Questions: A contractor is preparing a bid to install swimming pools at a new housing addition. The estimated time to build the first pool is 34 hours. The contractor estimates an 85 percent learning rate. Using POM for Windows or OM Explorer, how long do you estimate the time required to install the fifth pool? The time required to install the fifth pool is hours. (Enter your response rounded to two decimal places.) What is your estimate of the total time for all five pools? The total time for all five pools is hours. (Enter your response rounded to two decimal places.)

Solution Steps

To solve this problem, we will use the concept of a learning curve, which suggests that the time required to complete a task decreases at a constant rate as a worker gains experience. The learning rate is given as 85%, and the time for the first pool is 34 hours. We will calculate the time for the fifth pool using the learning curve formula and then sum the times for all five pools to find the total time.

Using the learning curve formula, the time required to install the \( n \)-th pool is given by:

\[ T_n = T_1 \cdot n^{\log_2(\text{learning rate})} \]

where \( T_1 = 34 \) hours and the learning rate is \( 0.85 \). For \( n = 5 \):

\[ T_5 = 34 \cdot 5^{\log_2(0.85)} \]

Calculating this gives:

\[ T_5 \approx 23.31 \text{ hours} \]

The total time for all five pools is the sum of the times for each individual pool:

\[ \text{Total Time} = T_1 + T_2 + T_3 + T_4 + T_5 \]

Calculating each term using the same formula for \( n = 1, 2, 3, 4, 5 \):

\[ \text{Total Time} \approx 137.06 \text{ hours} \]

Final Answer