Cherries cost ​$4​/lb. Grapes cost ​$​2.50/lb. You can spend no more than ​$15 on​ fruit, and you need at least 5 lb in all. Create a graph showing the amount of each fruit you can buy.

Answer 1

To create a graph showing the amount of each fruit you can buy, let's first define our variables:

Let x represent the number of pounds of cherries.

Let y represent the number of pounds of grapes.

We can write the following inequalities to represent the given constraints:

4x + 2.5y ≤ 15 (total cost cannot exceed $15)
x + y ≥ 5 (total weight must be at least 5 pounds)

To graph these inequalities, we can use a graphing calculator or manually plot points on a graph.

Let's start by graphing the line 4x + 2.5y = 15:

- To graph this line, we can first find the x-intercept and y-intercept.
- Let x = 0, then 2.5y = 15, so y = 6.
- Let y = 0, then 4x = 15, so x = 3.75.
- Plot these two points and draw a line through them.

Now, to determine which region of the graph satisfies the inequality 4x + 2.5y ≤ 15, we can choose a point that is not on the line (such as (0,0)) and substitute its coordinates into the inequality. If the inequality is true, then that point is a valid solution and the region that contains it will satisfy the inequality. Otherwise, we can shade the other side of the line.

- For (0,0), 4(0) + 2.5(0) ≤ 15 is true, so the origin (and the entire region that contains it) satisfies the inequality.
- Shade the region below the line.

Next, let's graph the line x + y = 5:

- To graph this line, we can find the x-intercept and y-intercept.
- Let x = 0, then y = 5.
- Let y = 0, then x = 5.
- Plot these two points and draw a line through them.

Now, to determine which region of the graph satisfies the inequality x + y ≥ 5, we can repeat the same process as before:

- For (0,0), 0 + 0 ≥ 5 is false, so shade the other side of the line.
- Shade the region above the line.

The shaded region that satisfies both inequalities is the feasible region that represents the amount of each fruit you can buy.

The graph should look something like this:


The feasible region is the shaded triangle below the line 4x + 2.5y = 15 and above the line x + y = 5. The vertices of the feasible region are (0, 5), (3.75, 1.0), and (3.33, 1.67).

To find the possible combinations of cherries and grapes that satisfy the constraints, we can substitute these vertex coordinates for x and y in the inequalities:

- For (0, 5), 4(0) + 2.5(5) ≤ 15 is true and 0 + 5 ≥ 5 is true, so we can buy 5 pounds of grapes but no cherries.
- For (3.75, 1.0), 4(3.75) + 2.5(1.0) ≤ 15 is true and 3.75 + 1.0 ≥ 5 is true, so we can buy 3.75 pounds of cherries and 1 pound of grapes.
- For (3.33, 1.67), 4(3.33) + 2.5(1.67) ≤ 15 is true and 3.33 + 1.67 ≥ 5 is true, so we can buy 3.33 pounds of cherries and 1.67 pounds of grapes.

Therefore, we can buy either 5 pounds of grapes, 3.75 pounds of cherries and 1 pound of grapes, or 3.33 pounds of cherries and 1.67 pounds of grapes, all of which satisfy the given constraints.

Answer 2

Cherries cost $4/Ib. Grapes cost $1.50/Ib. You can spend no more than $9 on fruit, and you need at least 4 Ib in all. Create a graph showing the amount of each fruit you can buy.

Answer 3

To create a graph showing the amount of each fruit you can buy, let's first define our variables:

Let x represent the number of pounds of cherries.

Let y represent the number of pounds of grapes.

We can write the following inequalities to represent the given constraints:

4x + 1.5y ≤ 9 (total cost cannot exceed $9)
x + y ≥ 4 (total weight must be at least 4 pounds)

To graph these inequalities, we can use a graphing calculator or manually plot points on a graph.

Let's start by graphing the line 4x + 1.5y = 9:

- To graph this line, we can first find the x-intercept and y-intercept.
- Let x = 0, then 1.5y = 9, so y = 6.
- Let y = 0, then 4x = 9, so x = 2.25.
- Plot these two points and draw a line through them.

Now, to determine which region of the graph satisfies the inequality 4x + 1.5y ≤ 9, we can choose a point that is not on the line (such as (0,0)) and substitute its coordinates into the inequality. If the inequality is true, then that point is a valid solution and the region that contains it will satisfy the inequality. Otherwise, we can shade the other side of the line.

- For (0,0), 4(0) + 1.5(0) ≤ 9 is true, so the origin (and the entire region that contains it) satisfies the inequality.
- Shade the region below the line.

Next, let's graph the line x + y = 4:

- To graph this line, we can find the x-intercept and y-intercept.
- Let x = 0, then y = 4.
- Let y = 0, then x = 4.
- Plot these two points and draw a line through them.

Now, to determine which region of the graph satisfies the inequality x + y ≥ 4, we can repeat the same process as before:

- For (0,0), 0 + 0 ≥ 4 is false, so shade the other side of the line.
- Shade the region above the line.

The shaded region that satisfies both inequalities is the feasible region that represents the amount of each fruit you can buy.

The graph should look something like this:


The feasible region is the shaded triangle below the line 4x + 1.5y = 9 and above the line x + y = 4. The vertices of the feasible region are (0, 4), (2, 2), and (2.25, 1.5).

To find the possible combinations of cherries and grapes that satisfy the constraints, we can substitute these vertex coordinates for x and y in the inequalities:

- For (0, 4), 4(0) + 1.5(4) ≤ 9 is true and 0 + 4 ≥ 4 is true, so we can buy 4 pounds of grapes but no cherries.
- For (2, 2), 4(2) + 1.5(2) ≤ 9 is true and 2 + 2 ≥ 4 is true, so we can buy 2 pounds of cherries and 2 pounds of grapes.
- For (2.25, 1.5), 4(2.25) + 1.5(1.5) ≤ 9 is true and 2.25 + 1.5 ≥ 4 is true, so we can buy 2.25 pounds of cherries and 1.5 pounds of grapes.

Therefore, we can buy either 4 pounds of grapes, 2 pounds of cherries and 2 pounds of grapes, or 2.25 pounds of cherries and 1.5 pounds of grapes, all of which satisfy the given constraints.