To solve these questions, we'll need to use the concept of relative speed. The relative speed of an object is the difference between its speed and the speed of another object it is moving with or against. Let's tackle each question step by step:
1. Let's assume the speed of the boat in still water is 'B' and the speed of the current is 'C'. When the boat is traveling downstream, it moves with the current, so its effective speed is (B + C). Similarly, when the boat is traveling upstream, it moves against the current, so its effective speed is (B - C).
To find the speed of the boat in still water (B), we can use the formula:
Distance = Speed * Time.
For the downstream trip, we have (B + C) * 10 = 210 miles.
For the upstream trip, we have (B - C) * 70 = 210 miles.
Simplifying these equations, we get:
B + C = 21 (Equation 1)
B - C = 3 (Equation 2)
Adding Equation 1 and Equation 2, we get:
2B = 24
Dividing both sides by 2, we find:
B = 12 mph.
Now, substituting the value of B in Equation 1, we can find the speed of the current (C):
12 + C = 21
C = 21 - 12
C = 9 mph.
Therefore, the speed of the boat in still water is 12 mph, and the speed of the current is 9 mph.
2. Let's assume the speed of the plane in still air is 'P' and the speed of the wind is 'W'. When the plane is flying with the tailwind, its effective speed is (P + W). Similarly, when it is flying against the headwind, its effective speed is (P - W).
To find the speed of the wind (W), we can use the same formula: Distance = Speed * Time.
For the journey with the tailwind, we have (P + W) * T = 158 km, where T is the time taken.
For the journey against the headwind, we have (P - W) * T = 112 km.
Since the distance traveled is the same in both cases, we can set these two equations equal to each other:
(P + W) * T = (P - W) * T
P + W = P - W
Simplifying, we find:
2W = 2P - 158 - 112
2W = 2P - 270
Dividing both sides by 2, we get:
W = P - 135
Since we are left with two unknowns, we need one more piece of information to solve for P and W.