An airplane is flying at a speed of
350 mi/h at an altitude of one mile. The plane passes directly
above a radar station at time t � 0.
(a) Express the distance s (in miles) between the plane and
the radar station as a function of the horizontal distance d
(in miles) that the plane has flown.
(b) Express d as a function of the time t (in hours) that the
plane has flown.
(c) Use composition to express s as a function of t.
To answer these questions, we will need to break down the problem step by step.
(a) Expressing the distance s as a function of the horizontal distance d:
We can use the Pythagorean theorem to find the relationship between s and d. The theorem states that in a right triangle, the square of the hypotenuse (s) is equal to the sum of the squares of the other two sides. In this case, the horizontal distance d is one of the other two sides.
Using the Pythagorean theorem, we have:
s^2 = d^2 + 1^2
Simplifying:
s^2 = d^2 + 1
Taking the square root of both sides:
s = √(d^2 + 1)
Therefore, the distance s is expressed as a function of the horizontal distance d as s = √(d^2 + 1)
(b) Expressing d as a function of time t:
To find the horizontal distance d, we need to know the time it takes for the plane to travel that distance. Because the plane is flying at a constant speed of 350 mi/h, we can use the formula:
Distance = Speed * Time
In this case, the distance is the horizontal distance d and the speed is 350 mi/h. Since the plane passed directly above the radar station at time t = 0, we can express the relationship between d and t as:
d = 350t
Therefore, d is expressed as a function of time t as d = 350t.
(c) Using composition to express s as a function of t:
Now that we have expressions for s as a function of d and d as a function of t, we can substitute the expression of d into the expression of s to get s as a function of t:
s = √(d^2 + 1)
s = √((350t)^2 + 1)
Therefore, s is expressed as a function of time t as s = √((350t)^2 + 1).
Note: In this problem, we assumed that the plane is flying in a straight line directly above the radar station.