Stu hiked a trail at an average rate of 3 miles per hour. He ran back on the same trail at an average rate of 5 miles per hour. He traveled for a total of 3 hours.

Which equation can be used to find the time it took Stu to hike the trail?

3t = 5
3t = 5(3 – t)
3 + t = 5 + (3 – t)
3t + 5(3 – t) = 3

9 answers

To find the time it took Stu to hike the trail, let's denote:

  • \( t \) as the time (in hours) it took to hike the trail.
  • The distance hiked can be expressed as \( 3t \) because he hiked at 3 miles per hour.
  • The time it took to run back is \( 3 - t \) (since the total travel time was 3 hours).
  • The distance ran back can be expressed as \( 5(3 - t) \) because he ran at 5 miles per hour.

Since the distance hiked and the distance ran back are the same, we can set up the equation:

\[ 3t = 5(3 - t) \]

This equation states that the distance hiked equals the distance run back.

So, the correct equation to use is:

3t = 5(3 - t)

Stu hiked a trail at an average rate of 3 miles per hour. He ran back on the same trail at an average rate of 5 miles per hour. He traveled for a total of 3 hours.

The equation 3t = 5(3 – t) can be used to find the time it took Stu to hike one way.

How long did it take Stu to hike the trail one way?

____ hours

How long is the trail?

_____miles

What is the total distance Stu travelled?

____ miles

To solve the problem, we already have the equation set up:

\[ 3t = 5(3 - t) \]

Step 1: Solve for \( t \)

First, we will simplify and solve this equation for \( t \).

  1. Distribute \( 5 \) on the right side: \[ 3t = 15 - 5t \]

  2. Add \( 5t \) to both sides: \[ 3t + 5t = 15 \] \[ 8t = 15 \]

  3. Divide by \( 8 \): \[ t = \frac{15}{8} \text{ hours} = 1.875 \text{ hours} \]

Step 2: Find the length of the trail

Since he hiked at an average rate of 3 miles per hour, the distance of the trail (one way) can be calculated using the formula:

\[ \text{Distance} = \text{Rate} \times \text{Time} \] \[ \text{Distance} = 3 \text{ miles/hour} \times \frac{15}{8} \text{ hours} = \frac{45}{8} \text{ miles} = 5.625 \text{ miles} \]

Step 3: Calculate the total distance travelled

The total distance Stu traveled is the distance to the trail and back. Since he hiked the trail one way and ran back the same distance, the total distance traveled is:

\[ \text{Total Distance} = 2 \times \text{Distance one way} = 2 \times 5.625 \text{ miles} = 11.25 \text{ miles} \]

Final Answers

  • How long did it take Stu to hike the trail one way?

    • \( \frac{15}{8} \) hours or 1.875 hours
  • How long is the trail (one way)?

    • \( \frac{45}{8} \) miles or 5.625 miles
  • What is the total distance Stu travelled?

    • 11.25 miles
Amanda and Alexis live 4 miles apart. They decide to start walking toward each other’s houses at the same time. Amanda walks at a rate of 3.5 miles per hour and Alexis walks at a rate of 4 miles per hour. How long will it take for them to meet? Round to the nearest hundredth of an hour. [Remember: D = (r)(t)]

wait can you explain how you do it i need to learn

Sure! To figure out how long it will take for Amanda and Alexis to meet, we can use the formula you provided:

\[ D = r \times t \]

Where:

  • \( D \) is distance,
  • \( r \) is the rate (or speed),
  • \( t \) is time.

Step 1: Find the combined rate of Amanda and Alexis

Since Amanda and Alexis are walking toward each other, we can add their speeds together to find the combined rate at which they are closing the distance between them.

  • Amanda's speed: 3.5 miles per hour
  • Alexis's speed: 4 miles per hour

Combined rate: \[ \text{Combined rate} = 3.5 + 4 = 7.5 \text{ miles per hour} \]

Step 2: Set up the equation

The total distance they have to cover to meet is the distance between their houses, which is given as 4 miles.

