Question

To make a profit, a company’s revenue must be greater than its operating costs. The company’s revenue is modeled by the expression 7.5x – 100, where x represents the number of items sold. The company’s operation costs are modeled by the expression 79.86 + 5.8x. How many items does the company need to sell to make a profit? The inequality that will determine the number of items that need to be sold to make a profit is ? The solution to the inequality is ? The company must sell at least?  items to make a profit.

Answer 1

Revenue = 7.5x - 100
Operation Costs = 5.8x + 79.86

To break even, operation cost = Revenue
⇒ 7.5x - 100 = 5.8x + 79.86
7.5x = 5.8x + 179.86 (Add 100 to both sides)

7.5x - 5.8x = 179.86
1.7x = 179.86
x = 105.8

This implies that the company will need to sell at least 106 items to make a profit.

The inequality that will determine the number of items at need to be sold to make a profit is x ≥ 106

The solution to the inequality is as follows

Revenue = 7.5x - 100
if x =106
Revenue = 7.5(106) - 100
Revenue = 695

Operational Cost = 5.8x + 79.86
if x = 106
Operational Cost = 5.8(106) + 79.86
Operational Cost = 694.66

Profit ≥ (695 - 694.66)
Profit ≥ 0.34

The company must sell at least 106 items to make a profit.

Answer 2

Using the profit concept, it is found that:

• The inequality is: $1.7x - 179.6 > 0$

• The solution is $x > 105.6$
• The company must sell at least 106 items to make a profit.

Profit is revenue subtracted by operations costs, that is:

$P(x) = R(x) - C(x)$

In this problem, the functions are:

$R(x) = 7.5x - 100$

$C(s) = 79.6 + 5.8x$

Thus, the profit function is:

$P(x) = R(x) - C(x)$

$P(x) = 7.5x - 100 - 79.6 - 5.8x$

$P(x) = 1.7x - 179.6$

It makes a profit if:

$P(x) > 0$

Thus, the inequality is:

$1.7x - 179.6 > 0$

The solution is:

$1.7x > 179.6$

$x > \frac{179.6}{1.7}$

$x > 105.6$

Thus, the company must sell at least 106 items to make a profit.

A similar problem is given at https://brainly.com/question/24373628