Economic Tasks Topic 7.6.2

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

Topic 7.6.2 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.

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

Topic 7.6.2 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 = –0.3x2 + 4.5x, where x is the number of people employed. According to the owner’s formula, how many full-time members of staff does the restaurant have to employ to make a profit of $15,000? Solution follows…

x: number of staff. P: profit in thousands of dollars. Topic 7.6.2 Economic Tasks Example 1 Solution P = –0.3x2 + 4.5x x: number of staff. P: profit in thousands of dollars. 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 –0.3x2 + 4.5x = 15. Solution continues…

7.6.2 Topic Economic Tasks Solution (continued) Example 1 Solution (continued) Rewriting –0.3x2 + 4.5x = 15 in the form ax2 + bx + c = 0 gives: 0.3x2 – 4.5x + 15 = 0 x2 – 15x + 50 = 0 Divide through by 0.3 (x – 10)(x – 5) = 0 x = 10 or x = 5 Solve using the zero property This means that the restaurant can employ either 5 people or 10 people and make a profit of $15,000.

7.6.2 Topic Economic Tasks Example 2 Clearly, the restaurant can’t employ 7.5 people — a good idea now is to draw the graph so that you can answer this question more realistically. 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 = –0.3x2 + 4.5x, where x is the number of people employed. According to the owner’s formula, how many full-time members of staff should the restaurant employ to make maximum profit? Find the x-intercepts by solving P = 0: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) Solution = –0.3x(x – 15) To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3x2 + 4.5x. = 0 at x = 0 and x = 15 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. So the graph looks like this: To do this, you can complete the square: 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. P = –0.3x2 + 4.5x = –0.3(x2 – 15x) 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. Solution follows…

7.6.2 Topic Economic Tasks Guided Practice 1. The profit p in cents per 10-minute period earned from driving a taxicab is given by p = 80x – 3x2, where x is the speed in mph. What speed would yield a profit of 512 cents per 10-minute period? x = 16 mph or x = 10.7 mph (1 d.p.) 2. An investor kept track of her portfolio profit, P, at time, t, measured in years after she began investing. If P = 4000t2 – 28000t + 3000 represents her profit, after how many years will she have made $150,000 profit? 10.5 years 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 = –t2 + 8t + 17 represents the money a customer spends, how long will it take before a customer decides to leave the store without spending any money? t = 9.74 minutes (2 d.p.) Solution follows…

7.6.2 Topic Economic Tasks Independent Practice Leo produces x pounds of salsa. The ingredients cost 0.1x2 – 30 dollars and he makes 2x dollars revenue from the sale of his salsa. 1. What is Leo’s maximum possible profit? 2. How many pounds of salsa would Leo need to sell to break even? $40 30 pounds The value in dollars, V, of a certain stock can be modeled by the equation V = –16t2 + 88t + 101, where t represents the time in months. 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? $101.00 $222.00 months 11 4 6.5 months (1 d.p.) Solution follows…

7.6.2 Topic Economic Tasks Independent Practice The value, V, of Juan’s investment portfolio can be modeled by the equation V = 16t2 – 256t + 16,000, where t is the time in months. 7. What was the original value of Juan’s portfolio? 8. What was the minimum value of Juan’s portfolio? 9. When will Juan’s investment portfolio be worth $16,576.00? $16,000 $14,976 After 18 months Solution follows…

7.6.2 Topic Economic Tasks Round Up Usually when you graph quadratic problems involving money, the vertex of the graph shows you the point where there’s maximum profit.

7.6.2 Topic Economic Tasks Example 2 Clearly, the restaurant can’t employ 7.5 people — a good idea now is to draw the graph so that you can answer this question more realistically. 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 = –0.3x2 + 4.5x, where x is the number of people employed. According to the owner’s formula, how many full-time members of staff should the restaurant employ to make maximum profit? Find the x-intercepts by solving P = 0: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) Solution = –0.3x(x – 15) To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3x2 + 4.5x. = 0 at x = 0 and x = 15 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. So the graph looks like this: To do this, you can complete the square: 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. P = –0.3x2 + 4.5x = –0.3(x2 – 15x) 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. Solution follows…

