Question

You have a budget of $48. You want to buy fruit and granola bars. Fruit costs $4 per pound, and the granola bars are $1 each. You need at least 10 granola bars. What combinations of fruit and granola bars can you buy?
1. Define the variables you will use in your model. (2 points)





2. Write an inequality representing the total cost of your purchase. (3 points)
a. Each pound of fruit costs $4. Write an expression that shows the total cost of the fruit. Use the variable you identified in question 1.





b. Each granola bar costs $1. Write an expression that shows the total cost of the granola bars. Use the variable you identified in question 1.





c. Combine the expressions from parts a and b to write an expression for the total cost. Then use this expression to write an inequality that compares the total cost with the amount you have to spend.





3. Write the inequality that models the number of granola bars you need to buy. (1 point)





4. Describe in words what each of your inequalities means. (2 points)
Inequality 1: ____________ means...





Inequality 2: ____________ means...





Graph the System of Inequalities
5. Graph your system of inequalities. Shade the half plane that represents the solution for each inequality, and then identify the area that represents the solution to the system of inequalities. (5 points: 1 point for each boundary line, 1 point for each correctly shaded half plane, 1 point for identifying the solution)

Interpreting the Graph
6. Identify one point on the graph that represents a viable solution to the problem, and then identify one point that does not represent a viable solution. Identify the number of granola bars and pounds of fruit represented by each point, and explain why the point is or is not viable. (4 points: 1 for each point and 1 for each explanation)

Answer 1

1)Define the variables you will use in your model.

Let

x------->total pounds of the fruit

y-------> the total number of granola bars

TC----> total cost

Tf------> total cost of the fruit

Tg----> total cost of the granola bars

2) Write an inequality representing the total cost of your purchase. (3 points)

a. Each pound of fruit costs $4. Write an expression that shows the total cost of the fruit. Use the variable you identified in question 1

Tf=4*x

b. Each granola bar costs $1. Write an expression that shows the total cost of the granola bars. Use the variable you identified in question 1.

Tg=1*y--------> Tg=y

c. Combine the expressions from parts a and b to write an expression for the total cost. Then use this expression to write an inequality that compares the total cost with the amount you have to spend.

TC=Tf+Tg

TC=4*x+y

inequality-------> $4x+y \leq 48$

3. Write the inequality that models the number of granola bars you need to buy.

You need at least 10 granola bars

so

$y\geq 10$

4. Describe in words what each of your inequalities means

inequality 1

$4x+y \leq 48$

the total cost of fruits plus the total cost of granola bars must be less than or equal to 48 dollars

inequality 2

$y\geq 10$

5. Graph your system of inequalities.

using a graph tool

see the attached figure

6. Identify one point on the graph that represents a viable solution to the problem, and then identify one point that does not represent a viable solution.

Let

A-------> one point on the graph that represents a viable solution

B-------> one point on the graph that does not represent a viable solution

A(4,20)

x=4-------> total pounds of the fruit

y=20------> the total number of granola bars

check point A

substitute the values of x and y in the inequality 1

$4x+y \leq 48$

$4*4+20\leq 48\\ 36\leq 48$-------> is ok

substitute the values of x and y in the inequality 2

$y\geq 10$

$20\geq 10$--------> is ok

the point A represents a viable solution, because satisfies both inequalities

B(10,30)

x=10-------> total pounds of the fruit

y=30------> the total number of granola bars

check point B

substitute the values of x and y in the inequality 1

$4x+y \leq 48$

$4*10+30\leq 48\\ 70\leq 48$-------> is not a viable solution

substitute the values of x and y in the inequality 2

$y\geq 10$

$30\geq 10$--------> is ok

the point B is not a viable solution because does not satisfy the inequality 1