1. A kangaroo hops 2 kilometers in 3 minutes. At this rate: a. How long does it take the kangaroo to travel 5 kilometers? b. How far does the kangaroo travel in 2 minutes? 2. Mai r... 1. A kangaroo hops 2 kilometers in 3 minutes. At this rate: a. How long does it take the kangaroo to travel 5 kilometers? b. How far does the kangaroo travel in 2 minutes? 2. Mai runs around a 400-meter track at a constant speed of 250 meters per minute. How many minutes does it take Mai to complete 4 laps of the track? Explain or show your reasoning. 3. At 10:00 a.m., Han and Tyler both started running toward each other from opposite ends of a 10-mile path along a river. Han runs at a pace of 12 minutes per mile, and Tyler runs at a pace of 15 minutes per mile. a. How far does Han run after a half hour? After an hour? b. Do Han and Tyler meet on the path within 1 hour? Explain or show your reasoning. Read more

Steps to Solve

1. Kangaroo's speed calculation The kangaroo hops 2 kilometers in 3 minutes. First, we find the speed:

   $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{2 \text{ km}}{3 \text{ min}} $$

2. Time to travel 5 kilometers To find how long it takes to travel 5 kilometers, we set up the ratio based on the speed:

   Let $t$ be the time in minutes to travel 5 km.

   $$ \frac{2 \text{ km}}{3 \text{ min}} = \frac{5 \text{ km}}{t \text{ min}} $$

   Cross-multiplying gives:

   $$ 2t = 15 \implies t = 7.5 \text{ minutes} $$

3. Distance traveled in 2 minutes We will calculate how far the kangaroo travels in 2 minutes using its speed:

   $$ \text{Distance} = \text{Speed} \times \text{Time} = \frac{2 \text{ km}}{3 \text{ min}} \times 2 \text{ min} $$

   Simplifying gives:

   $$ \text{Distance} = \frac{4}{3} \text{ km} \approx 1.33 \text{ km} $$

4. Mai's time for 4 laps Mai runs at a speed of 250 meters per minute. The total distance for 4 laps around a 400-meter track is:

   $$ \text{Distance} = 4 \times 400 = 1600 \text{ meters} $$

   To find the time:

   $$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1600 \text{ m}}{250 \text{ m/min}} $$

   This results in:

   $$ \text{Time} = 6.4 \text{ minutes} $$

5. Han's distance after half an hour Han runs at a pace of 12 minutes per mile. This means he runs at:

   $$ \text{Speed} = \frac{1 \text{ mile}}{12 \text{ min}} = \frac{60 \text{ miles}}{720 \text{ min}} = 5 \text{ mph} $$

   In half an hour:

   $$ \text{Distance} = \text{Speed} \times \text{Time} = 5 \text{ mph} \times 0.5 \text{ hours} = 2.5 \text{ miles} $$

6. Han's distance after an hour Using the same speed:

   $$ \text{Distance} = 5 \text{ mph} \times 1 \text{ hour} = 5 \text{ miles} $$

7. Meeting point calculation Tyler runs at a pace of 15 minutes per mile, translating to a speed of:

   $$ \text{Speed} = \frac{1 \text{ mile}}{15 \text{ min}} = 4 \text{ mph} $$

   In one hour, Tyler would cover:

   $$ \text{Distance} = 4 \text{ mph} \times 1 \text{ hour} = 4 \text{ miles} $$

   Together, Han and Tyler cover:

   $$ \text{Combined distance} = 2.5 \text{ miles} + 4 \text{ miles} = 6.5 \text{ miles} $$

   Since the total path is 10 miles, they will meet after:

   $$ 10 - 6.5 = 3.5 \text{ miles} $$

   They will meet in under an hour.