QuestionCallie's pillow company has monthly costs of \$3200 and makes pillows for \$50 each, selling them for \$18.
A. Write and solve an inequality for the number of pillows $p$ sold to ensure profit. B. Discuss which solutions are valid in this context.

Studdy Solution

STEP 1

Assumptions1. The fixed monthly expenses are \$3200. The cost to make each pillow is \$503. Each pillow sells for \$184. Profit is the amount of money left after paying all expenses

STEP 2

First, we need to write an inequality that represents the profit. Profit is the revenue (the amount of money made from selling the pillows) minus the expenses (the cost of making the pillows and the fixed monthly expenses).
$Profit = Revenue - Expenses$

STEP 3

Let's denote the number of pillows sold each month as $p$ . The revenue from selling $p$ pillows is $p \times \$18$ .
$Revenue = p \times \$18$

STEP 4

The cost of making $p$ pillows is $p \times \$50$ and the fixed monthly expenses are \ $3200. So, the total expenses are$ p \times \ $50 + \$3200$ .
$Expenses = p \times \$50 + \$3200$

STEP 5

Now, we can write the inequality for the profit. The company makes a profit when the revenue is greater than the expenses.
$p \times \$18 > p \times \$50 + \$3200$

STEP 6

To simplify the inequality, we can subtract $p \times \$50$ from both sides.
$p \times \$18 - p \times \$50 > \$3200$

STEP 7

implify the left side of the inequality by combining like terms.
$p \times (\$18 - \$50) > \$3200$

STEP 8

Calculate the value inside the parentheses.
$p \times (-\$32) > \$3200$

STEP 9

To solve for $p$ , we can divide both sides of the inequality by -\$32. Remember, when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes.
$p < -\frac{\$320}{\$32}$

STEP 10

Calculate the value on the right side of the inequality.
$p < -100$ This means that the company needs to sell less than -100 pillows to make a profit, which doesn't make sense in this context.

STEP 11

The error in our calculation comes from the fact that the selling price of the pillows is less than the cost to make them. This means that the company is losing money on each pillow sold, and therefore cannot make a profit regardless of how many pillows are sold. The correct inequality should be $p \times \$18 < p \times \$50 + \$3200$ This inequality states that for any number of pillows sold, the revenue will always be less than the expenses, resulting in a loss, not a profit.

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QuestionCallie's pillow company has monthly costs of \$3200 and makes pillows for \$50 each, selling them for \$18.
A. Write and solve an inequality for the number of pillows $p$ sold to ensure profit. B. Discuss which solutions are valid in this context.

Studdy Solution

STEP 1

Assumptions1. The fixed monthly expenses are \$3200. The cost to make each pillow is \$503. Each pillow sells for \$184. Profit is the amount of money left after paying all expenses

STEP 2

First, we need to write an inequality that represents the profit. Profit is the revenue (the amount of money made from selling the pillows) minus the expenses (the cost of making the pillows and the fixed monthly expenses).
$Profit = Revenue - Expenses$

STEP 3

Let's denote the number of pillows sold each month as $p$ . The revenue from selling $p$ pillows is $p \times \$18$ .
$Revenue = p \times \$18$

STEP 4

The cost of making $p$ pillows is $p \times \$50$ and the fixed monthly expenses are \ $3200. So, the total expenses are$ p \times \ $50 + \$3200$ .
$Expenses = p \times \$50 + \$3200$

STEP 5

Now, we can write the inequality for the profit. The company makes a profit when the revenue is greater than the expenses.
$p \times \$18 > p \times \$50 + \$3200$

STEP 6

To simplify the inequality, we can subtract $p \times \$50$ from both sides.
$p \times \$18 - p \times \$50 > \$3200$

STEP 7

implify the left side of the inequality by combining like terms.
$p \times (\$18 - \$50) > \$3200$

STEP 8

Calculate the value inside the parentheses.
$p \times (-\$32) > \$3200$

STEP 9

To solve for $p$ , we can divide both sides of the inequality by -\$32. Remember, when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes.
$p < -\frac{\$320}{\$32}$

STEP 10

Calculate the value on the right side of the inequality.
$p < -100$ This means that the company needs to sell less than -100 pillows to make a profit, which doesn't make sense in this context.

