Suppose that a company guarantees refrigerators and will replace with a new one that breaks while it is under the guarantee. However, the company does not want to replace more than 5% of the refrigerators under the guarantee. For how long should the guarantee be made? You can round to the nearest tenth of a year.

To determine how long the guarantee should be made, we need to consider the failure rate of the refrigerators. Since the company does not want to replace more than 5% of the refrigerators under the guarantee, we can set the failure rate to be less than or equal to 5%.

Let's break down the problem step by step:

1. Determine the failure rate:
The failure rate represents the percentage of refrigerators that break within a given period. Since the company doesn't want to replace more than 5% of the refrigerators, we set the failure rate to be less than or equal to 5%.

2. Calculate the average failure rate per year:
Next, we need to convert the failure rate to a yearly percentage. If the guarantee is for a certain number of years, we need to ensure that the average failure rate per year is less than or equal to 5%.

3. Set up an equation:
To find the appropriate guarantee length, we can use the formula:
(1 - failure rate)^(number of years) ≤ 0.05

4. Solve for the number of years:
Using the equation from step 3, solve for the number of years needed for the guarantee. This will give us the approximate length of the guarantee that satisfies the company's requirement.

Now, let's calculate the answer. Assuming the failure rate is constant over time, we can try different values for the number of years until we find a satisfactory solution. We can start with a relatively short duration, such as 1 year:

(1 - 0.05)^1 ≤ 0.05
0.95 ≤ 0.05

The condition is not met with 1 year. Now, let's try 2 years:

(1 - 0.05)^2 = 0.9025 ≤ 0.05

The condition is not yet met. Let's try 3 years:

(1 - 0.05)^3 = 0.857375 ≤ 0.05

The condition is still not met. Finally, let's try 4 years:

(1 - 0.05)^4 = 0.81450625 ≤ 0.05

The condition is met with 4 years. Therefore, the guarantee should be made for approximately 4 years to ensure that the failure rate remains below 5%.

Remember, this is an iterative process that starts with an initial guess. If a satisfactory solution is not found, you can continue increasing the number of years until the condition is met or use more advanced mathematical techniques to find a precise solution.