# Operations on Functions: Word Problems (Grades 11-12)

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## Operations on Functions: Word Problems

1.

Andrew is filling a swimming pool with a garden hose. The hose is releasing water at a rate of 13 gallons per minute but the pool is leaking at a rate of .8 gallons per minute. If the formula for filling the pool is [math]F(m)[/math] and the formula for the leaking water is [math]L(m)[/math], then what is the appropriate combined formula?

- [math](F(m))/(L(m))[/math]
- [math]F(m)-L(m)[/math]
- [math]F(m)+L(m)[/math]
- [math]F(L(m))[/math]

2.

Andrew is filling a swimming pool with a garden hose. The hose is releasing water at a rate of 13 gallons per minute but the pool is leaking at a rate of .8 gallons per minute. Combine the equations for the filling and leaking to find the rate at which the pool is filling.

- [math]16.25 \ "gpm"[/math]
- [math]13.8 \ "gpm"[/math]
- [math]12.2 \ "gpm"[/math]
- [math]10.4 \ "gpm"[/math]

3.

Adam works at a clothing store. He earns $14 per hour. Adam also earns a sales commission of 10% of whatever he helps to sell. Typically, Adam helps to sell about $50 worth of clothing in an hour. Let [math]t[/math] represent time in hours, [math]P[/math] how much base pay Adam earns in dollars, and [math]C[/math] how much Adam earns in commissions (in dollars). Which of the following correctly shows the sum of these two functions, [math]P(t) [/math] and [math]C(t)[/math], thus representing Adam's approximate total hourly pay?

- [math](P+C)(t) = 0.1(14+50)t = 6.4t[/math]
- [math](P+C)(t) = 0.1*14 + t = 1.4 + t[/math]
- [math](P+C)(t) = 14t + 0.1t = 14.1t[/math]
- [math](P+C)(t) = 14t + 5t = 19t[/math]

4.

One machine pours out salt at a rate of 10 grams per minute. Let the function [math]S(t) = 10t, \ \ t>0[/math] represent this, where [math]S[/math] is measured in grams, and [math]t[/math] in minutes. Another machine pours out water at a rate of 2 liters per minute. Let the function [math]W(t) = 2t , \ \ t>0[/math] represent this, where [math]W[/math] is in liters, and [math]t[/math] in minutes. If the output of both machines is pouring into the same container, which of the following combined functions shows the correct salinity of the water (how much salt is dissolved in the water in grams per liter)?

- [math](W/S)(t) = 1/5 , \ \ t>0[/math]
- [math](S/W)(t) = 5, \ \ t>0[/math]
- [math](W+S)(t) = 12t, \ \ t>0[/math]
- [math](W*S)(t) = 20t^2, \ \ t>0 [/math]

5.

Janet has a savings account which earns compound interest. She initially invested $500, and the account earns 1.5% interest, compounded semi-annually. At the same time as she opened her account, her uncle decided he would set aside some money in separate account for Janet. He put in $200. The account he started pays simple interest, with a rate of 2%. Using the formulas for compound and simple interest, which of the following correctly shows the total amount being saved for Janet (in both accounts) after t years?

- [math]500(1.0075)^(6t) + 200[/math]
- [math]100","000(1.0075)^(2t) (1+0.02t)[/math]
- [math]200 + 703.77^(2t)[/math]
- [math]500(1.0075)^(2t) + 200(1+0.02t)[/math]

6.

A very large container has 5 gallons of water already in it and more water being added to it. The function [math]W(t) = 0.05t^2 + 5, \ \ 0 <= t < 24[/math] describes this, where [math]W[/math] is measured in gallons and [math]t[/math] is measured in hours. Just as the additional water is beginning to be added, a small population of bacteria is introduced to the container. Its growth in the water can be modeled approximately by the function [math]B(t) = 50e^(0.1t), \ \ t>=0[/math], where [math]B[/math] is the number of bacteria present, and [math]t[/math] is the time, in hours. What is the approximate concentration of bacteria, in bacteria count per gallon, at a given time [math]t[/math] as the container fills with water?

- [math](W*B)(t) = (0.05t^2 + 5) * (50e^(0.1t)), \ \ 0 <= t < 24 [/math]
- [math](W/B)(t) = (0.01t^2 + 1) / (10e^(0.1t)), \ \ 0 <= t < 24[/math]
- [math](B/W)(t) = (1000e^(0.1t)) / (t^2 + 100), \ \ 0 <= t < 24[/math]
- [math](B/W)(t) = (10e^(0.1t)) / (0.05t^2 + 1), \ \ 0 <= t < 24[/math]

7.

