Question

Sarah consumes apples and oranges (the only foods he eats). He has decided that his monthly

Answer 1

Answer:

• a) 0.25x + 0.50y ≤ 50

• b) See the first graph attached

• c) See the second graph attached

• d) See the third graph attached

• e) Both remain unchanged

Explanation:

a. What is the expression for Sarah's budget constraint?

1. Call x the number of apples and y the number of oranges she can buy.

• 0.25x is the expense on apples

• 0.50y is the expense of oranges

• 0.25x + 0.50y is the total expense in fruits

2. The budget is the constraint: the maximum amount of money to spend on fruits:

• 0.25x + 0.50y ≤ $50

b. Draw a graph of Sarah' budget line (also called his budget set).

To draw Sarah´s budget line:

• Draw and label the axis: x-axis number of apples, y-axis number of oranges

• Choose the x-intercept and y-intercept of the border line as the two ending points of your line (because x ≥ and y ≥ 0 are natural constraints)

         x = 0 ⇒ 0.25(0) + 0.50y = 50

         ⇒ 0.50y = 50

         ⇒ y = 50/0.50 = 100

         ⇒ point (0,100): see it in the graph

             This means that she can buy 100 oranges when she buy 0 oranges.

         y = 0 ⇒ 0.25x + 0.50(0) = 50

          ⇒ 0.25x = 50

          ⇒ x = 0.50/0.25

          ⇒ x = 200

          ⇒ point (0,200): see it in the graph

              This means that she can buy 200 apples when she buy 0 apples.

• Shade the region between the your line and the axis (first quadrant). Make the line solid since the points on the line are in the solution set.

• See the first graph attached.

c. Show graphically how Sarah's budget line changes if the price of apples increases to $0.50.

Changing the price of apples changes the slope of the curve, and keeps one point.

The new inequality would be:

• 0.50x + 0.50y ≤ 50

The y-intercept is still y = 50/0.5 = 100 ⇒ (0, 100)

The new x-intercept is: x = 50/0.50 ⇒ (100,0)

Following the same procedure you can make the new graph, which I include in the second figure:

It is clear the change in the slope and the shaded region includes less apples. The maximum number of apples she can purchase is 100, when she purchases 0 oranges.

d. Show graphically how Sarah's budget line changes if the price of oranges decreases to $0.25.

The new constraint would be:

• 0.25x + 0.25y ≤ 50

The new graph (third graph attached) have intercepts:

• (0) + 0.25y = 50 ⇒ y = 50/0.25 = 200 ⇒ (0,200)

• 0.25x + 0 = 50 ⇒ x = 50/0.25 = 200 ⇒ (200,0)

Then, the y-intercept is the same and the x-intercept changed.

The maximum number of oranges that she can buy increased to 200, when she buys 0 apples, while the maximum number of apples that she can buy is still 200.

See the third graph attached.

e. Suppose Sarah decides to cut his monthly budget for fruit in half. Coincidentally, the next time she goes to the grocery store, she learns that oranges and apples are on sale for half price and will remain so for the next month. That is, the price of apples falls to $0.125 per apple and the price of oranges falls to $0.25 per orange. What happens to both the expression of Sarah's budget constraint and the graph of his budget line?

The new constraint is:

• 0.125x + 0.25y ≤ 25

Note that if you multiply both the left-hand side and the right-hand side by 2, the inequality does not change:

• 2(0.125x) + 2(0.25y) ≤ 2(25)

• 0.25x + 0.50y ≤ 50

Hence, both the expression of Sarah's budget constraint and the graph of her budget line are the same.

Hence, Sarah has the same set of possible purchases, i.e pairs (apples, oranges) that she can buy, when her budget is cut in half and the prices of the fruits are also cut in half.