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1 Topic Economic Tasks

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2 Topic Economic Tasks What it means for you: Youll model money problems using quadratic equations, and then solve the equations. Key words: economic profit quadratic vertex parabola completing the square California Standards: 23.0: Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

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3 Topic Economic Tasks As well as the motion tasks you saw in Topic 7.6.1, you can use quadratic equations to model real-life problems involving money.

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4 Topic Applications of Quadratics to Economics Economic Tasks The best way to introduce quadratic equations modeling money problems is to show you an example:

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5 Topic Economic Tasks Example 1 The owner of a restaurant wishes to graph the annual profit of his restaurant against the number of people he employs. He calculates that the annual profit in thousands of dollars ( P ) can be modeled by the formula P = –0.3 x x, where x is the number of people employed. According to the owners formula, how many full-time members of staff does the restaurant have to employ to make a profit of $15,000? Solution follows…

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6 Topic Economic Tasks Example 1 P = –0.3 x x x : number of staff. P : profit in thousands of dollars. Solution You have a formula for the profit P, and you have to find when this equals 15 (since the formula gives you the profit in thousands of dollars). So you need to solve the quadratic equation –0.3 x x = 15. Solution continues…

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7 Topic Economic Tasks Example 1 Rewriting –0.3 x x = 15 in the form ax 2 + bx + c = 0 gives: 0.3 x 2 – 4.5 x + 15 = 0 This means that the restaurant can employ either 5 people or 10 people and make a profit of $15,000. Solve using the zero property x = 10 or x = 5 ( x – 10)( x – 5) = 0 Divide through by 0.3 x 2 – 15 x + 50 = 0 Solution (continued)

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8 Topic Example 2 Solution follows… Economic Tasks The owner of a restaurant wishes to graph the annual profit of his restaurant against the number of people he employs. He calculates that the annual profit in thousands of dollars ( P ) can be modeled by the formula P = –0.3 x x, where x is the number of people employed. According to the owners formula, how many full-time members of staff should the restaurant employ to make maximum profit? Solution To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3 x x. To do this, you can complete the square: P = –0.3 x x = –0.3( x 2 – 15 x ) So the vertex of the parabola is at, which (in theory) means that the restaurant should employ 7.5 people to make the maximum possible profit of $16,875. Clearly, the restaurant cant employ 7.5 people a good idea now is to draw the graph so that you can answer this question more realistically. Find the x -intercepts by solving P = 0: = 0 at x = 0 and x = 15 = –0.3 x ( x – 15) = –0.3( x 2 – 15 x ) P = –0.3 x x So the graph looks like this: You can see from the symmetry of the graph (the line of symmetry is x = 7.5) that the maximum possible profit while employing a whole number of people is at x = 7 and x = 8, at which points the profit is $16,800. So, if the restaurant employs more than 8 people, profits decrease, possibly because there is not enough work for more than 8 people to do efficiently.

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9 Topic Guided Practice Solution follows… Economic Tasks 1. The profit p in cents per 10-minute period earned from driving a taxicab is given by p = 80 x – 3 x 2, where x is the speed in mph. What speed would yield a profit of 512 cents per 10-minute period? 2. An investor kept track of her portfolio profit, P, at time, t, measured in years after she began investing. If P = 4000 t 2 – t represents her profit, after how many years will she have made $150,000 profit? 3. The amount of money a customer is willing to spend at a store is related to t, the number of minutes they have to wait before being served. If M = – t t + 17 represents the money a customer spends, how long will it take before a customer decides to leave the store without spending any money? x = 16 mph or x = 10.7 mph (1 d.p.) 10.5 years t = 9.74 minutes (2 d.p.)

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10 Topic Independent Practice Solution follows… Economic Tasks Leo produces x pounds of salsa. The ingredients cost 0.1 x 2 – 30 dollars and he makes 2 x dollars revenue from the sale of his salsa. 3. What was the original value of the stock? 4. What was the maximum value of the stock? 5. When did the stock reach the maximum value? 6. When did the stock become worthless? The value in dollars, V, of a certain stock can be modeled by the equation V = –16 t t + 101, where t represents the time in months. 1. What is Leos maximum possible profit? 2. How many pounds of salsa would Leo need to sell to break even? $40 30 pounds $ $ months (1 d.p.) months 11 4

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Topic Independent Practice Solution follows… Economic Tasks The value, V, of Juans investment portfolio can be modeled by the equation V = 16 t 2 – 256 t + 16,000, where t is the time in months. 7. What was the original value of Juans portfolio? 8. What was the minimum value of Juans portfolio? 9. When will Juans investment portfolio be worth $16,576.00? $16,000 $14,976 After 18 months

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12 Topic Economic Tasks Round Up Usually when you graph quadratic problems involving money, the vertex of the graph shows you the point where theres maximum profit.

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13 Topic Example 2 Solution follows… Economic Tasks The owner of a restaurant wishes to graph the annual profit of his restaurant against the number of people he employs. He calculates that the annual profit in thousands of dollars ( P ) can be modeled by the formula P = –0.3 x x, where x is the number of people employed. According to the owners formula, how many full-time members of staff should the restaurant employ to make maximum profit? Solution To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3 x x. To do this, you can complete the square: P = –0.3 x x = –0.3( x 2 – 15 x ) So the vertex of the parabola is at, which (in theory) means that the restaurant should employ 7.5 people to make the maximum possible profit of $16,875. Clearly, the restaurant cant employ 7.5 people a good idea now is to draw the graph so that you can answer this question more realistically. Find the x -intercepts by solving P = 0: = 0 at x = 0 and x = 15 = –0.3 x ( x – 15) = –0.3( x 2 – 15 x ) P = –0.3 x x So the graph looks like this: You can see from the symmetry of the graph (the line of symmetry is x = 7.5) that the maximum possible profit while employing a whole number of people is at x = 7 and x = 8, at which points the profit is $16,800. So, if the restaurant employs more than 8 people, profits decrease, possibly because there is not enough work for more than 8 people to do efficiently.

