An airplane flies 2368 miles round trip in 7 hours. The speed of the wind is 20 mph. Determine the airplanes speed in still air.
339.5 mph /●~●/
To determine the airplane's speed in still air, we need to subtract the effect of the wind.
Let's assume that the airplane's speed in still air is represented by "x" mph.
When the airplane flies with the wind, its speed is increased by the speed of the wind, which is 20 mph. Therefore, the speed becomes x + 20 mph.
When the airplane flies against the wind, its speed is decreased by the speed of the wind, which is again 20 mph. Therefore, the speed becomes x - 20 mph.
We know that the airplane travels 2368 miles in total and it takes 7 hours in total for the round trip.
When the airplane flies with the wind on the first leg of the trip, it covers half the distance with the speed of x + 20 mph. So, the time taken for this leg is 2368/2(x + 20), which simplifies to 1184/(x + 20) hours.
When the airplane flies against the wind on the second leg of the trip, it covers the remaining half the distance with the speed of x - 20 mph. So, the time taken for this leg is also 1184/(x - 20) hours.
The total time for the round trip is 7 hours. So, we can write the equation as:
1184/(x + 20) + 1184/(x - 20) = 7
To solve this equation and find the value of x, we need to cross-multiply and simplify:
1184(x - 20) + 1184(x + 20) = 7(x + 20)(x - 20)
Now, let's solve the equation step-by-step:
Expand the equation:
1184x - 23680 + 1184x + 23680 = 7(x^2 - 400)
Combine like terms:
2368x = 7x^2 - 2800
Rearrange the equation:
7x^2 - 2368x - 2800 = 0
Divide all terms by 7 to simplify the equation:
x^2 - 338.286x - 400 = 0
Now, we can use the quadratic formula to find the value of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values into the formula:
x = (-(-338.286) ± √((-338.286)^2 - 4(1)(-400))) / (2(1))
Simplifying the equation:
x = (338.286 ± √(114253.446 - (-1600))) / 2
x = (338.286 ± √115853.446) / 2
Using a calculator to find the square root value:
x = (338.286 ± 339.994) / 2
Now, solving for both solutions:
x = (338.286 + 339.994) / 2 ≈ 339.14 mph
or
x = (338.286 - 339.994) / 2 ≈ -0.85 mph
Since speed cannot be negative, we discard the negative solution.
Therefore, the airplane's speed in still air is approximately 339.14 mph.
To determine the airplane's speed in still air, we need to find the speed of the airplane when there is no wind affecting its motion.
Let's denote the airplane's speed in still air as "S" (in mph).
Given that the airplane flies 2368 miles round trip in 7 hours, we can break it down into two parts: the time it takes to travel one way and the time it takes to travel back.
Let's calculate the time it takes to travel one way:
Distance = Speed × Time
2368 miles = (S + 20 mph) × T
(where T is the time taken to travel one way)
Similarly, the time it takes to travel back is:
Distance = Speed × Time
2368 miles = (S - 20 mph) × (7 - T)
(where 7 - T is the time taken to travel back)
Now, we have two equations with two unknowns (S and T). Let's solve them simultaneously to find the value of S, the airplane's speed in still air.
Equation 1: 2368 miles = (S + 20 mph) × T
Equation 2: 2368 miles = (S - 20 mph) × (7 - T)
First, let's simplify Equation 2:
2368 miles = (S - 20 mph) × (7 - T)
2368 miles = 7S - ST - 20T + 400 mph (expand using distributive property)
2368 miles = 7S - ST + 400 mph - 20T
2368 miles = 7S - T(S - 20) + 400 mph - 20T
2368 miles = 7S - T(S - 20) - 20(T - 20) (apply difference of squares)
Now we have two equations:
Equation 1: 2368 miles = (S + 20 mph) × T
Equation 2: 2368 miles = 7S - T(S - 20) - 20(T - 20)
Now we can solve for S by eliminating T.
Let's simplify Equation 2 further:
2368 miles = 7S - T(S - 20) - 20(T - 20)
2368 miles = 7S - ST + 20T + 20S - 400 mph - 20T
2368 miles = (7S + 20S) - ST + (20T - 20T) - 400 mph
2368 miles = 27S - ST - 400 mph
Now we can substitute this simplified Equation 2 into Equation 1:
2368 miles = (S + 20 mph) × T
2368 miles = (S + 20)T
Now we have:
27S - ST - 400 mph = (S + 20)T
Next, let's distribute the T to S and -20:
27S - ST - 400 mph = ST + 20T
Combining like terms:
27S - 400 mph = 2ST + 20T
Now, let's isolate the terms with S on one side:
27S - 2ST = 400 mph + 20T
S(27 - 2T) = 400 mph + 20T
Divide both sides by (27 - 2T):
S = (400 mph + 20T)/(27 - 2T)
Now we have the equation for the airplane's speed in still air (S) in terms of T, the time taken to travel one way. To find the exact value of S, we need to know the value of T (time taken to travel one way). Once we have that information, we can substitute it into the equation to get the airplane's speed in still air.