A speed boat is travelling from Florida to Cuba (distance of approximately 380 km), it is directly affected by
the water current. Thus, the actual speed of the boat with respect to the ground is the speed of the boat (125
km/h) plus the water current speed, c.
a) Develop an equation of the total time it takes to make a round trip if the current is c. HINT: Distance = velocity x time
Isolate for current with respect to time.
b) Then state the domain & range of this application.
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a) Let's start by finding the time it takes for the boat to travel from Florida to Cuba.
Distance = Velocity × Time
We have the distance as 380 km and the velocity as the speed of the boat (125 km/h) plus the water current speed (c). Let's call the time it takes for the boat to travel from Florida to Cuba as T1.
So, we have the equation:
380 = (125 + c) × T1
Next, let's find the time it takes for the boat to travel back from Cuba to Florida. The distance is the same (380 km), but this time the velocity will be the speed of the boat (125 km/h) minus the water current speed (c). Let's call the time it takes for the boat to travel back from Cuba to Florida as T2.
We have the equation:
380 = (125 - c) × T2
To find the total time for the round trip, we sum T1 and T2:
Total Time = T1 + T2
Now, let's isolate c with respect to time. We'll solve the first equation for T1 and substitute it into the equation for the total time:
380 = (125 + c) × T1
T1 = 380 / (125 + c)
Total Time = 380 / (125 + c) + T2
We can substitute the second equation into the equation for the total time:
Total Time = 380 / (125 + c) + 380 / (125 - c)
b) The domain of this application would be all possible values of the water current speed (c) that are realistic and feasible. For example, we could consider a reasonable range of currents, such as -10 km/h (flowing in the opposite direction of the boat) to +10 km/h (flowing in the same direction as the boat).
The range would be the total time it takes for the round trip, which could vary depending on different water current speeds. The range would include all possible total times (measured in hours) within the given domain.
a) To develop the equation for the total time it takes to make a round trip, we need to consider the speed of the boat with respect to the ground. Given that the boat's actual speed is the sum of its speed in still water (125 km/h) and the water current speed (c), the equation speed = 125 + c represents the boat's speed relative to the ground.
Now, let's calculate the time it takes for the boat to cover the distance from Florida to Cuba:
Distance = Velocity x Time
380 km = (125 + c) km/h x t
To isolate for t, we divide both sides of the equation by (125 + c):
t = 380 km / (125 + c) km/h
This equation gives us the time it takes for the boat to travel from Florida to Cuba, considering the water current speed (c).
To calculate the total time for the round trip, we need to account for the time it takes to travel from Cuba back to Florida. Since the water current direction will be opposite on the return trip, the effective speed of the boat will be reduced by the current speed. Therefore, the equation for the total time (T) is:
T = t + (380 km / (125 - c) km/h)
b) The domain of this application refers to the possible values for the water current speed (c). The current can have any real value, but it is limited practically by the maximum speed the boat can handle and the water current conditions.
The range of this application is the total time (T) it takes for the boat to make a round trip. Since the total time cannot be negative, the range is T ≥ 0.
a) To develop an equation for the total time it takes to make a round trip, we can use the formula Distance = Velocity x Time.
Let's consider the boat's speed with respect to the ground as the actual speed. So, the actual speed of the boat during the round trip will be the speed of the boat (125 km/h) plus the water current speed (c).
During the trip from Florida to Cuba, the distance covered is 380 km.
Let's calculate the time taken for this one-way trip:
Time taken = Distance / Velocity
Time taken = 380 km / (125 km/h + c)
Since the boat has to make a round trip, the total time taken (round trip time) will be twice that of the one-way trip:
Round trip time = 2 * Time taken
Round trip time = 2 * (380 km / (125 km/h + c))
Round trip time = 760 km / (125 km/h + c)
b) Now, let's discuss the domain and range of this application:
Domain: The domain refers to the possible values or restrictions on the variables involved. In this case, the variable "c" represents the water current speed. The domain of "c" would typically be any non-negative number, as a negative current speed would imply the current is going in the opposite direction, which is not realistic in this scenario. Therefore, the domain for "c" is c >= 0.
Range: The range refers to the possible values or restrictions on the output of the function. In this case, the output of the function is the round trip time. The round trip time should be a positive number, as it represents the time taken for the boat to complete the journey. Therefore, the range for the round trip time is time > 0.
In summary, the domain for the water current speed "c" is c >= 0, and the range for the round trip time is time > 0.