4.3. Model With Formulas. Investigate Use Formulas to Solve Problems

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1 4.3 Model With Formulas The game of volleyball was invented in the late 19th century as an alternative to basketball. Six players on each team hit the ball back and forth over the net. The players try to hit the ball to the floor in the opposing team s court. In volleyball, an attack is any offensive hit of the ball and a kill is an attack that results in an immediate point. Investigate Use Formulas to Solve Problems The attack percent, Pct, in volleyball is calculated using this formula : Pct = K - E, where K is the number of kills, E is the number of attack TA errors, and TA is the total number of attacks. formula describes an algebraic relationship between two or more variables 1. Calculate the attack percent for each player. a) Kristi has 7 kills, 3 errors, and 25 attacks. b) Jessica has 11 kills, 2 errors, and 22 attacks. 2. Jade s attack percent is She has 1 error in 7 attacks. Find the number of kills Jade has made. Describe the steps you used to solve this problem. 3. Merella s attack percent is She has 1 kill and 2 errors. Find the total number of attacks Merella has made. How did you calculate the answer? 4. Use your answer to question 3 as a guide. a) Rewrite the formula to isolate total attacks. b) Reflect How are the steps you followed in part a) similar to the steps you followed in question 3? 174 MHR Chapter 4

2 Example 1 Rearrange Formulas Rearrange each formula to solve for the indicated variable. a) y = mx + b, solve for x b) A = P (1 + rt), solve for t c) w = u + at 2, solve for a Solution In each formula, follow the same steps you would to solve an equation. a) y = mx + b Subtract b from both sides of the equation. y - b = mx + b - b y - b = mx Divide both sides of the equation by m to y - b isolate x. m = mx m y - b m = x b) A = P (1 + rt) Multiply to remove the brackets. A = P + Prt Subtract P from both sides. A - P = P + Prt - P A - P = Prt Divide both sides by Pr to isolate t. A - P = Prt Pr Pr A - P = t Pr c) w = u + at 2 Subtract u from both sides. w - u = u + at 2 - u w = at 2 Divide both sides by t 2 to isolate a. w - u = at 2 t 2 w - u = a t 2 t Model With Formulas MHR 175

3 Example 2 Distance, Speed, and Time With One Object At a constant speed, the distance, (d), an object travels when it moves at a constant speed depends on how long it travels, (t), and its speed (s). The formula relating distance, speed, and time is d = st. Depending on their positions in orbit, the distance from Earth to Mars varies from a minimum of km to a maximum of km. A space probe travels km/h. The mean distance between Earth and Mars is km. How long does it take the probe to travel from Earth to Mars, on average? Solution Method 1: Substitute Then Solve Substitute d = , s = Then, solve for t. d = st = t Divide both sides by to isolate t = t t Since the values of d and s are known, isolate t. It would take the space probe approximately 5769 hours, or about 240 days to reach Mars. Method 2: Rearrange the Formula First d = st d s = st s d s = t Substitute d = and s = = t t It would take the space probe about 5769 hours, or approximately 240 days to reach Mars. Some formulas contain many variables. You must be sure what each variable represents as you solve the equation that results once values are substituted into the formula. 176 MHR Chapter 4

4 Example 3 Simple Interest This formula shows how the amount of simple interest, I, earned on an investment is related to the amount invested (also called the principal) in dollars, P, the interest rate, r, (expressed as a decimal) and the time, t, of the investment in years. I = Prt Damon deposits $500 into a savings account that pays simple interest at a rate of 0.65% per year. How long will it take Damon to earn $130 in interest? Math Connect A savings account is the easiest way to earn interest on your money. To learn more about how to earn interest with your money, go to www. mcgrawhill.ca /links/ foundations10 and follow the links. Solution Method 1: Substitute Then Solve I = 130, P = 500, and r = 0.65% or Substitute these values into the equation. 130 = 500 (0.0065) t 130 = 3.25t = 3.25t = t It will take Damon 40 years to earn $130 in interest. Method 2: Rearrange First I = Prt Since the values of I, P and r are known, isolate t. I Pr = Prt Pr I Pr = t Substitute I = 130, P = 500, and r = = t t = 40 It will take Damon 40 years to earn $130 in interest. 4.3 Model With Formulas MHR 177

5 Example 4 Taxes on a Meal A local restaurant features a live band. The bill for food and beverages has 6% GST, 8% PST, and a 12% service charge added. The restaurant also adds a cover charge of $25. If x represents the cost for food and beverages, the total cost in dollars, C, can be calculated using this equation: C = x x x x + 25 Rana s total bill was $ How much was the bill for Rana s food and beverages? Solution Method 1: Use Paper and Pencil Collect like terms to simplify the equation. C = x x x x + 25 C = 1.26x + 25 Substitute C = into the equation = 1.26x = 1.26x = 1.26x = 1.26x x The bill for Rana s food and beverages was $ Method 2: Use a Computer Algebra System 1. Press 2nd 1, to clear 1-character variables. 178 MHR Chapter 4

6 2. Press, 8, to clear the home screen and the command prompt line. 3. Type and press. 4. Type and press. 5. Type and press. 6. Press 2nd, 5 for Number. Then Right Arrow, 3 to paste round( to the command prompt line. 7. Type Press 2,. The bill for Rana s food and beverages was $ Model With Formulas MHR 179

