The first unit of a product took 832 hours to build, and the learning curve is 75%. How long will it take to make the 30th unit? (Use at least three decimals in the exponent if you use the logarithmic approach.)

A) less than 200 hours
B) from 200 to 225 hours
C) from 225 to 250 hours
D) from 2501 to 275 hours
E) 275 or more hours

The time to produce the nth unit in a learning curve is given by $T_n=T_1\times n^{{\log }_b(a)}$ , where $T_1$ is the time to produce the first unit, $a$ is the learning percentage (as a decimal), and $b$ is the base of the learning curve (usually 2 for a 75% learning curve, indicating a 25% reduction in time per doubling of output). Here, $T_1=832$ hours, $a=0.75$ , and $n=30$ . The exponent is ${\log }_2(0.75)$ , which is negative because $0.75<1$ . To find the time to produce the 30th unit, we calculate: $$T_{30}=832\times 30^{{\log }_2(0.75)}$$ First, find the exponent: $${\log }_2(0.75)\approx -0.415$$ Then, calculate the time for the 30th unit: $$T_{30}=832\times 30^{-0.415}\approx 832\times 0.234\approx 194.688$$ Thus, it will take just under 200 hours to produce the 30th unit, which falls into the range of "from 200 to 225 hours."