Topic 7.6.2 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 = –0.3x2 + 4.5x, where x is the number of people employed. According to the owner’s formula, how many full-time members of staff should the restaurant employ to make maximum profit? Solution follows…

P: profit in thousands of dollars. P = –0.3x2 + 4.5x P = –0.3x2 + 4.5x x: number of staff. P: profit in thousands of dollars. Topic 7.6.2 Economic Tasks Example 2 Solution To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3x2 + 4.5x. To do this, you can complete the square: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) 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. Solution follows…

7.6.2 Topic Economic Tasks Example 2 Clearly, the restaurant can’t 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: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) Solution = –0.3x(x – 15) = 0 at x = 0 and x = 15 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. Solution follows…

P: profit in thousands of dollars. P = –0.3x2 + 4.5x x: number of staff. P: profit in thousands of dollars. Topic 7.6.2 Economic Tasks Example 2 Solution To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3x2 + 4.5x. To do this, you can complete the square: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) = –0.3 x – – 15 2 = –0.3 x – + 0.3 15 2 225 4 3 10 = –0.3 x – + 15 2 225 4 135 8 15 2 = –0.3 x – + Solution follows…

P: profit in thousands of dollars. P = –0.3x2 + 4.5x x: number of staff. P: profit in thousands of dollars. Topic 7.6.2 Economic Tasks Example 2 Solution To do this, you can complete the square: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) = –0.3 x – – 15 2 = –0.3 x – + 0.3 15 2 225 4 3 10 = –0.3 x – + 15 2 225 4 135 8 15 2 = –0.3 x – + So the vertex of the parabola is at . 135 8 15 2 , This means that (in theory) the restaurant should employ 7.5 people to make the maximum possible profit of $16,875. Solution follows…

, P = –0.3x2 + 4.5x = –0.3 x – + 0.3 = –0.3 x – + = –0.3(x2 – 15x) 135 8 15 2 , P = –0.3x2 + 4.5x = –0.3(x2 – 15x) 2 15 2 225 4 P = –0.3x2 + 4.5x = –0.3 x – + 0.3 2 15 2 3 10 225 4 = –0.3 x – + = –0.3(x2 – 15x) 2 2 2 15 2 15 2 15 2 135 8 = –0.3 x – – = –0.3 x – +

Topic 7.6.2 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 = –0.3x2 + 4.5x, where x is the number of people employed. According to the owner’s formula, how many full-time members of staff should the restaurant employ to make maximum profit? Solution follows…

P: profit in thousands of dollars. P = –0.3x2 + 4.5x x: number of staff. P: profit in thousands of dollars. Topic 7.6.2 Economic Tasks Example 2 Solution To find the maximum profit, you need to find the maximum value of the quadratic P = –0.3x2 + 4.5x. To do this, you can complete the square: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) = –0.3 x – – 15 2 = –0.3 x – + 0.3 15 2 225 4 3 10 = –0.3 x – + 15 2 225 4 135 8 15 2 = –0.3 x – + Solution follows…

7.6.2 Topic Economic Tasks Solution P = –0.3 x – + , Example 2 Solution P = –0.3 x – + 135 8 15 2 So the vertex of the parabola is at . 135 8 15 2 , This means that (in theory) the restaurant should employ 7.5 people to make the maximum possible profit of $16,875. Clearly, the restaurant can’t employ 7.5 people — a good idea now is to draw the graph so that you can answer this question more realistically. Solution follows…

7.6.2 Topic Economic Tasks Solution Example 2 Solution Clearly, the restaurant can’t 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: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) = –0.3x(x – 15) So P = 0 at x = 0 and x = 15 Solution follows…

7.6.2 Topic Economic Tasks Solution Example 2 Solution Find the x-intercepts by solving P = 0: P = –0.3x2 + 4.5x = –0.3(x2 – 15x) = –0.3x(x – 15) So P = 0 at x = 0 and x = 15 Solution follows…

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

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

7.6.2 Topic Economic Tasks Solution Example 2 Solution 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. Solution follows…

7.6.2 Topic Economic Tasks Solution Example 2 Solution 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. Solution follows…