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2.3 - Applications of Equations

This text language guide provides problem-solving steps, examples, and equations for various equation application problems such as number relations, dimensions/volumes, interest rate/time/distance, and mixture problems.

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2.3 - Applications of Equations

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  1. 2.3 - Applications of Equations Problem types we will do: Number Relations Dimensions/Volumes Interest Rate/Time/Distance Mixture Steps: Read/Label (know what you are solving for) Draw (if appropriate) Translate Solve Check

  2. Solve: ex: Number Relations: The average of 2 numbers is 41.125 and their product is 1683. find the two numbers. Average of two numbers a & b: • Key Info: • average • product • “is” Product of two numbers a & b:

  3. Check:

  4. 2x Draw: The rectangle is 3.5 in. x 7 in. x ex: Dimensions: A rectangle is twice as wide as it is high. If it has an area of 24.5 square inches (in2), what are its dimensions? • Area of any square object = side x side • Translate with what we have: 24.5 = (2x)(x) Positives only for physical dimensions

  5. Draw: h b b ex: Dimensions: A rectangular box with a square base and no top is to have a volume of 30,000 cm3. If the surface area of the box is 6000cm2, what are the dimensions of the box? • Keys: • Rectangular box • Square base • Open (No) top • Volume = 30,000 • Surface Area = 6000 • Translate: • Surface area is the areas of each individual side added together: How many sides do we have? • Volume for a square object = side x side x side representing length, width and height…

  6. Most convenient variable to solve for: Surface Area Equation: • Surface area for this box: • 4 sides of area (h x b) • 1 side of area (b x b) • Volume for this box: • (h x b x b) Volume Equation:

  7. 2 solutions

  8. Surface Area Equation: Find h: For b = 21.704: For b = 64.293: Check for b = 21.704 h = 63.686 Check for b = 64.293 h = 7.258 Box is either 21.704 x 21.704 x 63.686 OR 64.293 x 64.293 x 7.258

  9. ex: Interest • Interest (I) = A fee paid for the use of money. • Principal (P) = An initial amount of $ loaned or deposited. • Rate ( r )= A percentage used to determine interest. • Simple Interest Formula: I = Prt • ex: How much interest will you make on a $1000 deposit @ 8% • for one year? • I = (1000)(.08)(1) = $80 • General technique: when splitting a principal into two different • Amounts: • ex: $5000 is to be split between savings and checking • Designate a letter for one amount: savings = s • The other amount is the principal minus the letter you picked: • checking = (5000 - s)

  10. ex: A $9000 principal is to be split between a stock and a savings account. The stock pays a 12% dividend and the savings account pays interest @ 6%. How much of the $9000 should be put into the stock and the savings account to earn an 8% return on the principal after one year? • Keys: • $9000 principal • 2 different amounts with different rates • Should total 8% on $9000 = (.08)(9000) = $720 • Designate: • Amount into stock = s • Amount into savings = (9000 - s) • Stock dividend = 0.12 • Savings interest = 0.06

  11. Return on stock Return from savings 8% of $9000 Build equation: Solve: • Answer: • $3000 in stock • $6000 in savings

  12. ex: Distance/Rate/Time • Basic equation: D=rt • ex: If you drive 60mph for 2 hours how far have you gone? • (60)(2) = 120 miles • ex: If you drive 50mph how long do you have to drive to go 150 mi? • D = rt --> D/r = t --> 150/50 = 3 hours • ex: Splitting a distance between 2 rates and times: • You drive 30 mph for part of a trip and 60 mph for part of a trip - how long do you need to do each speed to cover 240 miles in 5 hours? • Total D = (D1) + (D2) • Total D = (r1t1) + (r2t2) • Total time (T) = t1 + t2 --> t1 = (T - t2) OR t2 = (T - t1) • 240 = (30t1) + (60t2)

  13. (r - 30) Start End (r + 40) • 240 = (30t1) + (60(5 - t1)) • 240 = (30t1) + (300 - 60 t1) • 240 = 30t1 - 60t1 + 300 • 240 = -30t1 + 300 • -60 = -30t1 • t1 = 2 • t2 = 3 • 240 = (30t1) + (60t2) • 2 hours @ 30 mph and 3 hours @ 60mph ex: A pilot wants to fly an 840 mi round trip in 5 hours. If there is a 30mph headwind on the way to the destination and a 40mph tailwind on the way back, what constant speed should the plane be flown at?

  14. Total Distance = 840 • Total time = 5 • Constant rate = r • 840 = (r - 30)(t1) + (r + 40)(t2) • 840 = (r - 30)(t1) + (r + 40)(5 - t1) • Still have 2 unknowns - have to solve for one in terms of the • other… • Distance for each leg of trip = 840/2 = 420 • 420 = (r - 30)(t1) --> Substitute:

  15. 0 200 • Check: • First leg time: • Second leg time: t1 + t2 = 5 hours @ 170

  16. Mixture: • A car radiator contains 12 quarts of fluid, 20% of which is antifreeze. How much fluid should be drained and replaced with pure antifreeze so that the resulting mixture is 50% antifreeze? • Tabular method:

  17. IC 2.3-A, pg 105: #’s 1-4 ALL, 5-7 ALL (setup only), 9, 10. IC 2.3-B, pg 105: #’s 11-17 all, 19, 25, 26.

  18. 2.3 Practice Problems Pg 135, #’s 17-24 all, 26.

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