Cherries cost ​$2​/lb. Grapes cost ​$0.50​/lb. You can spend no more than ​$3 on​ fruit, and you need at least 3 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 that can be bought, we need to determine the maximum number of pounds of each fruit that can be purchased within the given constraints.

Let's assume x represents the number of pounds of cherries and y represents the number of pounds of grapes.

The cost of cherries, c, is given by c = $2 * x.
The cost of grapes, g, is given by g = $0.50 * y.

According to the budget constraint, the total cost of fruit cannot exceed $3, so we have the inequality: c + g ≤ $3.
Substituting the values of c and g, we obtain 2x + 0.50y ≤ 3.

Additionally, we need at least 3 pounds of fruit, so we have the constraint: x + y ≥ 3.

Now, we can create a graph by graphing the inequalities on the xy-plane.

First, let's rearrange the inequality 2x + 0.50y ≤ 3 to be y ≤ (3 - 2x) / 0.50.

The graph will have a solid boundary line for the inequality x + y = 3 as we need at least 3 pounds of fruit.

Next, consider the inequality y ≤ (3 - 2x) / 0.50.
To graph this, we can rewrite the inequality as y ≤ 6 - 4x since (3 - 2x) / 0.50 = (6 - 4x) / 2 = 3 - 2x.

Let's plot the graph:
- Draw a solid line for x + y = 3, which passes through the points (3, 0) and (0, 3).
- Shade the area below the line x + y = 3.
- Draw the line y = 6 - 4x.
- Shade the area below the line y = 6 - 4x.

The intersection of the shaded areas represents the region where the cost constraint and weight constraint are satisfied.

On this graph, the x-axis represents the number of pounds of cherries, and the y-axis represents the number of pounds of grapes. The points within the shaded region indicate the different options for the number of pounds of cherries and grapes that can be bought.

Note: The graph cannot be directly displayed within this text-based format, but you can create it on a graphing software or manually sketch it on a piece of paper using the instructions provided.

Answer 2

are you sure?

Answer 3

Apologies for the confusion. Since I cannot directly display the graph here, I will describe the steps to create it manually on a piece of paper:

1. Draw an x-axis labelled "Pounds of Cherries (x)", and a y-axis labelled "Pounds of Grapes (y)".

2. Mark the points (3,0) and (0,3) on the axes. These points represent the constraint x + y = 3.

3. Draw a solid line passing through the points (3,0) and (0,3). This line represents the constraint x + y = 3.

4. Shade the area below this line to represent the region where x + y ≥ 3.

5. Now, to graph the inequality y ≤ 6 - 4x, we need to determine two points on this line.

a. Pick any value for x (let's say 0), then find the corresponding value for y: y = 6 - 4(0) = 6. Plot the point (0,6).
b. Pick another value for x (let's say 1), then find the corresponding value for y: y = 6 - 4(1) = 2. Plot the point (1,2).

With these two points, you can draw a line passing through them.

6. Shade the area below the line y = 6 - 4x to represent the region where y ≤ 6 - 4x.

7. The intersection of the shaded regions from steps 4 and 6 represents the feasible region showing the amount of each fruit you can buy while satisfying the cost and weight constraints.

Remember, the x-axis represents the pounds of cherries, and the y-axis represents the pounds of grapes. Points within the shaded region indicate the different options for the number of pounds of cherries and grapes that can be bought.