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.
I really don't know... would it be something like d= t . a + b?
b) Express the distance s between the plane and the radar station as a function of d.
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.
Sure! Let's break it down step by step.
a) To express the horizontal distance d (in miles) that the plane has flown as a function of time t, we need to consider the speed of the plane.
The speed of the plane is given as 350 mi/h. This means that for every hour the plane flies, it covers a distance of 350 miles. Therefore, we can say that the distance covered by the plane in t hours is 350t.
However, we are interested in expressing this distance in terms of t, not in hours. To do that, we need to consider the relationship between time and distance. Since the plane is flying at a constant speed, this relationship is linear.
Let's assume that at time t = 0, the plane is directly above the radar station. This means that the plane has not yet started flying horizontally. Therefore, the initial horizontal distance d is 0.
So, we can write the equation for the horizontal distance as a function of time:
d(t) = 350t
b) To express the distance s between the plane and the radar station as a function of the horizontal distance d, we need to consider the altitude of the plane.
The altitude of the plane is given as 1 mile. Since the plane is directly above the radar station at time t = 0, the initial vertical distance s is 1 mile.
The horizontal distance d and the vertical distance s can be related using the Pythagorean theorem:
s^2 + d^2 = altitude^2
Substituting the values, we get:
s^2 + (350t)^2 = 1^2
s^2 + 122,500t^2 = 1
Taking the square root, we obtain:
s = √(1 - 122,500t^2)
c) To express the distance s as a function of time t, we can substitute the expression for d(t) from part a) into the equation for s:
s = √(1 - 122,500t^2)
Therefore, we can say that the distance s between the plane and the radar station is a function of time t given by:
s(t) = √(1 - 122,500t^2)
And that's the solution! Let me know if you have any more questions.
a) To express the horizontal distance d that the plane has flown as a function of time t, we can use the formula: d = v * t, where v is the velocity or speed of the plane. In this case, the speed of the plane is 350 mi/h, so the expression becomes:
d = 350 * t
b) To express the distance s between the plane and the radar station as a function of d, we can use the Pythagorean theorem. According to the theorem, the square of the distance s is equal to the sum of the squares of the horizontal distance d and the altitude h. Mathematically, this can be expressed as:
s^2 = d^2 + h^2
Since the altitude h is given as one mile (which is equivalent to 1^2 = 1), the equation becomes:
s^2 = d^2 + 1
c) To express s as a function of t using composition, we substitute the expression for d from part (a) into the equation from part (b):
s^2 = (350 * t)^2 + 1
By taking the square root of both sides, we can express s as a function of t:
s = √[(350 * t)^2 + 1]
Therefore, the distance s between the plane and the radar station can be expressed as a function of time t.