The Answer is C

UpStudy Free Solution By Steps

Step 1: Define the rates and distances

- The first part of the climb: 1,000 feet at a rate of \(x\) feet per hour.

- The second part of the climb: 5,000 feet at a rate of \(2x - 10\) feet per hour (since the rate is 10 feet per hour less than twice the first rate).

Step 2: Write the expressions for the time taken for each part of the climb

- Time for the first part of the climb:

\[t_ 1 = \frac { 1000} { x} \]

- Time for the second part of the climb:

\[t_ 2 = \frac { 5000} { 2x - 10} \]

Step 3: Combine the times to get the total time

- Total time \(t\) is the sum of the times for the two parts of the climb:

\[t = t_ 1 + t_ 2\]

\[t = \frac { 1000} { x} + \frac { 5000} { 2x - 10} \]

Final Answer

The expression that represents the number of hours the mountaineer climbed is:

\[{ \frac { 1000} { x} + \frac { 5000} { 2x - 10} } \]

Key Concepts:

1. Rate: The speed at which an activity is performed, often expressed as distance per unit of time (e.g., feet per hour).

2. Distance: The length of the path traveled, usually measured in units such as feet or meters.

3. Time: The duration taken to cover a certain distance at a given rate, calculated as the distance divided by the rate.

Explanation:

- Defining Rates and Distances: Break the problem into pieces: identify the distances covered and rates at which each part of a distance is covered. That is, for example, if one leg of the trip goes at a different rate than the other legs, compute each part separately.

- Calculating Time for Each Part: Use the formula:

\[\text { Time} = \frac { \text { Distance} } { \text { Rate} } \]

to find the time taken for each part of the journey. For instance, if the first part of the journey is 1000 feet at a rate of \(x\) feet per hour, the time for this part is:

\[t_ 1 = \frac { 1000} { x} \]

- Combining Times for Total Time: Add the times for each part of the journey to get the total time. If the second part of the journey is 5000 feet at a rate of \(2x - 10\) feet per hour, the time for this part is:

\[t_ 2 = \frac { 5000} { 2x - 10} \]

The total time is the sum of the times for each part:

\[t = t_ 1 + t_ 2 = \frac { 1000} { x} + \frac { 5000} { 2x - 10} \]

Having these principles in mind, complex rate and distance problems can be solved by disintegrating them into the smaller parts of the problem and then deriving their result, subsequently reuniting them. Get personalized help by solving fully worked solutions to problems from your homework using UpStudy AI Math Homework Solver at this link. With full explanations and guided assistance, learn the topics well with UpStudy. Make your discovery of learning different—explore UpStudy!