A delivery business plans to serve three towns on a stretch of highway. For every trip they make to town A, which we'll place at position x=0, they expect to make two trips to town B (at position x=1) and 3 trips to town C (at position x=4). If they build their delivery centre at position x, then, their daily driving time should be proportional to d(x) = |x| + 2|x - 1| + 3|x - 4|. The owners are thinking of building the center in town C, since it's the biggest but overhead costs would be lower in town B. Where should they build?
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Well, I think I should sketch the function too, but I don't know how...
Thanks for any help.
Using the function
d(x) = |x| + 2|x - 1| + 3|x - 4|
make a table of values of
x and d(x)
for x = 0, x=1 and x=4.
If the company decides to build the delivery centre along a highway, then intermediate values of x should also be considered.
I have an impression that the owners will have quite a fight over the location.
Tell me what you think.
To sketch the function d(x) = |x| + 2|x - 1| + 3|x - 4|, we can start by breaking it down into separate pieces based on the absolute value notation.
For x < 0, the function becomes d(x) = -x + 2(-x - 1) + 3(-x - 4).
For 0 <= x < 1, the function becomes d(x) = x + 2(x - 1) + 3(x - 4).
For 1 <= x < 4, the function becomes d(x) = x + 2(x - 1) + 3(4 - x).
For x >= 4, the function becomes d(x) = x + 2(4 - x) + 3(4 - x).
Now, we can simplify each piece:
For x < 0, d(x) = -6x - 11.
For 0 <= x < 1, d(x) = 10 - 4x.
For 1 <= x < 4, d(x) = -2x + 13.
For x >= 4, d(x) = -2x + 20.
To sketch the graph, we can plot some key points and then connect them with appropriate curves.
1. For x < 0, we can choose some values, such as x = -2, -1, -0.5, 0, and plot the corresponding values of d(x).
2. For 0 <= x < 1, we can choose some values, such as x = 0, 0.5, 0.75, 1, and plot the corresponding values of d(x).
3. For 1 <= x < 4, we can choose some values, such as x = 1, 2, 3, and plot the corresponding values of d(x).
4. For x >= 4, we can choose some values, such as x = 4, 5, 6, and plot the corresponding values of d(x).
Once we have plotted these key points, we can connect them with smooth curves to create the graph of the function d(x). The lowest point on the graph will indicate the location of the delivery center with the minimum driving time.