Question

A hiker on the Appalachian Trail planned to increase the distance covered by 20% each day. After 7 days, the total distance traveled is 51.664 miles.Part A: How many miles did the hiker travel on the first day? Round your answer to the nearest mile and show all necessary math work. (4 points)Part B: What is the equation for Sn? Show all necessary math work. (3 points)Part C: If this pattern continues, what is the total number of miles the hiker will travel in 10 days? Round your answer to the hundredths place and show all necessary math work. (3 points)

Answer 1

Given:

There are given that the Appalachain trail planned to increase the distance covered by 20%.

Step-by-step explanation:

(a):

According to the question:

We need to find the number of miles did the hiker travel on the first.

So,

To find the number of miles, we use the below equation:

`51.664=((x(1.2)^7-1))/(0.2)`

Then,

`\begin{gathered} 51.664=((x(1.2)^(7)-1))/(0.2) \\ 10.33=(x(1.2)^7-1) \\ 10.33=2.6x \\ x=(10.33)/(2.6) \\ x=4 \end{gathered}`

Hence, he started with 4 miles.

Part B:

According to the question:

There are given 7 days.

So,

We get the second part of the equation because each day is increased by 20%.

So, we have:

`r=(1.2x)/(x)=1.2`

So,

Our top half will be x multiplied by:

`((1.2)^7-1)`

Because it is the 7th day and we are increasing by:

`1.2x-1`

So,

The equation is shown below:

`51.664=((x(1.2)^(7)-1))/(0.2)`

Part C:

We now have x which is 4.

So,

We plug x into the equation above and remove the 51.664.

Then,

We also change the expression:

`(1.2)^7,to,(1.2)^(10)`

Because it's the 10th day, not the 7th.

So,

Our equation now looks like this:

`((4(1.2)^(10)-1))/(0.2)=103.83`

Hence, the total number of miles the hiker will travel in 10 days is 103.83.