Mathematics, 05.10.2019 16:00 zurfluhe

The town librarian bought a combination of new-release movies on dvd for $20 and classic movies on dvd for $8. let x represent the number of new releases, and let y represent the number of classics. if the librarian had a budget of $500 and wanted to purchase as many dvds as possible, which values of x and y could represent the number of new-release and classic movies bought? x = 8, y = 45 x = 10, y = 22 x = 16, y = 22 x = 18, y = 18

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Answer from: dumbdumbkylee

x = 16, y = 22.

Further explanation

We will solve the problem of linear programming. The case is simple, but we must still be accurate in doing so.

Given:

The cost for new-release movies on DVD for $20 and classic movies on DVD for $8. The librarian had a budget of $500 and wanted to purchase as many DVDs as possible.Let x represent the number of new releases and let y represent the number of classics.

Question:

Which values of x and y could represent the number of new-release and classic movies bought?

The Process:

Because the number of goods must be positive, then

$\boxed{ \ x \geq 0 \ }$ $\boxed{ \ y \geq 0 \ }$

The total number of DVDs purchased is 20x + 8y (in USD) and must not exceed $500. Thus, we have the inequality

$\boxed{ \ 20x + 8y \leq 500 \ }$

Let's examine one by one for each available option, whether it meets the inequality above.

Option A

$\boxed{ \ x = 8, y = 45 \ } \rightarrow \boxed{ \ 20(8) + 8(45) = 520 \ }$

This did not meet and was rejected.

Option B

$\boxed{ \ x = 10, y = 22 \ } \rightarrow \boxed{ \ 20(10) + 8(22) = 376 \ }$

It fulfills but then we will examine it again.

Option C

$\boxed{ \ x = 16, y = 22 \ } \rightarrow \boxed{ \ 20(16) + 8(22) = 496 \ }$

It fulfills but then we will examine it again.

Option D

$\boxed{ \ x = 18, y = 18 \ } \rightarrow \boxed{ \ 20(18) + 8(18) = 504 \ }$

This did not meet and was rejected.

Comparison of Option B and Option C

$\boxed{ \ x = 10, y = 22 \ } \rightarrow \boxed{ \ x + y = 10 + 22 = 32 \ DVDs \ }$ $\boxed{ \ x = 16, y = 22 \ } \rightarrow \boxed{ \ x + y = 16 + 22 = 38 \ DVDs \ },$ this means more DVDs.

The librarian wanted to purchase as many DVDs as possible. Thus, values of x and y could represent the number of new-release and classic movies bought is $\boxed{ \ x = 16, y = 22 \ }$ .

Learn moreWhich expression represents the amount of change Huang should receive? Anna has the most money Does Marcus pay tax?

Keywords: linear programming, the town librarian, a combination of new-release movies on DVD, classic, purchases, budget, as many DVDs as possible

The town librarian bought a combination of new-release movies on dvd for $20 and classic movies on d

Answer from: Quest

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.

Me if you can and if you finished this lesson on edg- comment you.

Answer from: Quest

Do something that you've always wanted to do. do something crazy, out of the ordinary but stay in your lanejaja hope it : d

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