Description
1. The Area of a rectangle is represented by and its length is represented by .
- What is the width of the rectangle?
- If you double the length of the rectangle but keep the same width, what is the new area of the rectangle?
2. Consider the following expression:
- Subtract the rational expressions
- Given the equation , solve the equation using your solution from part a.
- Solve the equation using LCD.
- Compare the methods used to solve the equation in parts c and d. Which method do you think is better and why?
3. The average cost of electricity used for an 18 cubic ft. refrigerator that is 10 years old is $128 per year. A new refrigerator the same size costs about $81 per year for electricity. (Source: https://www.efficiencyvermont.com)
- Suppose a new refrigerator costs $500. Determine the total cost of having either refrigerator for 12 years, assuming the only costs associated with the refrigerator are its purchase cost and electricity.
- Determine the total annual cost for a new refrigerator that lasts 12 years, assuming the only costs associated with the refrigerator are its purchase cost and electricity.
- Write a function that gives the annual cost of a new refrigerator as a function of the number of years you own the refrigerator.
- What are the asymptotes of the rational function in part d? What does the horizontal asymptote mean in the context of this problem?
- Sketch the graph of the rational function from part c.
- If you plan on owning your refrigerator for 12 years, would it be more cost effective to keep your old refrigerator or purchase a new one? Explain.
Explanation & Answer
The solutions are ready, please read them and ask if something is unclear.docx and pdf files are identical
1. Area is
𝑥 4 −𝑦 4
8𝑥𝑦
, length is
𝑥 2 −𝑦 2
2𝑥𝑦
.
a. The width is area divided by length, i.e.
𝑥4 − 𝑦4
𝑥 4 − 𝑦 4 2𝑥𝑦 (𝑥 2 − 𝑦 2 )(𝑥 2 + 𝑦 2 ) 1 𝒙𝟐 + 𝒚𝟐
8𝑥𝑦
=
∙
=
∙ =
.
𝑥 2 − 𝑦 2 𝑥 2 − 𝑦 2 8𝑥𝑦
𝑥2 − 𝑦2
4
𝟒
2𝑥𝑦
b. If we double the length of a rectangle keeping the same width, the area will become
twice greater. In this case it will be
𝑥 4 − 𝑦 4 𝒙 𝟒 − 𝒚𝟒
2∙
=
.
8𝑥𝑦
𝟒𝒙𝒚
2. The expression is
1
3𝑥
𝑥−4
− 𝑥 2 −16.
a. Subtract probably means make one fraction:
1
3𝑥
1
3𝑥
𝑥+4
3𝑥
− 2
=
−
=
−
=
𝑥 − 4 𝑥 − 16 𝑥 − 4 (𝑥 − 4)(𝑥 + 4) (𝑥 − 4)(𝑥 + 4) (𝑥 − 4)(𝑥 + 4)
𝑥 + 4 − 3𝑥
4 − 2𝑥
𝟒 − 𝟐𝒙
=
=
= 𝟐
.
(𝑥 − 4)(𝑥 + 4) (𝑥 − 4)(𝑥 + 4) 𝒙 − 𝟏𝟔
b. If it is equal to
2
𝑥+4
, we get
4−2𝑥
(𝑥−4)(𝑥+4)
2
= 𝑥+4. To solve, fir...
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