A small fast-food restaurant invests $4,000 to produce a new food item that will sell for $4.59. Each item can be produced for $2.19.

(a) How many items must be sold to break even? (Round your answer to the nearest whole number.)

items

(b) How many items must be sold to make a profit of $5,500? (Round your answer to the nearest whole number.)

items

the cost for x items is 4000+2.19x

the income for x items is 4.59x

Breakeven is where income and cost are equal:

4.59x = 4000 + 2.19x

For 5500 profit, you need

4.59x = 4000 + 2.19x + 5500

To find the number of items that need to be sold to break even, we need to calculate the break-even point. The break-even point is when the revenue equals the total cost.

(a) First, let's calculate the total cost:
Total Cost = Investment + (Cost per item * Number of items)

Given:
Investment = $4,000
Cost per item = $2.19

Let's assume the number of items to be sold as 'x':
Total Cost = $4,000 + ($2.19 * x)

Next, we calculate the revenue:
Revenue = Price per item * Number of items
Revenue = $4.59 * x

To find the break-even point, we set the revenue equal to the total cost:
$4.59 * x = $4,000 + ($2.19 * x)

To solve for 'x', we can rearrange the equation and isolate 'x':
$4.59 * x - $2.19 * x = $4,000
($4.59 - $2.19) * x = $4,000
$2.40 * x = $4,000
x = $4,000 / $2.40

Now, let's calculate the value of 'x':
x = 1666.6667

Since the number of items must be a whole number, we round it to the nearest whole number:
x ≈ 1667

Therefore, to break even, the small fast-food restaurant must sell approximately 1667 items.

(b) To find the number of items that need to be sold to make a profit of $5,500, we can use a similar approach.

Let's assume the number of items to be sold as 'y':
Revenue = Price per item * Number of items
Revenue = $4.59 * y

Given:
Profit = Revenue - Total Cost
Profit = $5,500

Total Cost = $4,000 + ($2.19 * y)
Revenue = $4.59 * y

Profit = $5,500
$4.59 * y - $4,000 - ($2.19 * y) = $5,500

To solve for 'y', we can rearrange the equation and isolate 'y':
($4.59 - $2.19) * y = $5,500 + $4,000
$2.40 * y = $9,500
y = $9,500 / $2.40

Now, let's calculate the value of 'y':
y ≈ 3958.333333

Since the number of items must be a whole number, we round it to the nearest whole number:
y ≈ 3958

Therefore, to make a profit of $5,500, the small fast-food restaurant must sell approximately 3958 items.