A steady wind blows a kite due west. The kite's height above ground from horizontal position x = 0 to x = 80 ft is given by the following.
y = 150 - (1/40)(x-50)^2
Find the distance traveled by the kite.
y = 150 - (1/40)(x-50)^2
y = 150 - (1/40)(x-50)(x-50)
y = 150 - (1/40)x^2 + (5/2)x + 125/2
y = (-1/40)x^2 + (5/2)x + 425/2
y' = (-1/20)x + 5/2
(y')^2 = ((-1/20)x + 5/2)^2
(y')^2 = (1/400)x^2 - (1/4)x + 25/4
Length = Integral from 0 to 80 of:
Sqrt[1+(1/400)x^2 - (1/4)x + 25/4]
Sqrt[(1/400)x^2 - (1/4)x + 29/4]
How would I integrate this? Is this the correct procedure? Thanks.
You can calculate the derivative directly as:
y' = -2 (1/40)(x-50)
using the chain rule. Then you find:
1 + y'^2 = 1 + 1/400 (x-50)^2
Which is the same as what you got. However, to compute the integral, you need to write it in this form anyway:
Integral sqrt[1 + 1/400 (x-50)^2] dx
put x = 20 y + 50:
Integral 20 sqrt[1 + y^2] dy
Substitute y = Sinh(t) in here. The square root beomes a hyperbolic cosine, you get another hyperbolic cosine from the integration measure dy.
The integral of cosh^2 can be computed by using that it is a sum of exp(t) and
exp(-t) The square of this is just a sum of exponentials which you can integrate term by term.
47 becuase that is my luky number
U CAN'T C ME
I apologize for the confusion in the previous explanation. Let's go through the correct procedure step by step to integrate the expression.
Given:
Length = ∫[0,80] √[(1/400)x^2 - (1/4)x + 29/4] dx
To integrate this expression, we'll first simplify the expression inside the square root as follows:
Length = ∫[0,80] √[(1/400)x^2 - (1/4)x + 29/4] dx
= ∫[0,80] √[(1/400)(x^2 - (1/100)x) + 29/4] dx
= ∫[0,80] √[(1/400)(x^2 - (1/100)x + 1/400 - 1/400) + 29/4] dx
= ∫[0,80] √[(1/400)(x^2 - (1/20)x + 1/400) + 29/4 - 1/400] dx
= ∫[0,80] √[(1/400)(x - 1/20)^2 + (29/4 - 1/400)] dx
= ∫[0,80] √[(1/400)(x - 1/20)^2 + (116 - 1)/400] dx
= ∫[0,80] √[(1/400)(x - 1/20)^2 + 115/400] dx
Now we can make a substitution: Let (x - 1/20) = t. Then dx = dt.
Substituting these values into the integral expression:
Length = ∫[0,80] √[(1/400)(t^2) + 115/400] dt
= (1/20) ∫[0,80] √[t^2 + 115] dt
At this point, the integral cannot be expressed in terms of elementary functions, so we need to use numerical methods or approximation techniques to compute the result. You can use numerical integration methods such as Simpson's rule or the trapezoidal rule to calculate an approximate value for the integral.
Please note that the steps outlined above are a simpler way to integrate the expression compared to the method mentioned in the previous response. I apologize for any confusion caused.