Question

A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose that each week the crew digs 95% of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.
a. How far does the crew dig in:
i. 10 weeks
ii. 20 weeks
iii. N weeks
b. What is the longest tunnel the crew can build at this rate?

Answer 1

The successive distance dug through the mountain can be modeled by the recursive sequence,
\[d_{n+1}=0.95d_n,~d_1=100\]
where \(n\) denotes the week number.

Note that this sequence is decreasing, since 95% of some number is smaller than that number, which means this sequence cannot and does not represent the total soil removed. So, the total amount of soil and earth removed must be modeled by the series,
\[\sum_{n=1}^\infty d_n\]
Does this make sense so far?

Answer 2

yes

Answer 3

To find the distance dug by the crew in 10 weeks, you'd have to find
\[\sum_{n=1}^{10}d_n\]
which is just a matter of summing up the first 10 terms of the sequence.

Answer 4

Similarly, you'll then have to find
\[\sum_{n=1}^{20}d_n~~\text{and}~~\sum_{n=1}^Nd_n\]
and finally, the infinite sum.

Answer 5

Thanks :)