Suppose the rental cost for a snowboard is $25 for the first d...

Question

Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).

Evaluate lim t→2.9 f(t).

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with limits. This problem says, a library finds $$0.50 for the first day or any part of the first day. A book is overdue and 0.25 $$for each additional day or any part of the day. The book is not returned. We need to determine the limit as T approaches 4.5 of FFTFFT represents the total fine as a function of the number of overdue days. T we're given four possible choices as our answers. For choice A, we have $$1.38 for choice B, we have 1.50 $$for choice C, we have $$1.75 and for choice D, we have 1.13. $$Now, before we can take the limit, we first need to define our function F of T and so F of T gonna represent the fine after the um number of T overdue days. So we'll have FT is equal to and here we're gonna have a piece wise function. This is gonna be equal to 0.50 if zero is less than T is less than, or equal to one and 0.50 plus 0.25 multiplying the quantity of T minus one if T is greater than one. So with this, um, function is saying that if T is equal to or less than one day, then the fine is just gonna be $$0.50. But if it's greater than one day, we'll still have the 0.50 $$fine for the first day. But we'll have to add to that a $$0.25 fine for each additional day. So that's why we have T minus one. Since the 0.50 $$fine takes into account the first day and we only need to add the $$0.25 uh fine for each additional day. So that's why we subtract one. So we have our function defined, we now can take the limit. So we're looking at the limit as T approaches 4.5 of F of T. Now, in this case, T is going to be greater than one. So we can substitute our expression from the function definition into this limit. So we'll have the limit as T approaches 4.5 of the quantity of 0.50 plus 0.25 multiplying the quantity of E minus one. Now, we can take the limit or determine the limit here by setting T equal to five in our expression. That's because we're told the problem that any part of the day gets rounded up the next day for the time that the book is not returned. So here we have 4.5. So that would round up to the next day of five. So substituting T equal to five into our expression to evaluate this limit, we will have 0.50 plus 0.25 multiplying the quantity of five minus one. So evaluating that expression, this is gonna be equal to 1.50. It's gonna be my fine in dollars. So that is 1.50. $$And that corresponds to the correct answer choice of B for this problem. So I hope that this explanation has been useful and I'll see you in the next video.

Suppose the rental cost for a snowboard is \$25 for the first day (or any part of the first day) plus \$15 for each additional day (or any part of a day).
Evaluate lim t→2.9 f(t).

Key Concepts

Limits

Piecewise Functions

Continuity

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