Applications of Equations and

1
Chapter 1

• Applications of Equations and
• Inequalities

2
Example 3 (p.52)

• Anderson Company produces a product for which the
  variable cost per unit is 6 and fixed cost is
  80,000. Each unit has a selling price of 10.
  How many units must be sold for the company to
  earn a profit of 60,000?
• Variable costs are those that vary with amount of
  output
• Fixed costs are those which must be paid even if
  no output is produced

3
Solution

• Profit total revenue total cost
• 60,000 10q (6q 80,000)
• Solving for q yields
• 60,000 4q - 80,000
• 4q 140,000
• q 35,000
• 35,000 units must be sold to earn a profit of
  60,000

4
Example 4 (p. 53)

• Sportcraft manufactures womens sportswear to
  sell in its new line of slacks to retail outlets.
  The cost to the retailer will be 33/pr. What
  price should be marked on the price tag so that
  the retailer can reduce this price by 20 during
  a sale and still make a profit of 15 over cost?

5
Solution

• Selling price cost/pr. profit/pr.
• p 0.2p 33 (0.15)(33)
• 0.8p 1.15(33)
• p (1.15)(33)/(0.8)
• P 47.44
• Mark the price tag at 47.44

6
Example 5 (p.53)

• A total of 10,000 was invested in two business
  ventures, A and B. At the end of the first year,
  A and B yielded returns of 6 and 5.75,
  respectively, on the original investments. How
  was the original amount allocated if the total
  amount earned was 588.75?

7
Solution

• Let x be the total amount invested in A at 6.
• Then,
• (0.06)x (0.0575)(10,000 x) 588.75
• (0.0025)x (0.0575)(10,000) 588.75
• (0.0025)x 575 588.75
• (0.0025)x 13.75
• x 5500

8
Example 6 (p. 53-54)

• The board of directors of Maven Corporation
  agrees to redeem some its bonds in 2 years. At
  that time, 1,102,500 will be needed. Suppose
  that the firm sets aside 1,000,000 today. At
  what rate of interest, compounded annually, will
  this money have to be invested so that its future
  value will be sufficient to redeem the bonds?

9
Solution

• Let r equal the annual rate of return
• Then,
• At end of second year, the accumulated amount
  will be 1,000,000(1r) plus the interest on this
  amount of 1,000,000(1r)r, or
• 1,000,000(1r) 1,000,000(1r)r
• 1,000,000 (1r)(1r) 1,000,000(1r)2

10
Solution (cont.)

• So,
• 1,000,000(1r)2 1,102,500
• (1r)2 1.1025
• (1r) SQRT(1.1025)
• (1r) 1.05
• r 0.05
• The funds must invested at 5

11
Example 7 (p.54)

• A real-estate firm owns the Parklane Garden
  Apartments, which has 96 units. At 550/month
  each of the apartments can be rented, but each
  25 increase in rent will result in 3 vacancies.
  To receive 54,600 each month what rental price
  should be charged for each apartment?

12
Solution

• Let n number of 25 increases in rent (i.e., if
  n 2, then monthly rent is 600)
• So,
• 54,600 (550 25n)(96 3n)
• 54,600 52,800 2400n 1650n 75n2
• 75n2 750n 1800 0
• n2 - 10n 24 0
• (n-4)(n-6) 0, thus the rent charged either
  could be 650 or 700

13
Solution (cont.)
revenue
54,675
54,600
4
n
5
6
14
Homework

• P. 55-59, 9,11,13,15,17,19,21,29,31,33

15
Inequalities

• Definition An inequality is a statement that
  one number is less than another number
• Rules for inequalities
• If a lt b, then (a c) lt (b c)
• If a lt b, and c gt0, then ac lt bc
• If a lt b, and c lt 0, then ac gt bc
• If a lt b, then (1/a) gt (1/b)

16
Solving Inequalities

• Solve 2(x-3) lt 4
• X-3 lt 2
• X lt 5
• Solve (3/2)(s-2) 1 gt -2(s-4)
• 3(s-2) 2 gt -4(s-4)
• 7s gt 16 4
• s gt 20/7

17
More Inequalities

• Solve 3 2x ? 6
• -2x ? 3
• X ? -(3/2)
• Solve 2(x-4) 3 gt 2x 1
• 2x 11 gt 2x 1
• -11 gt -1 ???
• No solution

18
Homework

• P. 64, 9,11,15,19
• It would be a good idea to do a few more of the
  odd numbered ones too, just to make sure that you
  have all of this clearly in mind

19
Applications of Inequalities

• For a company that produces aquarium heaters, the
  combined cost for labor and material is
  21/heater. Fixed costs are 70,000. If the
  selling price of a heater is 35, how many must
  be sold for the company to earn a profit?

