Arc length = ∫(1 + [dy/dx]^2)^(1/2) dx, where the integral is taken over the range [0, 16]

Calculating the integrand, we get:

1 + [dy/dx]^2 = 1 + [x/32]^2

Taking the square root and simplifying, we get:

(1 + [dy/dx]^2)^(1/2) = (1 + [x/32]^2)^(1/2)

Substituting this back into the arc length formula and integrating, we get:

Arc length = ∫(1 + [x/32]^2)^(1/2) dx, from x = 0 to x = 16

This integral can be evaluated by making the substitution x = 16tanθ, dx = 16sec^2θ dθ. After simplifying and using the identity sec^3θ = secθ tanθ, we get:

Arc length = 32∫sec^3θ dθ, from θ = 0 to θ = tan^(-1)(2)

This integral can be evaluated using integration by parts or by using a table of integrals. Either way, the result is:

Arc length = 32(1/2)sec(tan^(-1)(2))(tan(tan^(-1)(2)) + ln|sec(tan^(-1)(2)) + tan(tan^(-1)(2))|) ≈ 156.532

Therefore, the statement is true. The distance traveled by the prey from the time it is dropped until it hits the ground is approximately 156.532 m.

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A Hawk Flying at an Altitude Of $121$ M Accidentally Drops Its Prey