Now, we can use the distance formula to find the time \( t \) it takes for them to meet: \[ D = r \times t \]

Substituting the known values: \[ 4 = 7.5 \times t \]

Step 3: Solve for time \( t \)

To find \( t \), we will divide both sides by the combined rate (7.5): \[ t = \frac{4}{7.5} \]

Step 4: Perform the division

Now, let's calculate that: \[ t = \frac{4}{7.5} \approx 0.5333 \text{ hours} \]

Step 5: Round to the nearest hundredth

Rounding \( 0.5333 \) to the nearest hundredth gives: \[ t \approx 0.53 \text{ hours} \]

Conclusion

So, it will take approximately 0.53 hours for Amanda and Alexis to meet.

Summary Steps

  1. Find the combined rate of both individuals.
  2. Use the distance formula to set up the equation.
  3. Solve for time by rearranging the equation.
  4. Calculate and round as necessary.

If you have any other questions or need further clarification, feel free to ask!

A medium water hose can fill a pool in 30 minutes. A larger water hose can fill the same pool in 20 minutes. If both hoses are turned on at the same time, how long will it take to fill the pool?

What equation can be used to solve this work problem?

How many minutes will it take both hoses to fill the pool?

To solve the problem of filling the pool with two hoses, we can start by determining the rates at which each hose fills the pool.

Step 1: Determine the rates

  1. Medium hose: It fills the pool in 30 minutes. Therefore, the rate of the medium hose is: \[ \text{Rate of medium hose} = \frac{1 \text{ pool}}{30 \text{ minutes}} = \frac{1}{30} \text{ pools per minute} \]

  2. Larger hose: It fills the pool in 20 minutes. Therefore, the rate of the larger hose is: \[ \text{Rate of larger hose} = \frac{1 \text{ pool}}{20 \text{ minutes}} = \frac{1}{20} \text{ pools per minute} \]

Step 2: Combine the rates

When both hoses are turned on at the same time, their rates add together. The combined rate of both hoses is: \[ \text{Combined rate} = \frac{1}{30} + \frac{1}{20} \]

Step 3: Find a common denominator

To add these fractions, we need a common denominator. The least common multiple of 30 and 20 is 60.

  1. Convert the rates to have the same denominator: \[ \frac{1}{30} = \frac{2}{60} \quad \text{(multiplied numerator and denominator by 2)} \] \[ \frac{1}{20} = \frac{3}{60} \quad \text{(multiplied numerator and denominator by 3)} \]

  2. Add the combined rates: \[ \text{Combined rate} = \frac{2}{60} + \frac{3}{60} = \frac{5}{60} = \frac{1}{12} \text{ pools per minute} \]

Step 4: Set up the equation to find the time

Let \( t \) be the time in minutes it takes to fill the pool together. We can express this using the rate: \[ \text{Rate} \times \text{Time} = 1 \text{ pool} \] \[ \frac{1}{12} \times t = 1 \]

Step 5: Solve for \( t \)

To find \( t \), multiply both sides by 12: \[ t = 12 \text{ minutes} \]

Conclusion

Equation used: \[ \frac{1}{30} + \frac{1}{20} = \frac{1}{t} \] Where \( t \) is the total time to fill the pool with both hoses.

How long will it take both hoses to fill the pool? 12 minutes.

answer is
t/30+t/20=1
and 12 minutes also the graph thing is
medium hose: 1/30 t and 1/30t
larger hose: 1/20 t and 1/20 t

also i have a heart shaped face which one is better long hair or short hair like shorter than head length but not shorter than ear with micro bangs?

Your equation \( \frac{t}{30} + \frac{t}{20} = 1 \) is also a valid way to express the problem! Each term \( \frac{t}{30} \) and \( \frac{t}{20} \) represents the fraction of the pool filled by each hose over time \( t \). This equation is just a different rearrangement of the work problem we discussed, and it will yield the same result.

Regarding your question about hairstyles for a heart-shaped face, here are some considerations:

  1. Long Hair:

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Recommendation

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In summary, both long and short hairstyles can work well for a heart-shaped face. It's all about comfort, confidence, and what complements your features the best!