20
Solution

• Profit Total revenue Total cost, so profit is
  positive if TR gt TC.
• 35q gt 70,000 21q
• 14q gt 70,000
• q gt 5000
• So, more than 5,000 heaters must be sold for the
  company to earn a positive profit

21
Example 2 (p.65)

• A builder must decide whether to rent or buy an
  excavating machine. If he were to rent the
  machine, the rental fee would be 3000/month.
  And the daily cost of gas oil and labor would be
  180 for each day the machine is used. If he
  were to buy it, fixed cost would be 20,000 and
  daily costs would be 230. When is the cost of
  renting less than the cost of purchasing?

22
Solution

• Cost of renting lt cost of purchasing
• 12(3000) 180d lt 20,000 230d
• 36,000 180d lt 20,000 230d
• 16,000 lt 50d
• 320 lt d
• So, the rental cost is less expensive if the
  machine is rented for 321 days/yr or more

23
Example 3 (p. 65-66)

• Ace Sports Equipment Company has current assets
  (cash, inventory, accounts receivable) and
  current liabilities (short term loans, taxes
  payable). The company wishes to expand inventory
  by taking out a short-term loan. How much can be
  borrowed and maintain the current ratio above 2.5?

24
Solution

• Current ratio current assets ? current
  liabilities
• So we need to solve for the most that can be
  borrowed (x)
• (350,000 x) ? (80,000 x) ? 2.5
• (350,000 x) ? 2.5(80,000 x)
• 150,000 ? 1.5x
• 100,000 ? x
• Company can borrow up to 100,000

25
Example 4 (p. 66)

• A publishing company finds that the cost of
  publishing each copy of a certain magazine is
  1.50. The advertising revenue is 10 of revenue
  received from dealers for all copies sold beyond
  10,000. What is the least number of copies that
  must be sold to earn a profit for the company?

26
Solution

• Let q equal the number of copies sold and solve
  for total revenue total cost gt 0
• So, write
• 1.40q (0.10)1.40(q-10,000) 1.5q gt 0
• 0.04q 1400 gt 0
• q gt 1400/0.04 35,000
• Must sell more than 35,000 copies to earn a
  profit

27
Homework

• P. 67 1,3,5,7,9

28
Absolute Value

• Definition The absolute value of a real number,
  written ?x?, is defined as ?x? x if x ? 0 and
  ?x? -x if x lt 0
• Example Suppose x5, then ?x? 5. Suppose x
  -5, then ?x? -(-5) 5
• Need to be able to work with both absolute value
  equalities and inequalities

29
Example 1 (p.68)

• Absolute value equalities
• ?x-3? 2 means that x-3 either could equal 2 or
  2. Therefore, x either could be 5 or 1
• ?7 - 3x? 5 means that 7 3x either could be
  equal to 5 or 5. Therefore, if we solve 7 3x
  5 we get 2/3. If we solve 7 3x -5 we 4
• ?x-4? -2 has no solution

30
Example 2 (p.69)

• Solve ?x -2? lt4. Solve first as an equality.
  This means that x could either be x -2 or x
  6. So, for the absolute value inequality to hold
  2 ltx lt 6.
• Solve ?3 -2x? ? 5. Solve first as an equality.
  This means that 3 2x either could equal 5 or 5
• If equal to 5, then 3 2x 5 ? x -1
• If equal to 5, then 3 2x -5 ? x 4
• So 1 ? x ? 4

31
Homework

• P. 71 11, 19, 29, 31. Do additional problems
  from the odd numbered exercises too if needed to
  get the ideas clearly in mind.