A company is preparing a bid on a government contract for 40 units of a certain product. The operations manager estimates the assembly time required for the first two units to be 10.4 hours and 8.8 hours. Which unit, if any, will require approximately one-half the time of the first unit? A) 18^th unit B) 19^th unit C) 20^th unit D) 21^st unit E) 22^nd unit

Question

Quizplus · Accepted Answer

We can use the formula for exponential growth/decay to estimate the assembly time for the nth unit:
t = a*r^(n-1)
where t is the assembly time, a is the assembly time for the first unit, r is the common ratio between assembly times, and n is the unit number.

We can set up an equation for when the assembly time is half of the first unit's time:
5.2 = 10.4*r^(n-1)
0.5 = r^(n-1)

To solve for n, we can take the logarithm of both sides:
log(0.5) = (n-1)*log(r)
(n-1)*log(r) = -log(2)
n-1 = -log(2)/log(r)

We can use the assembly time for the second unit to find the common ratio between assembly times:
8.8 = 10.4*r
r = 0.846

Now we can plug in r and solve for n:
n-1 = -log(2)/log(0.846)
n = 18.95

Since n should be an integer unit number, we can round up to 19. Therefore, the 19th unit will require approximately one-half the time of the first unit, which corresponds to choice B.

A Company Is Preparing a Bid on a Government Contract