Question

A hiker on the Appalachian Trail planned to increase the distance covered by 10% each day. After 7 days, the total distance traveled is 75.897 miles.

Answer 1

part A. We are given that a hiker will increase the distance covered by 10% each day. Let "S" be the distance, then on the first day the distance is:

`S_1`

On the second day, we must add 10% of the first day, we get:

`S_1=S_1+(10)/(100)S_1`

Simplifying we get:

`S_2=S_1+0.1S_1=1.1S_1`

On the third day, we add 10% of the second day, we get:

`S_3=S_2+0.1S_2=1.1S_2=(1.1)(1.1)S_1=(1.1)^2S_1`

On the fourth day, we add 10% of the third day, we get:

`S_4=S_3+0.1S_3=1.1S_3=(1.1)^3S_1`

If we continue this pattern and we set "n" as the number of days, then a formula for the distance after "n" days is:

`S_n=(1.1)^(n-1)S_1`

Now, we are given that for n = 7 the distance is 75897, therefore, we substitute n = 7 in the formula:

`S_7=(1.1)^(7-1)S_1`

Substituting the value of the distance:

`75897=(1.1)^(7-1)S_1`

Now we can solve for S1, we do that by dividing both sides by 1.1 together with its

exponent:

`(75897)/((1.1)^(7-1))=S_1`

Now we solve the operations:

`(75897)/((1.1)^6)=S_1`

Solving the operations:

`42842=S_1`

Therefore, the distance the first day was 42842 miles.

part B. The formula for Sn is the given previously but we replace the known value of S1:

`S_n=42842(1.1)^(n-1)`

Part C. To determine the distance after 10 days, we substitute the value n = 10 in the formula, we get:

`S_(10)=42842(1.1)^(10-1)`

Solving the operations we get:

`S_(10)=101019.19`

Therefore, the distance after 10 days is 101019.19 miles.