The 4200 km trip from New York to San Francisco take 6 h flying against the wind but only 5 h returning. Find the speed of the plane in still air and the speed of the wind.
To solve this problem, we can use the concept of relative speed. Let's say the speed of the plane in still air is represented by "p" (in km/h) and the speed of the wind is represented by "w" (in km/h).
When the plane is flying against the wind, its effective speed is decreased due to the counteracting force of the wind. So, the effective speed against the wind is given by:
Effective speed against the wind = p - w
Similarly, when the plane is flying with the wind, its effective speed is increased due to the assistance of the wind. So, the effective speed with the wind is given by:
Effective speed with the wind = p + w
According to the information provided, the distance between New York and San Francisco is 4200 km. We can set up the following equation for the first leg of the trip (against the wind):
Distance = Speed × Time
4200 = (p - w) × 6
We can set up the following equation for the second leg of the trip (with the wind):
4200 = (p + w) × 5
Now, we have a system of two equations with two variables. We can solve this system of equations to find the values of "p" and "w". Let's solve it step by step.
Step 1: Simplify the equations
6p - 6w = 4200 (Equation 1)
5p + 5w = 4200 (Equation 2)
Step 2: Divide both equations by their respective coefficients
p - w = 700 (Equation 3)
p + w = 840 (Equation 4)
Step 3: Add Equation 3 and Equation 4 to eliminate "w"
2p = 1540
Step 4: Solve for "p"
p = 1540 / 2 = 770
So, the speed of the plane in still air (p) is 770 km/h.
Step 5: Substituting the value of "p" into Equation 4 to solve for "w"
770 + w = 840
w = 840 - 770 = 70
So, the speed of the wind (w) is 70 km/h.
Therefore, the speed of the plane in still air is 770 km/h, and the speed of the wind is 70 km/h.
To find the speed of the plane in still air (let's call it "P") and the speed of the wind (let's call it "W"), we can use the following formula:
P + W = Speed of the plane flying with the wind (5 hours)
P - W = Speed of the plane flying against the wind (6 hours)
Let's solve this system of equations using the method of substitution:
1. Solve the first equation for P:
P = Speed of the plane flying with the wind (5 hours) - W
2. Substitute P in the second equation:
Speed of the plane flying with the wind (5 hours) - W - W = Speed of the plane flying against the wind (6 hours)
Simplify the equation:
5 - 2W = 6
3. Subtract 5 from both sides:
-2W = 1
4. Divide both sides by -2:
W = -1/2
The speed of the wind is -1/2 km/h. However, since speed cannot be negative, we ignore this negative value.
5. Substitute W = -1/2 back into any of the original equations to solve for P:
P + (-1/2) = Speed of the plane flying with the wind (5 hours)
P - 1/2 = Speed of the plane flying against the wind (6 hours)
Simplify each equation:
P = 5 + 1/2
P = 5.5
The speed of the plane in still air is 5.5 km/h.
Therefore, the speed of the plane in still air is 5.5 km/h and the speed of the wind is 0.5 km/h.
4200/(p-w) = 6
4200/(p+w) = 5
p = 770
w = 70
check:
4200/840 = 5
4200/700 = 6