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14 Topic Economic Tasks Example 2 The owner of a restaurant wishes to graph the annual profit of his restaurant against the number of people he employs. He calculates that the annual profit in thousands of dollars ( P ) can be modeled by the formula P = –0.3 x x, where x is the number of people employed. According to the owners formula, how many full-time members of staff should the restaurant employ to make maximum profit? Solution follows…

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15 Topic Example 2 Solution follows… Economic Tasks Solution To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3 x x. To do this, you can complete the square: P = –0.3 x x = –0.3( x 2 – 15 x ) So the vertex of the parabola is at, which (in theory) means that the restaurant should employ 7.5 people to make the maximum possible profit of $16,875. P = –0.3 x x x : number of staff. P : profit in thousands of dollars. P = –0.3 x x

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16 You can see from the symmetry of the graph (the line of symmetry is x = 7.5) that the maximum possible profit while employing a whole number of people is at x = 7 and x = 8, at which points the profit is $16,800. So, if the restaurant employs more than 8 people, profits decrease, possibly because there is not enough work for more than 8 people to do efficiently. Topic Example 2 Solution follows… Economic Tasks Solution Clearly, the restaurant cant employ 7.5 people a good idea now is to draw the graph so that you can answer this question more realistically. Find the x -intercepts by solving P = 0: = 0 at x = 0 and x = 15 = –0.3 x ( x – 15) = –0.3( x 2 – 15 x ) P = –0.3 x x So the graph looks like this:

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17 Topic Example 2 Solution follows… Economic Tasks Solution To do this, you can complete the square: P = –0.3 x x x : number of staff. P : profit in thousands of dollars. P = –0.3 x x = –0.3( x 2 – 15 x ) = –0.3 x – = –0.3 x – – = –0.3 x – = –0.3 x – + 2 To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3 x x.

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18 Topic Example 2 Solution follows… Economic Tasks Solution To do this, you can complete the square: P = –0.3 x x x : number of staff. P : profit in thousands of dollars. P = –0.3 x x = –0.3( x 2 – 15 x ) = –0.3 x – = –0.3 x – – = –0.3 x – = –0.3 x – + 2 So the vertex of the parabola is at , This means that (in theory) the restaurant should employ 7.5 people to make the maximum possible profit of $16,875.

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19 = –0.3 x – P = –0.3 x x = –0.3( x 2 – 15 x ) P = –0.3 x x = –0.3( x 2 – 15 x ) = –0.3 x – – = –0.3 x – = –0.3 x – ,

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20 Topic Economic Tasks Example 2 The owner of a restaurant wishes to graph the annual profit of his restaurant against the number of people he employs. He calculates that the annual profit in thousands of dollars ( P ) can be modeled by the formula P = –0.3 x x, where x is the number of people employed. According to the owners formula, how many full-time members of staff should the restaurant employ to make maximum profit? Solution follows…

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21 Topic Example 2 Solution follows… Economic Tasks Solution To do this, you can complete the square: P = –0.3 x x x : number of staff. P : profit in thousands of dollars. P = –0.3 x x = –0.3( x 2 – 15 x ) = –0.3 x – = –0.3 x – – = –0.3 x – = –0.3 x – + 2 To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3 x x.

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22 Topic Example 2 Solution follows… Economic Tasks Solution P = –0.3 x – So the vertex of the parabola is at , This means that (in theory) the restaurant should employ 7.5 people to make the maximum possible profit of $16,875. Clearly, the restaurant cant employ 7.5 people a good idea now is to draw the graph so that you can answer this question more realistically.

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23 Topic Example 2 Solution follows… Economic Tasks Solution Clearly, the restaurant cant employ 7.5 people a good idea now is to draw the graph so that you can answer this question more realistically. Find the x -intercepts by solving P = 0: So P = 0 at x = 0 and x = 15 = –0.3 x ( x – 15) = –0.3( x 2 – 15 x ) P = –0.3 x x

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24 Topic Example 2 Solution follows… Economic Tasks Solution Find the x -intercepts by solving P = 0: So P = 0 at x = 0 and x = 15 = –0.3 x ( x – 15) = –0.3( x 2 – 15 x ) P = –0.3 x x

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25 You can see from the symmetry of the graph (the line of symmetry is x = 7.5) that the maximum possible profit while employing a whole number of people is at x = 7 and x = 8, at which points the profit is $16,800. So, if the restaurant employs more than 8 people, profits decrease, possibly because there is not enough work for more than 8 people to do efficiently. Topic Example 2 Solution follows… Economic Tasks Solution So the graph looks like this:

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26 Topic Example 2 Solution follows… Economic Tasks Solution So the graph looks like this:

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27 You can see from the symmetry of the graph (the line of symmetry is x = 7.5) that the maximum possible profit while employing a whole number of people is at x = 7 and x = 8, at which points the profit is $16,800. So, if the restaurant employs more than 8 people, profits decrease, possibly because there is not enough work for more than 8 people to do efficiently. Topic Example 2 Solution follows… Economic Tasks Solution

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28 You can see from the symmetry of the graph (the line of symmetry is x = 7.5) that the maximum possible profit while employing a whole number of people is at x = 7 and x = 8, at which points the profit is $16,800. So, if the restaurant employs more than 8 people, profits decrease, possibly because there is not enough work for more than 8 people to do efficiently. Topic Example 2 Solution follows… Economic Tasks Solution

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