Question
Jesse and Mark are jogging along the route shown at a rate of 12 miles per hour. They start by jogging south along Capitol Street for 1 mile, then turn east on H Street and jog for 1.75 miles. At that point, Jesse is tired and decides to walk home along Florida Avenue at a rate of 5 miles per hour. Mark plans to jog back the way they came. Jesse wants to find out who will arrive home first and by how much time. Which statements should he consider when solving the problem? Check all that apply.
A triangle. The sides are Capitol Street, H Street, and they hypotenuse is Florida Avenue.
Jesse will walk home a distance of approximately 2.0 miles as found by evaluating 1 squared + 1.75 squared = d squared and then 1 + 3.0625 = d squared.
Jesse will take about 0.4 hours to walk the rest of the way home as found by 2.0 = 5 (t).
Jesse will walk home a distance of approximately 2.25 miles as found by 1 squared + 1.75 squared = d squared and then 2 + 3.0625 = d squared.
Mark will jog home a distance of 2.75 miles.
It will take Mark about 0.23 hours to jog home as found by 2.75 = 12 (t).
Mark will jog home a distance of 2.25 miles.
Mark will get home about 0.17 hours (or 10 minutes) sooner.
Mark will get home about 0.21 hours (or 12.5 minutes) sooner.
A triangle. The sides are Capitol Street, H Street, and they hypotenuse is Florida Avenue.
Jesse will walk home a distance of approximately 2.0 miles as found by evaluating 1 squared + 1.75 squared = d squared and then 1 + 3.0625 = d squared.
Jesse will take about 0.4 hours to walk the rest of the way home as found by 2.0 = 5 (t).
Jesse will walk home a distance of approximately 2.25 miles as found by 1 squared + 1.75 squared = d squared and then 2 + 3.0625 = d squared.
Mark will jog home a distance of 2.75 miles.
It will take Mark about 0.23 hours to jog home as found by 2.75 = 12 (t).
Mark will jog home a distance of 2.25 miles.
Mark will get home about 0.17 hours (or 10 minutes) sooner.
Mark will get home about 0.21 hours (or 12.5 minutes) sooner.
Answers
GPT-4o mini
To determine who arrives home first and by how much time, Jesse should consider the following statements:
1. **A triangle. The sides are Capitol Street, H Street, and the hypotenuse is Florida Avenue.** (True) - This statement indicates that the relationship between the distances forms a right triangle, which is relevant for calculating the straight-line distance Jesse needs to walk.
2. **Jesse will walk home a distance of approximately 2.0 miles as found by evaluating 1 squared + 1.75 squared = d squared and then 1 + 3.0625 = d squared.** (False) - The calculation for the distance is not correct. The correct calculation should use the formula \( d = \sqrt{1^2 + 1.75^2} \).
3. **Jesse will take about 0.4 hours to walk the rest of the way home as found by 2.0 = 5 (t).** (False) - If Jesse has a distance of approximately 2.0 miles, the calculation does not match since \( t = \frac{d}{r} = \frac{2.0}{5} = 0.4 \) hours is correct but the distance needs to be verified first.
4. **Jesse will walk home a distance of approximately 2.25 miles as found by 1 squared + 1.75 squared = d squared and then 2 + 3.0625 = d squared.** (True) - The correct calculation is \( d = \sqrt{1^2 + 1.75^2} = \sqrt{1 + 3.0625} = \sqrt{4.0625} \approx 2.0156 \) which leads to Jesse walking approximately 2.25 miles when rounded or approximated slightly.
5. **Mark will jog home a distance of 2.75 miles.** (True) - This is correct as he jogs the same distance back on Capitol and H Streets, which would be 1 mile + 1.75 miles = 2.75 miles.
6. **It will take Mark about 0.23 hours to jog home as found by 2.75 = 12 (t).** (True) - The time calculation is correct as \( t = \frac{2.75}{12} \approx 0.2292 \) hours, or approximately 0.23 hours.
7. **Mark will jog home a distance of 2.25 miles.** (False) - Mark jogs 2.75 miles back home.
8. **Mark will get home about 0.17 hours (or 10 minutes) sooner.** (False) - This could be determined through calculations but requires verification.
9. **Mark will get home about 0.21 hours (or 12.5 minutes) sooner.** (True) - Comparing Jesse and Mark’s times, if Jesse takes about 0.4 hours and Mark takes about 0.23 hours, Mark is indeed arriving home sooner.
In summary, Jesse should consider the statements regarding the triangle relationship of their route, the calculation of distances they will travel, the jogging speed of Mark, and the comparative time calculation between Jesse's walking time and Mark's jogging time.
1. **A triangle. The sides are Capitol Street, H Street, and the hypotenuse is Florida Avenue.** (True) - This statement indicates that the relationship between the distances forms a right triangle, which is relevant for calculating the straight-line distance Jesse needs to walk.
2. **Jesse will walk home a distance of approximately 2.0 miles as found by evaluating 1 squared + 1.75 squared = d squared and then 1 + 3.0625 = d squared.** (False) - The calculation for the distance is not correct. The correct calculation should use the formula \( d = \sqrt{1^2 + 1.75^2} \).
3. **Jesse will take about 0.4 hours to walk the rest of the way home as found by 2.0 = 5 (t).** (False) - If Jesse has a distance of approximately 2.0 miles, the calculation does not match since \( t = \frac{d}{r} = \frac{2.0}{5} = 0.4 \) hours is correct but the distance needs to be verified first.
4. **Jesse will walk home a distance of approximately 2.25 miles as found by 1 squared + 1.75 squared = d squared and then 2 + 3.0625 = d squared.** (True) - The correct calculation is \( d = \sqrt{1^2 + 1.75^2} = \sqrt{1 + 3.0625} = \sqrt{4.0625} \approx 2.0156 \) which leads to Jesse walking approximately 2.25 miles when rounded or approximated slightly.
5. **Mark will jog home a distance of 2.75 miles.** (True) - This is correct as he jogs the same distance back on Capitol and H Streets, which would be 1 mile + 1.75 miles = 2.75 miles.
6. **It will take Mark about 0.23 hours to jog home as found by 2.75 = 12 (t).** (True) - The time calculation is correct as \( t = \frac{2.75}{12} \approx 0.2292 \) hours, or approximately 0.23 hours.
7. **Mark will jog home a distance of 2.25 miles.** (False) - Mark jogs 2.75 miles back home.
8. **Mark will get home about 0.17 hours (or 10 minutes) sooner.** (False) - This could be determined through calculations but requires verification.
9. **Mark will get home about 0.21 hours (or 12.5 minutes) sooner.** (True) - Comparing Jesse and Mark’s times, if Jesse takes about 0.4 hours and Mark takes about 0.23 hours, Mark is indeed arriving home sooner.
In summary, Jesse should consider the statements regarding the triangle relationship of their route, the calculation of distances they will travel, the jogging speed of Mark, and the comparative time calculation between Jesse's walking time and Mark's jogging time.