Consider Mr. Smith has an apple orchard. He hires his daughter, Lucy, to pick apples and offers her two payment options:
Option A: $1.50 per bushel of apples picked
Option B: 1¢ cent for picking one bushel, 3¢ cents for picking two bushels, 9¢ cents for picking three bushels, and so on, with the amount of money tripling for each additional bushel picked
1.Write a function to model each option.
2.If Lucy picks six bushels of apples, which option should she choose?
3.If Lucy picks 12 bushels of apples, which option should she choose?
4.How many bushels of apples does Lucy need go pick to make Option B better for than Option A?
A: y = 1.50x
B: y = .013^(x-1)
Now you can answer the questions...
2.)option a
3.)option b
4.)7-12
5.)yes
1. To model each option, we can use the following functions:
Option A: f(x) = 1.50x, where x is the number of bushels picked and f(x) is the total payment.
Option B: g(x) = 0.01(3^x - 1), where x is the number of bushels picked and g(x) is the total payment.
2. If Lucy picks six bushels of apples, we can calculate the payment for each option:
Option A: f(6) = 1.50 * 6 = $9
Option B: g(6) = 0.01(3^6 - 1) = $5.67
Lucy should choose Option A as it offers a higher payment.
3. If Lucy picks twelve bushels of apples, we can calculate the payment for each option:
Option A: f(12) = 1.50 * 12 = $18
Option B: g(12) = 0.01(3^12 - 1) = $40.95
Lucy should choose Option B as it offers a higher payment.
4. To find the number of bushels of apples Lucy needs to pick for Option B to be better than Option A, we need to set the two functions equal to each other and solve for x:
1.50x = 0.01(3^x - 1)
Expanding and rearranging the equation:
1.50x = 0.01(3^x) - 0.01
0.01(3^x) = 1.50x + 0.01
Dividing both sides by 0.01:
3^x = 150x + 1
To solve this equation, you would need to use numerical methods or graphing techniques.
To answer these questions, we need to model the payment options with mathematical functions. Let's define the functions first:
Option A: $1.50 per bushel of apples picked
Function for Option A: f(x) = 1.5x
Option B: 1¢ cent for picking one bushel, 3¢ cents for picking two bushels, 9¢ cents for picking three bushels, and so on, with the amount of money tripling for each additional bushel picked
Function for Option B: g(x) = 0.01(3^(x-1))
Now, let's proceed to answer each question:
1. Write a function to model each option:
We've defined the functions above.
2. If Lucy picks six bushels of apples, which option should she choose?
To determine the payment for each option, we need to evaluate the functions at x = 6:
For Option A: f(6) = 1.5 * 6 = $9
For Option B: g(6) = 0.01 * (3^(6-1)) = $0.81
Option A pays $9, while Option B pays $0.81 for picking six bushels. Lucy should choose Option A in this case.
3. If Lucy picks 12 bushels of apples, which option should she choose?
Again, evaluate the functions at x = 12:
For Option A: f(12) = 1.5 * 12 = $18
For Option B: g(12) = 0.01 * (3^(12-1)) = $8.19
Option A pays $18, while Option B pays $8.19 for picking twelve bushels. Lucy should choose Option A in this case as well.
4. How many bushels of apples does Lucy need to pick to make Option B better than Option A?
To find the point at which Option B becomes better than Option A, we need to set the payment of Option B equal to the payment of Option A and solve for x:
0.01 * (3^(x-1)) = 1.5x
To solve this equation, we can use numerical methods or graphing software. By doing so, we find that Lucy would need to pick approximately 19 bushels for Option B to become better than Option A.
Therefore, if Lucy picks 19 or more bushels of apples, she should choose Option B. If she picks fewer than 19 bushels, Option A is the better choice.