STEP 11

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QuestionCallie's pillow company has monthly costs of \$3200 and makes pillows for \$50 each, selling them for \$18. A. Write and solve an inequality for the number of pillows ppp sold to ensure profit. B. Discuss which solutions are valid in this context.

Studdy Solution

STEP 1

Assumptions1. The fixed monthly expenses are \$3200. The cost to make each pillow is \$503. Each pillow sells for \$184. Profit is the amount of money left after paying all expenses

STEP 2

STEP 3

Let's denote the number of pillows sold each month as ppp. The revenue from selling ppp pillows is p×$18p \times \$18p×$18.Revenue=p×$18Revenue = p \times \$18Revenue=p×$18

STEP 4

The cost of making ppp pillows is p×$50p \times \$50p×$50 and the fixed monthly expenses are \3200.So,thetotalexpensesare3200. So, the total expenses are 3200.So,thetotalexpensesarep \times \50+$320050 + \$320050+$3200.Expenses=p×$50+$3200Expenses = p \times \$50 + \$3200Expenses=p×$50+$3200

STEP 5

Now, we can write the inequality for the profit. The company makes a profit when the revenue is greater than the expenses.p×$18>p×$50+$3200p \times \$18 > p \times \$50 + \$3200p×$18>p×$50+$3200

STEP 6

To simplify the inequality, we can subtract p×$50p \times \$50p×$50 from both sides.p×$18−p×$50>$3200p \times \$18 - p \times \$50 > \$3200p×$18−p×$50>$3200

STEP 7

implify the left side of the inequality by combining like terms.p×($18−$50)>$3200p \times (\$18 - \$50) > \$3200p×($18−$50)>$3200

STEP 8

Calculate the value inside the parentheses.p×(−$32)>$3200p \times (-\$32) > \$3200p×(−$32)>$3200

STEP 9

To solve for ppp, we can divide both sides of the inequality by -\$32. Remember, when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes.p<−$320$32p < -\frac{\$320}{\$32}p<−$32$320​

STEP 10

Calculate the value on the right side of the inequality.p<−100p < -100p<−100This means that the company needs to sell less than -100 pillows to make a profit, which doesn't make sense in this context.

STEP 11

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionCallie's pillow company has monthly costs of \$3200 and makes pillows for \$50 each, selling them for \$18.
A. Write and solve an inequality for the number of pillows $p$ sold to ensure profit. B. Discuss which solutions are valid in this context.

Let's denote the number of pillows sold each month as $p$ . The revenue from selling $p$ pillows is $p \times \$18$ .
$Revenue = p \times \$18$

The cost of making $p$ pillows is $p \times \$50$ and the fixed monthly expenses are \ $3200. So, the total expenses are$ p \times \ $50 + \$3200$ .
$Expenses = p \times \$50 + \$3200$

Now, we can write the inequality for the profit. The company makes a profit when the revenue is greater than the expenses.
$p \times \$18 > p \times \$50 + \$3200$

To simplify the inequality, we can subtract $p \times \$50$ from both sides.
$p \times \$18 - p \times \$50 > \$3200$

implify the left side of the inequality by combining like terms.
$p \times (\$18 - \$50) > \$3200$

Calculate the value inside the parentheses.
$p \times (-\$32) > \$3200$

To solve for $p$ , we can divide both sides of the inequality by -\$32. Remember, when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes.
$p < -\frac{\$320}{\$32}$

Calculate the value on the right side of the inequality.
$p < -100$ This means that the company needs to sell less than -100 pillows to make a profit, which doesn't make sense in this context.