Problem Solving/Modeling
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1 Prepared by: Sa diyya Hendrickson Problem Solving/ I. General Strategies S1 S Understand the problem. It may be helpful to draw charts or diagrams. Identify the following: 1 What is given and record all relevant formulas What is unknown: Give a let statement for the variable of your choosing and try to express all other unknowns in terms of this variable S3 S4 Connect the given to the unknown through formulas, equations, etc. Solve the equation(s using relevant strategies II. Toolbox Formulas are very helpful when doing problem solving. Below are some examples of common formulas that are usually learned in grade school. 1 Perimeter Formulas e.g. P parallelogram =` +w = (` + w where ` = length and w = width P circle = circumference = r where r = radius Area Formulas e.g. A parallelogram = b h where b = base and h = height A triangle = 1 b h where b = base and h = height Surface Area (cylinder: SA cylinder = (area of circle + (circumference height 3 Volume Formulas e.g. V rectangular prism = (area of base height = ` w h where ` = length, w = width and h = height 4 Pythagorean Theorem Given a right triangle, with side lengths a and b and hypotenuse c: a + b = c 5 Formula for Distance, Rate and Time For D = distance, r = rate/speed and t = time, D = r t ( t = D r ( r = D t Making Math Possible 1 of 5 c Sa diyya Hendrickson
2 III. Exercises from Section (Sect. 1.7, #30 A woman earns 15% more than her husband. Together they make $69,875 per year. What is the husbands annual salary? (1 woman s salary = husband s salary + 15% of husband s salary ( Combined salary = The desired unknown is the husband s salary. Let h = husband s salary. Then, from (1, we can express the woman s salary as follows: woman s salary = h + (15% of h 15 = h + h 100 = h = h 0 expressing a percentage as a fraction; factoring out h = 3 0 To connect what s given to what s unknown, we use the combined salary = 69875, given in (. We also know that: combined salary = woman s salary + husband s salary. This gives: 3 h + h = a linear equation! h = factoring out h 43 h = h = dividing both sides by 43/0 h = since = 165 h = 3500 Therefore, the husband s salary is $3,500. Making Math Possible of 5 c Sa diyya Hendrickson
3 . (Sect. 1.7, #4 A rectangular bedroom is 7 ft longer than it is wide. Its area is 8 ft. What is its width? Consider the following diagram on the right. (1 area = 8 ft ( length = width + 7 (units in ft The desired unknown is the width of the rectangular bedroom. Let w =width. Then, from (, we can express the length follows: length = w + 7 To connect what s given to what s unknown, consider the area = 8 ft, given in (1. We also know that: area = length width. This gives the following results: (w + 7 w = 8 a quadratic equation! w +7w 8 = 0 getting a zero on one side (w 1(w + 19 = 0 factoring w 1 = 0 or w + 19 = 0 by ZPP w = 1 or w = 19 solving the linear equations Recall that w denotes the width of the room. Thus, w > 0, making w = Therefore, the width of the room is 1 ft. 19 invalid. 3. (Sect. 1.7, #50 A poster has a rectangular printed area 100 cm by 140 cm and a blank strip of uniform width around the edges. The perimeter of the poster is 1 1 times the perimeter of the printed area. What is the width of the blank strip? Consider the following diagram on the right. 3 (1 Perimeter (poster = Perimeter (printed ( Dimensions of the printed area: 100 cm 140 cm With these dimensions, we have: Perimeter (printed = (length + (width Perimeter (poster = = (140 + (100 = = 480 cm 3 Perimeter (printed by (1 = by above result = 3(40 cm since 480 = 40 Making Math Possible 3 of 5 c Sa diyya Hendrickson
4 3. Continued The desired unknown is the uniform width of the blank strip. Let w = width of the strip (in cm. Consider the diagram to the right. To connect what s given to what s unknown, we recall that the Perimeter (poster = 3(40 cm, given in (. Also, from the diagram: Perimeter (poster = (140 + w + (100 + w Thus, we have the following results: (140 + w + (100 + w = 3(40 a linear equation! (140 + w + (100 + w = 3(40 dividing through by w w = 3(10 since 40 = 10 4w + 40 = 360 collecting 4w = w = 10 w = 10 4 = 30 Therefore, the width of the blank strip is 30 cm. 4. (Sect. 1.7, #64 Stan and Hilda can mow the lawn in 40 min if they work together. If Hilda works twice as fast as Stan, how long does it take Stan to mow the lawn alone? (1 Together, they mow one lawn in 40 min. ( Hilda mows twice as fast as Stan. The desired unknown is the time it takes Stan to mow one lawn. Let f = fraction of the lawn mowed by Stan in 40 min. Then, from (, we know that: Hilda s fraction = f To connect what s given to what s unknown, we use the fact that one lawn consists of Stan s fraction plus Hilda s fraction. This gives the following results: f +f =1 alinear equation! 3f = 1 collecting f = 1 3 Hence, Stan mows one-third of the lawn in 40 minutes. It then follows that Stan mows one lawn (three-thirds in 3(40 = 10 minutes, which is equal to hours. Making Math Possible 4 of 5 c Sa diyya Hendrickson
5 5. (Sect. 1.7, #74 Kiran drove from Tortula to Cactus, a distance of 50 mi. She increased her speed by 10 mi/h for the 360-mi trip from Cactus to Dry Junction. If the total trip took 11 h, what was her speed from Tortula to Cactus? (1 Distance from Tortula (T to Cactus (C is 50 mi ( Distance from Cactus (C to Dry Junction (DJ is 360 mi (3 Speed (C to DJ = Speed (T to C + 10 (4 Time (Total trip = 11 hrs The desired unknown is the speed from Tortula (T to Cactus (C. Let s = speed from T to C. To connect what s given to what s unknown, consider the Time (Total trip = 11 hrs (given in (4 and the fact that: Time (Total trip =Time(TtoC+Time(CtoDJ To determine Time (T to C and Time (C to DJ, we consider the following chart: Distance Rate Time = D r TtoC 50 mi s 50 s CtoDJ 360 mi s s+10 Using the formulas given in the chart, we have: Time (Total trip = Time (T to C + Time (C to DJ 11 = 50 s s = 50(s (s s(s (s(s + 10 = 50(s (s s(s + 10 (s(s + 10 since s =speed6= 0, 10 11s(s + 10 = 50s s a quadratic equation! 11s + 110s 50s s = 0 getting a zero on one side 11s 500s 500 = 0 collecting (s 50(11s + 50 = 0 factoring s 50 = 0 or 11s + 50 = 0 by ZPP s = 50 or s = 50/11 solving the linear equations Because s denotes the speed from Tortula to Cactus, we know that s > 0. This gives that s = invalid, leaving us to conclude that Kiran s speed from Tortula to Cactus was 50 mi/hr. 50/11 is Making Math Possible 5 of 5 c Sa diyya Hendrickson
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