7 Key Concepts By reversing the order of operations, formulas can be rearranged in the same way as equations are solved. When values for variables are given, the values can be substituted first and then the resulting equation solved, or the formula can be rearranged and then the values substituted. There is more than one way to solve most problems. Choose the method you are most comfortable with. Discuss the Concepts D1. Describe the steps required to rearrange each formula to obtain the form shown below. a) V = adt b) E = mc 2 c) s = w - 10e t w = V gh e = st - w -10 E c 2 = m D2. Mina and Francesco are solving this problem: An airplane travels 990 km in 4.5 h. How fast is the plane flying? Mina s solution started d = st 990 = 4.5t Who is correct? Explain. Francesco s solution started d = st d t = s = s Practise the Concepts A For help with question 1, refer to Example Rearrange each formula to solve for the indicated variable. a) A = lw, solve for w b) P = 2l + 2w, solve for l c) y = mx + b, solve for b d) C = 2πr, solve for r e) V = lwh, solve for h f) A = bh, solve for h 2 For help with question 2, refer to Example a) A car travels at 45 km/h for 2.5 h. How far does the car travel? b) Rearrange the formula d = st to solve for s. Use this formula to find the speed of a truck that travels km in 3.5 h. c) Rearrange the formula d = st to solve for t. Use this formula to find how long it would take a boat to travel 59.5 km at a speed of 34 km/h. 180 MHR Chapter 4

8 For help with questions 3 to 5, refer to Example The formula for the amount of simple interest earned on an investment is I = Prt, where I is the interest earned, P is the principal, or amount invested, r is the interest rate as a decimal, and t is the time the investment is left in the bank (in years). Find the amount of interest earned on an investment of $4000 at 0.85% interest after 4 years. 4. Use the simple interest formula I = Prt. Find the amount that needs to be invested at 8% per year for 10 years in order to earn $2000 in interest. 5. a) Rearrange the formula I = Prt to solve for t. b) Rearrange the formula I = Prt to solve for r. c) Rearrange the formula I = Prt to solve for P. d) Copy and complete the table. I P r t For help with question 6, refer to Example An all-inclusive resort vacation charges an airport tax of $150 plus 8% tax and 10% gratuity. If the total cost for a vacation is $3926, what is the cost of the vacation before the extra fees and taxes? Apply the Concepts B 7. Graham and Colin leave the same place at the same time and drive in opposite directions. Colin drives 10 km/h faster than Graham does. After 2 h, they are 200 km apart. How fast is each man driving? 8. Jenna and Maya have walkie-talkies with a range of 5 km. They leave the park on their bicycles, at the same time. Jenna rides east at 14 km/h and Maya rides west at 12 km/h. After half an hour, will they be able to use their walkie-talkies? How do you know? 4.3 Model With Formulas MHR 181

9 9. a) Describe a situation in which you would rearrange a formula before substituting the known values and solving. b) Describe a situation in which you would substitute known values into a formula before rearranging and solving. 10. The equation s = w - 10e models the speed in words per minute, t s, at which someone types. The speed, s, is related to the number of words typed, w, the number of errors, e, and the time spent typing in minutes, t. Alex types 525 words in 5 min, with 10 errors. What is Alex s typing speed? 11. Use the equation for typing speed from question 10. Melanie s typing speed is 100 word/min. She types 800 words in 7 min. How many errors did Melanie make? 12. The formula F = 9 C + 32 relates temperature measured in 5 degrees Fahrenheit, F, to temperature measured in degrees Celsius, C. Nalini s air conditioner is broken. Her thermostat, which is calibrated in degrees Fahrenheit, reads 88 F. a) Rearrange the formula to isolate C. b) Use the formula from part a) to find the temperature inside Nalini s house in degrees Celsius. c) A quick way to approximate the conversion from degrees Celsius to degrees Fahrenheit is to double the temperature in degrees Celsius and add 30. This can be expressed using the formula F = 2C Rearrange this formula to isolate C. d) Use your answer to part c) to find the approximate temperature inside Nalini s house in degrees Celsius. e) Use a graphing calculator. Graph your equations from parts a) and c). Use the graph to explain why the short-cut formula gives a good approximation to the actual temperature. 13. Underwater pressure in the ocean increases by about 51 kpa for every 5 m of depth. The Serafina submersible is designed to descend to 3000 m. How much pressure will the Serafina need to withstand at that depth? 182 MHR Chapter 4

10 Chapter Problem 14. a) A banquet hall charged $ for an event that 250 people attended. If the hall has a flat fee of $4000 for an event plus a charge per person attending, what is the charge per person? b) A second hall charged $ for an event that 300 people attended. If this hall does not have a flat fee, how much does it charge per person? c) Which hall would be a better deal for an event where 400 people are expected to attend? Explain your answer. Achievement Check 15. With every increase in altitude of 1000 m, the temperature decreases by about 6 C. At the base of a mountain the temperature is 10 C. a) Write an equation to model the temperature on the mountain. b) The temperature outside the tents at the first camp for climbers is -10 C. How high up the mountain is the camp? c) An airplane flies over the mountain at a height of 7.5 km. What is the temperature outside the plane? Extend the Concepts C 16. One measure of a baseball pitcher s performance is WHIP, walks and hits per inning pitched. This statistic relates the number of runners who get on base per inning, r, to the total number of walks, w, the total number of hits, h, and the total number of innings pitched, i, according to the formula r = w + h. i The value of r is calculated to two decimal places. A high school coach is trying to decide which pitcher to use in the final game. Copy and complete the table. Which pitcher should the coach choose? Why? Pitcher Total Walks (w) Total Hits (h) Total Innings (i) Raymond Jesse Tran Harvinder Igor WHIP (r) 4.3 Model With Formulas MHR 183