In a certain state, the maximum speed permitted on freeways is 65 mi/h and the minimum speed is 40 mi/h. The fine for violating these limits is $15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed x and graph F(x) for 1≤ x≤100.
The speed midway between the allowed values is (65+40)/2 = 52.5
with the allowable variation 12.5 mph
F(x) =
15(40-x) for 1 ≤ x < 40
0 for 40 ≤ x ≤ 65
15(x-65) for 65 < x ≤ 100
Extra credit: what if it's a step function, like postage?
Why did the cheetah get a ticket on the freeway? Because it was caught going way too fast and paw-ting the rules!
Now, let's express the fine F as a function of the driving speed x. Since the fine is $15 for every mile per hour above the maximum speed or below the minimum speed, we can define the function as follows:
F(x) = $15 * (|x - 65|) if x > 65
F(x) = $15 * (|x - 40|) if x < 40
F(x) = 0 if 40 ≤ x ≤ 65
To graph F(x) for 1 ≤ x ≤ 100, we'll consider the different regions separately. Please keep in mind that this graph is just for illustrative purposes!
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$15/mph |
__________.________|
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$15/mph | |
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$15/mph | |
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$15/mph | |
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$15/mph | |
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$0/mph | |
_____________.___.__|_____________________
40 65 100
Remember, this is just a graphical representation, and the actual fine amounts and speeds may differ. Drive safely, my friend!
To express the fine F as a function of the driving speed x, we need to consider two cases: when x is above the maximum speed and when x is below the minimum speed.
1. When x > 65 mph:
The fine for exceeding the maximum speed is $15 for every mile per hour above the limit. So, the fine for x > 65 mph would be F(x) = 15(x - 65).
2. When x < 40 mph:
The fine for driving below the minimum speed is $15 for every mile per hour below the limit. So, the fine for x < 40 mph would be F(x) = 15(40 - x).
For the range 1 ≤ x ≤ 100, we need to consider both cases and take the maximum of the two fines. So, the expression for the fine F(x) would be:
F(x) = max(15(x - 65), 15(40 - x))
To graph F(x) for 1 ≤ x ≤ 100, you can plot the points by substituting different x values into the expression for F(x).
To express the amount of the fine F as a function of the driving speed x, we can use a piecewise function since the fine depends on whether the speed is above the maximum limit, below the minimum limit, or within the permissible range.
Let's start by defining the piecewise function:
F(x) = 15(x - 65), if x > 65
F(x) = 15(40 - x), if x < 40
F(x) = 0, if 40 ≤ x ≤ 65
Explanation of the piecewise function:
1. If the driving speed (x) is greater than 65 mph, the fine is calculated by multiplying the excess speed (x - 65) by $15.
2. If the driving speed (x) is less than 40 mph, the fine is calculated by multiplying the shortfall in speed (40 - x) by $15.
3. If the driving speed (x) is within the range of 40 to 65 mph, there is no fine imposed; hence, the fine is 0.
Now let's graph F(x) for the range of 1 ≤ x ≤ 100:
First, create a graph with the x-axis representing the driving speed (x) ranging from 1 to 100, and the y-axis representing the fine amount (F).
For x > 65:
- Draw a straight line starting from the point (65, 0) with a slope of 15. This line represents the fine amount for speeds above the maximum limit.
For x < 40:
- Draw a straight line starting from the point (40, 0) with a slope of -15. This line represents the fine amount for speeds below the minimum limit.
For 40 ≤ x ≤ 65:
- Draw a horizontal line at y = 0, representing no fine for speeds within the limit range.
The resulting graph will show increasing fines as the speed exceeds the maximum limit or falls below the minimum limit.
Please note that the graph will be a combination of straight lines, and the slopes of these lines represent the fine rate of $15 per mph over the limit or under the limit.