Carol is looking to start a business that sells bicycles. She knows from doing research that, in her area, the price of a bike can be modeled by the function [math]P(x) = 3000 - 4x[/math], where [math]x[/math] is the number of bikes. The function [math]Q(x) = x[/math] shows how many bikes Carol's business may produce. Which of the following combined functions is the correct revenue function for Carol's business?

- [math]R(x) = P(x)Q(x) = 3000x - 4x^2[/math]
- [math]R(x) = P(x) + Q(x) = 3000 - 3x[/math]
- [math]R(x) = P(x) - Q(x) = 3000 - 5x[/math]
- [math]R(x) = P(x)Q(x) = -4x^2 + 3000[/math]

8.

Abraham sells wool sweaters. Each month he pays $500 in rent for his store, and has to pay about $75 in wool for each sweater he makes. These represent his only costs. He knows from experience that his revenue can be modeled by the function [math]R(q) = -q^2 + 150q[/math], where [math]R[/math] is measured in dollars, and [math]q[/math] is the number of sweaters produced per month. Which of the following combined functions, where [math]C(q)[/math] is the cost function, shows Abraham's total monthly profit?

- [math]P(q) = C(q) - R(q) = q^2 - 75q + 500[/math]
- [math]P(q) = (R(q))/(C(q)) = (-q(q - 150))/(25(3q + 20))[/math]
- [math]P(q) = R(q) - C(q) = -q^2 + 75q - 500[/math]
- [math]P(q) = R(q)C(q) = -75q^3 + 10","750q^2 + 75","000q[/math]

9.

A small town is growing. Its population is increasing and its borders are expanding. The town's population growth can be modeled by the function [math]P(t) = 100e^(0.8t)[/math], where [math]t[/math] is the number of years that have passed since the current date and [math]P[/math] is the approximate number of people living in the town. The land area of the town can be approximately modeled by the function [math]A(t) = 0.25 pi (1+t)^2[/math], where [math]t[/math] is the number of years since the current date and [math]A[/math] is the land area in square kilometers. Which combined function shows the population density of the town?

- [math](P/A)(t) = (100e^(0.8t)) / (0.25 pi (1+t)^2) [/math]
- [math](A/P)(t) = (0.25 pi (1+t)^2) / (100e^(0.8t))[/math]
- [math](A/P)(t) = 100e^( (0.8t) / (0.25 \ pi \ (1 \ + \ t)^2)) [/math]
- [math](A*P)(t) = 25 pi e^(0.8t) (1+t)^2[/math]

10.

In general, an airplane's ground speed is equal to its airspeed plus the wind speed. A certain airplane is traveling due north, and its airspeed is approximately given by the function [math]A(t) = -3.515t^6 + 63.267t^5 - 445.43t^4+1549.1t^3 - 2774t^2 +2425t + 17[/math], [math] \ 0.5 <= t <= 5.5[/math], where [math]t[/math] is the time in hours since it left the airport, and [math]A(t)[/math] is measured in km/h. The wind speed during this time, which is directly in a southerly direction, is approximately modeled by the function [math]W(t) = 0.0069t^6 - 0.4t^5 + 5t^4 - 24.7t^3 + 52.3t^2 - 42t + 70[/math], [math] \ 0 <= t <= 6[/math], where [math]t[/math] is the time in hours since the plane left the airport, and [math]W(t)[/math] is measured in km/h. What is the ground speed of the plane for this period of time?

- [math](A+W)(t) = -3.5081t^6 + 63.667t^5 - 440.43t^4 + 1573.8t^3 - 2721.7t^2 + 2467t + 87[/math], [math] \ 0 <= t <= 6[/math]
- [math](A-W)(t) = -3.5219t^6 + 62.867t^5 - 450.43t^4 + 1524t^3 - 2826.3t^2 + 2383t - 53[/math], [math] \ 0 <= t <= 6[/math]
- [math](A+W)(t) = -3.5081t^6 + 62.867t^5 - 440.43t^4 + 1524.4t^3 - 2721.7t^2 + 2383t + 87[/math], [math] \ 0.5 <= t <= 5.5[/math]
- [math](A-W)(t) = -3.5219 t^6 + 63.667t^5 - 450.43t^4 + 1573.8t^3 - 2826.3t^2 + 2467t - 53[/math], [math] \ 0.5 <= t <= 5.5[/math]

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