An airplane is flying at a speed of 350 mi/h at an altitude of one mile and passes directly over a radar station at time t=0.
a) Express the horizontal distance d (in miles) that the plane has flown as a function of t.
b) Express the distance s between the plane and the radar station as a function of d.
I used the pythagorean theorem to get 1^2 + (350t)^2= sqrt of (1^2+d^2)^2
Is that right? If I solve for t it gives me zero.
c) Use composition to express s as a function of t.
Please Help. Explanations also would be nice, so I can see how to do it.
To solve this problem, let's break it down step by step.
a) Express the horizontal distance d (in miles) that the plane has flown as a function of t.
Since the plane is flying horizontally at a constant speed of 350 mi/h, we can express the distance traveled by the plane as d = 350t, where t is the time in hours.
b) Express the distance s between the plane and the radar station as a function of d.
To find the distance between the plane and the radar station, we can use the Pythagorean theorem. Let's call the distance between the plane and radar station as s and the vertical distance as h.
Using the Pythagorean theorem, we have the equation:
s^2 = d^2 + h^2
In this case, h is the altitude of the plane, which is constant at 1 mile. So, the equation becomes:
s^2 = d^2 + 1^2
s^2 = d^2 + 1
Taking the square root of both sides, we get:
s = √(d^2 + 1)
c) Use composition to express s as a function of t.
To express s as a function of t, we need to substitute the expression for d from part a into the equation from part b.
Substituting d = 350t into the equation s = √(d^2 + 1), we have:
s = √((350t)^2 + 1)
s = √(122,500t^2 + 1)
Therefore, the distance s between the plane and the radar station can be expressed as a function of t using composition:
s(t) = √(122,500t^2 + 1)
I hope this explanation helps! Let me know if you have any further questions.