A steady wind blows a kite due west. The kite's height above ground from horizontal position x = 0 to x = 60 ft is given by the following.
y= 155-(1/40)(x-50)^2
Find the distance traveled by the kite.
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To find the distance traveled by the kite, we need to calculate the length of the path it travels from x = 0 to x = 60 ft.
The given equation is y = 155 - (1/40)(x - 50)^2, which represents the height of the kite above the ground at any given horizontal position x.
To find the distance traveled, we need to calculate the arc length of the curve defined by this equation from x = 0 to x = 60.
The arc length formula for a function y = f(x) on an interval [a, b] is given by:
L = ∫[a,b] √(1 + (f'(x))^2) dx
First, let's find the derivative of the given equation:
y = 155 - (1/40)(x - 50)^2
y' = 0 - (1/20)(x - 50)
Next, we substitute this derivative into the arc length formula:
L = ∫[0,60] √(1 + (-(1/20)(x - 50))^2) dx
L = ∫[0,60] √(1 + (1/400)(x - 50)^2) dx
Simplifying the expression under the square root:
L = ∫[0,60] √((400 + (x - 50)^2)/400) dx
L = (1/20)∫[0,60] √(400 + (x - 50)^2) dx
Integrating this expression might be difficult, but we can use numerical methods or technology (like graphing calculators or software) to find a close approximation for the definite integral.
Once the definite integral is computed, the resulting value will give us the distance traveled by the kite from x = 0 to x = 60 ft.