A bake sale sells cupcakes and cookies. One day they sell:
up to 50 cupcakes
up to 60 cookies
no more than a total of 90 cupcakes and cookies.
Let x be the number of cupcakes sold. Let y be the number of cookies sold.
a. Show the region that satisfies these inequalities.
b. If cookies are sold for $1 and cupcakes are sold for $1.50, what is the
maximum amount of profit they can make?
To solve this problem, we need to set up a system of inequalities based on the given conditions. Let's start with the first condition:
1. "Up to 50 cupcakes": This means the number of cupcakes sold, represented by x, must be less than or equal to 50. So the first inequality is: x <= 50.
Next, let's consider the second condition:
2. "Up to 60 cookies": This means the number of cookies sold, represented by y, must be less than or equal to 60. So the second inequality is: y <= 60.
Finally, let's incorporate the third condition:
3. "No more than a total of 90 cupcakes and cookies": This means the combined total of cupcakes and cookies sold must be less than or equal to 90. Mathematically, this can be expressed as: x + y <= 90.
Now, let's graph these inequalities to visualize the region that satisfies them:
a. Show the region that satisfies these inequalities:
To graph these inequalities, we'll start by creating a coordinate system with x-axis representing the number of cupcakes and y-axis representing the number of cookies. We'll mark the points (50, 0), (0, 60), and (90, 0) on the graph.
1. Graph the inequality x <= 50:
Draw a vertical line passing through the point (50, 0) and shade the region to the left of this line, including the line itself.
2. Graph the inequality y <= 60:
Draw a horizontal line passing through the point (0, 60) and shade the region below this line, including the line itself.
3. Graph the inequality x + y <= 90:
Rearrange the equation to y <= -x + 90 to get it in slope-intercept form. Plot this line by finding two points (0, 90) and (90, 0), then draw a line through them. Shade the region below this line, including the line itself.
The shaded region that overlaps all three inequalities represents the region that satisfies the given conditions. The intersection of all shaded regions forms a triangular shape in the first quadrant of the coordinate system. This region represents the valid combinations of cupcakes and cookies.
b. Now, let's determine the maximum amount of profit they can make:
Since each cookie is sold for $1 and each cupcake is sold for $1.50, we need to determine the combination of cupcakes and cookies that maximizes their profit while still falling within the feasible region.
To find the maximum profit, we need to examine the vertices (corners) of the feasible region. We can calculate the profit for each vertex by multiplying the number of cupcakes (x) by $1.50 and the number of cookies (y) by $1, and summing those values.
We can calculate the profit for each vertex as follows:
1. (0, 0): Profit = (0 * $1.50) + (0 * $1) = $0
2. (0, 60): Profit = (0 * $1.50) + (60 * $1) = $60
3. (50, 0): Profit = (50 * $1.50) + (0 * $1) = $75
4. Point of intersection: Calculate the profit at the point of intersection of the lines to find the maximum profit.
Comparing the profits, we see that the maximum profit occurs at the point of intersection. So, to find the maximum profit, we need to determine the coordinates of the point of intersection.
By solving the system of inequalities, we can find the point of intersection.
a. To show the region that satisfies these inequalities, we can plot them on a graph.
The given inequalities are:
1. x ≤ 50 (up to 50 cupcakes)
2. y ≤ 60 (up to 60 cookies)
3. x + y ≤ 90 (no more than a total of 90 cupcakes and cookies)
On a graph, the x-axis represents the number of cupcakes (x) and the y-axis represents the number of cookies (y).
The first inequality, x ≤ 50, represents a vertical line passing through the point (50,0). This line extends infinitely downwards.
The second inequality, y ≤ 60, represents a horizontal line passing through the point (0,60). This line extends infinitely to the right.
The third inequality, x + y ≤ 90, represents a diagonal line passing through the points (90,0) and (0,90). This line forms a triangle with the other two lines.
The region that satisfies all three inequalities is the shaded region inside this triangle.
b. To find the maximum profit they can make, we need to consider the equation for profit. Since cookies are sold for $1 and cupcakes are sold for $1.50, we can calculate the profit as follows:
Profit = (Number of cookies sold × Price per cookie) + (Number of cupcakes sold × Price per cupcake)
Profit = (y × $1) + (x × $1.50)
To maximize the profit, we need to find the maximum value of this equation within the region we obtained in part a.
Note: Without specific information on the prices of cupcakes and cookies as well as any cost factors, it is not possible to provide an exact value for the maximum profit in this scenario.