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Problem 65 Hard Difficulty

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 dollars for every mile per hour above the maximum speed or below the maximum speed. Express the amount of the fine $ F $ as a function of the driving speed $ x $ and graph $ F(x) $ for $ 0 \le x \le 100 $.

Answer

$F(x)=\left\{\begin{array}{ll}15(40-x) & \text { if } 0 \leq x<40 \\ 0 & \text { if } 40 \leq x \leq 65 \\ 15(x-65) & \text { if } x>65\end{array}\right.$

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Video Transcript

in a certain state. The maximum speed permitted on freeways is sixty five miles per hour and the minimum speed is forty miles per hour. The fine for violating these limits is fifty dollars for every mile per hour above the max speed and below the I think that's a mistake. This supposed be minimum men's speed express the amount of the fine F as a function for the driving speed acts and graph capital s for excess between zero and one hundred. Okay, so it looks like it looks like we'LL have three different costs for a CE Forrest finds go in The easiest one to know is that we'LL pay zero fine if X is between forty and sixty five Okay, you notice that lets look at if we were to drive below the minimum speed. Okay, so it's fifty dollars for every miles per hour below forty miles per hour. So it will be fifteen well supplied by however many miles power we're going. So that will be represented by forty minus X if X is below forty. Okay. And so going above the max speed that'LL be fifteen dollars multiplied by however fast we're going minus sixty five if X is greater than sixty five. So this will be our function for our finds there and to draw a rough sketch. This is the mouse Brauer. And this will be the cost for fine. Okay? And so, if x word to be zero, it looks like we're paying. Ah, we're going zero mas per hour. That's forty, uh, forty miles per hour below the minimum speeds. Who were doing fifteen times forty. Don't do that in my calculator. Really quick. That's fifteen times forty, which is six hundred. So we're going zero miles per hour. Then I guess we're gonna pay six hundred bucks all the way until we're going forty miles per hour. That's a forty and then So this is sixty there. So we're paying zero dollars going that way. Just blue there. And for a hundred miles per hour. Okay, Looks like we're going to go one hundred minus sixty five times fifteen, which is five hundred twenty five bucks, which is right below six hundred. Do that green there. So maybe missile. This's a rough sketch of what the graph will look like