Problem 129 - College Algebra

Question

The percentage, \(p,\) of defective products manufactured by a company is given by \(|p-0.3 \%| \leq 0.2 \% .\) If \(100,000\) products are manufactured and the company offers a \(\$ 5\) refund for each defective product, describe the company's cost for refunds.

Step-by-Step Solution

Answer

The company's cost for refunds ranges from \$500 to \$2500.

Step 1: Identifying the range of defective products

The expression \(|p-0.3 \%| \leq 0.2 \% \) presents the range for the defective products percentage. Subtracting and adding \(0.2 \% \) from \(0.3 \% \) gives us the range from \(0.1%\) to \(0.5%\).

Step 2: Convert the percentages to quantities

To get a clear view of how many products are defective, converting the percentages to quantities is necessary. For \(0.1%\) defective products, there would be \(0.001 * 100,000 = 100\) defective products, and for \(0.5%\) defective products, there would be \(0.005 * 100,000 = 500\) defective products.

Step 3: Compute the refund costs

The company offers a $5 refund for each defective item. Therefore, the minimum refund cost would be \(100 * $5 = $500\), and the maximum refund cost would be \(500 * $5 = $2500\).