Taxicab fees A taxicab ride costs $3.50 plus ...

Question

Taxicab fees A taxicab ride costs $3.50 plus $2.50 per mile for the first 5 miles, with the rate dropping to $1.50 per mile after the fifth mile. Let m be the distance (in miles) from the airport to a hotel. Find and graph the piecewise linear function c(m) that represents the cost of taking a taxi from the airport to a hotel m miles away.

Accepted Answer

Welcome back, everyone. In this problem, there is an entry fee for a shop of $5. The costs of chocolate per kilogram for the 1st 3 kg is $12 per kilogram. After 3 kg, the cost becomes $10 per kilogram. What would be the total price if a child buys 10 kg of chocolates? Also, draw the graph of the function. Here each answer choice has the possible. Function and sorry, the functions and the graph and A says when X is between 0 and 3, Y equals 15 X + 5, and when X is greater than 3, Y equals 12 x + 11. B says that when X is between 0 and 3, Y E equals 12 x + 5, and when X is greater than 3, Y E equals 12 x + 11, where this is our graph of B. C says that when X is between 0 and 3, Y equals 10 x + 5, and when X is greater than 3 Y equals 10 X + 11, and here is the graph. And D says that when X is between 0 and 3, Y equals 12 x + 5, and when X is greater than 3, Y E equals 10 X + 11. And again, here is the graph. Now below that I've added a small graph here just to help us to draw or represent our function, OK? So this is gonna be our graph, OK. Where our vertical axis is gonna be the Y axis and our horizontal axis is going to be the X axis. Now, what do we know from our problem statement that can help us to draw the graph of the function? Well, let's start with the first part of our problem statement. It says it has an entry fee for a shop of $5 and the cost of chocolate per kilogram for the 1st 3 kg is $12 per kilogram. OK? So when OK, when X is between 0 and 3, OK, where X would only be a positive number as the value it represents, the weight cannot be negative, OK. Then we know. That why or, sorry, that would be our cost is going to be equal to our initial cost of $5 OK? Plus. The additional cost for $12 per kilogram. So if X equals 1, it's going to be 5 + 12 multiplied by 1. If X equals 2, it's gonna be 5 + 12 multiplied by 2. Generally, generally it's gonna be 5 + 12 multiplied by X. OK? In other words, on our graph, Y equals 12 X + 5. Now for a second condition, that is when X is greater than 3, we're told that after 3 kg, the cost becomes $10 per kilogram, OK? So no, that means then that Y is going to be equal to the initial cost of 5 plus 12 X. In this case, X is 3 because this is happening. After 3 kg. Plus 10 multiplied by the difference between whatever the weight is X and 3, OK? Our first initial weight, OK. Oh sorry, or the, the, the point at which it stops being $12. Now if we were to expand our expression here, that's going to be 5 + 36 + 10 X minus 30, which when we simplify 5 + 36 minus 30 equals 11. So Y is going to be equal to 10 X plus 11 when X is greater than 3. Now that we have our functions for Y or our equations for Y rather, depending on the domain, we can plot it on our graph. Now, for when X is between 0 and 3, we can plot two points on the graph, OK? So, for example, When X is 0, OK, then Y would be equal to 12 multiplied by 0 + 5, which is 5. So 05 is on the graph. Likewise, when X is 3, OK, uh X is 3, then Y is going to be equal to 12 multiplied by 336 + 5, which is 41. So we know that 05 and 341 are on our graph. So now when you put that on our graph, so we have 0 here, notice that this is 10, so 5 would be here, so that's 05. And then we have 34 to 1, which would be somewhere right here, OK. And know when X is greater than 3, OK? We know that we could still use 341 just to point out where, where it stops, OK? Because if we think about it, when X is greater than 3, if X were to be 3, then Y would be 10 multiplied by 3 + 11, which is 41. And then if we take another value for X, let's say that X is equal to 4, OK. Then, then multiplied by 4 equals 40 plus 11, that equals 51. So, 451 is going to be on the graph. Yeah. So if we go back to our our our graph here, 451 is probably going to be somewhere right here, OK. So now with that in mind, we can draw our graph. It's gonna be for when sorry, Y is when X is between 0 and 3 and then no, we have it when X is between. Or sorry, when X is greater than 3, OK? So this is going to be our graph for our scenario. Now if we were to look back at our answer choices, then we can tell that B is the correct answer. Thanks for watching everyone. I hope this video helped.

Key Concepts

Piecewise Functions

Linear Functions

Graphing Functions

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