A spacecraft is approaching a space station that is orbiting Earth. When the craft is 1000 km from the space station, reverse thrusters must be applied to begin braking. The time, t, in hours, required to reach a distance, d, in km, from the space station while the thrusters are being fired can be modelled by t=log_0.5(d/1000). The docking sequence can be initalized once the craft is within 10 km of the station's docking bay.
a) how long after the reverse thrusters are first fired should docking procedures begin.
I don't need help with this one.
b) What are the domain and range of this function and what do these represent?
I need help with this.
I am not sure if my graph of this function looks right because I don't have graphing software or a calculator. If someone could show me a picture of it that, I would appreciate it. But otherwise, could you just explain to me why the domain and range are what they are.
The domain is supposed to be 0<d< or = 1000
but why isn't it 10<d< or = 1000
Doesnt it say in the question that "the docking sequence can be initialized once the craft is within 10 km of the station's docking bay", hence the 10 and not 0 in the domain?????
You continue to approach the station and brake after your sequence begins at 10 km. The craft does not stop until d = 0
The domain of a function represents the set of all possible input values for the function. In this case, the variable "d" represents the distance from the space station. According to the information given in the question, the reverse thrusters are first fired when the craft is 1000 km from the station. Therefore, the minimum value for "d" is 1000 km.
However, the question specifies that the docking sequence can be initialized once the craft is within 10 km of the station's docking bay. This means that the craft needs to be at least 10 km away from the docking bay to initiate docking procedures. Since the craft cannot be closer than 10 km to the docking bay, the domain of the function should exclude any values of "d" less than 10 km.
Therefore, the correct domain for this function is 10 ≤ d ≤ 1000. It includes all distances from 10 km to 1000 km.
Regarding the range of the function, the range represents the set of all possible output values of the function. In this case, "t" represents the time required to reach a certain distance from the space station. The function given, t= log_0.5(d/1000), does not have any restrictions on the output, so there are no limitations on the range. Therefore, the range is all real numbers.
To determine the domain and range of a function, it's important to consider the constraints and the interpretation of the variables involved.
In this case, the function t = log_0.5(d/1000) represents the time required to reach a certain distance "d" from the space station while the reverse thrusters are being fired.
Let's start with the domain. The domain of a function represents all the possible valid inputs for the function. In this context, "d" represents the distance from the space station. However, the docking sequence can only be initialized once the craft is within 10 km of the station's docking bay. This means that the craft cannot start braking when it is further than 10 km away.
So why isn't the domain 10 < d ≤ 1000 instead of 0 < d ≤ 1000?
The reason is that the given function, t = log_0.5(d/1000), accounts for the entire braking process as defined in the question. The logarithmic function models the continuous decrease in time as the distance "d" decreases. If we were to restrict the domain to 10 < d ≤ 1000, we would essentially detach the mathematical model from the given scenario outlined in the question.
By keeping the domain as 0 < d ≤ 1000, we ensure that we include the entire braking process, starting from when the craft is 1000 km away until it reaches the docking bay. This allows us to analyze the function over the entire relevant range of distances.
Now, let's determine the range. The range of a function represents all the possible output values for the function. In this case, "t" represents the time it takes to travel a certain distance. Since time cannot be negative, the range of the function is t > 0.
To summarize:
Domain: 0 < d ≤ 1000 (0 km < d ≤ 1000 km)
Range: t > 0 (time is positive)
Unfortunately, I am unable to provide you with a graph since I am a text-based AI. However, you can visualize the graph either using graphing software such as Desmos or by using a scientific calculator that has graphing capabilities. Simply input the function t = log_0.5(d/1000) and adjust the view window to match the domain and range mentioned above.