A grocer sells bananas and apples. In one day he sells
up to 80 bananas
up to 90 apples
no more than a total of 110 pieces of fruit
Let x be the number of bananas sold Let y be the number of apples sold.
a. Show the region below that satisfies these inequalities
b. If apples are sold for $1 and bananas are sold for $0.50, what the
maximum amount of money he made in a day?
130 dollars
To graph the region that satisfies the given inequalities, we first need to set up the inequalities based on the given information.
Given:
- The grocer sells up to 80 bananas (x)
- The grocer sells up to 90 apples (y)
- The grocer sells no more than a total of 110 pieces of fruit (x + y ≤ 110)
a. To graph the region, we will plot the points that satisfy these inequalities.
1. Graphing x ≤ 80:
- Draw a horizontal line at x = 80.
- Shade the region to the left of the line, including the line itself.
- This represents the range of values for x (bananas) sold up to 80.
2. Graphing y ≤ 90:
- Draw a vertical line at y = 90.
- Shade the region below the line, including the line itself.
- This represents the range of values for y (apples) sold up to 90.
3. Graphing x + y ≤ 110:
- Rearranging the inequality to y ≤ -x + 110, we can graph y = -x + 110.
- Find the x-intercept (when y = 0): -x + 110 = 0 → x = 110.
- Find the y-intercept (when x = 0): y = -0 + 110 → y = 110.
- Plot the x-intercept (110, 0) and the y-intercept (0, 110).
- Draw a line passing through these two points.
- Shade the region below the line, including the line itself.
- This represents the range of values for the total number of bananas and apples sold up to 110.
The shaded region that satisfies all three inequalities is the intersection of all the shaded regions from the above steps.
b. To find the maximum amount of money the grocer made in a day, we need to determine how many of each fruit he sold within the shaded region and then calculate the total amount of money made.
Since we don't have specific values for x and y in the shaded region, we can consider the extreme points to get the maximum amount.
1. The corners of the shaded region are the extreme points. Let's find these points:
- Corner 1: (0, 90) - 0 bananas and 90 apples
- Corner 2: (80, 30) - 80 bananas and 30 apples
- Corner 3: (80, 0) - 80 bananas and 0 apples
2. Calculate the money made at each corner:
- At Corner 1: Number of apples (90) * Price per apple ($1) + Number of bananas (0) * Price per banana ($0.50)
- At Corner 2: Number of apples (30) * Price per apple ($1) + Number of bananas (80) * Price per banana ($0.50)
- At Corner 3: Number of apples (0) * Price per apple ($1) + Number of bananas (80) * Price per banana ($0.50)
3. Calculate the maximum amount by comparing the money made at each corner and selecting the highest value.
Therefore, to find the maximum amount of money the grocer made in a day, solve this problem by determining the total amount of money made at each corner and selecting the highest value.