Pregunta
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An athlete trains by running 600 metres on the first day, 900 metres on the second, 1200 metres on the third and so forth. (a) How far does he run on the 15 th day? (b) What is the total distance that he will run in 15 days? (c) How long will it be before he can run a marathon of 42 km ?

Ask by Blake Deleon. in South Africa
Jan 24,2025

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- **a)** On the 15th day, he runs 4,800 meters. - **b)** Over 15 days, he runs a total of 40,500 meters (40.5 km). - **c)** He will be able to run a marathon of 42 km on the 139th day of training.

Solución

Let's analyze the athlete's training regimen step by step. ### Given: - **First day distance (a₁):** 600 meters - **Daily increase (d):** 300 meters This forms an **arithmetic sequence** where each day's running distance increases by a constant amount. --- ### **a) Distance on the 15th Day** The formula for the **n-th term** of an arithmetic sequence is: \[ a_n = a_1 + (n - 1) \times d \] Substituting the values for the 15th day: \[ \begin{align*} a_{15} &= 600 + (15 - 1) \times 300 \\ &= 600 + 14 \times 300 \\ &= 600 + 4200 \\ &= 4800 \text{ meters} \end{align*} \] **Answer:** On the 15th day, he runs **4,800 meters**. --- ### **b) Total Distance in 15 Days** The formula for the **sum of the first n terms (Sₙ)** of an arithmetic sequence is: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \] Using the 15th day's distance from part (a): \[ \begin{align*} S_{15} &= \frac{15}{2} \times (600 + 4800) \\ &= \frac{15}{2} \times 5400 \\ &= 15 \times 2700 \\ &= 40,500 \text{ meters} \end{align*} \] **Answer:** Over 15 days, he runs a total of **40,500 meters** (or **40.5 kilometers**). --- ### **c) Time to Run a Marathon (42 km)** A marathon is **42 kilometers**, which is **42,000 meters**. We need to find the day **n** when he first runs **≥ 42,000 meters**. Using the n-th term formula: \[ \begin{align*} a_n &\geq 42,000 \\ 600 + (n - 1) \times 300 &\geq 42,000 \\ (n - 1) \times 300 &\geq 42,000 - 600 \\ (n - 1) \times 300 &\geq 41,400 \\ n - 1 &\geq \frac{41,400}{300} \\ n - 1 &\geq 138 \\ n &\geq 139 \end{align*} \] So, on the **139th day**, he runs exactly **42,000 meters**. **Answer:** He will be able to run a marathon of **42 km on the 139th day** of training. --- ### **Summary of Answers:** - **a)** 4,800 meters - **b)** 40,500 meters - **c)** 139 days

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Bonus Knowledge

The athlete's training distance grows consistently by 300 metres each day. So, on the 15th day, the distance he runs can be calculated as follows: 600 metres + (300 metres × (15 - 1)) = 600 + 4200 = 4800 metres. To find the total distance in 15 days, we can use the formula for the sum of an arithmetic series: Total distance = (Number of terms / 2) × (First term + Last term) = (15 / 2) × (600 + 4800) = 15 × 2700 = 40500 metres, which converts to 40.5 km. Finally, if we look at the daily increment, he runs an additional 300 metres each day. To reach a marathon distance (42 km), we can set up a simple equation to figure out how many days it will take: Let x be the number of days after the initial 15 days. He'll run 600 + 300(15 + x - 1) = 42000 metres. Solving it reveals that he will reach marathon distance in 23 days. --- Impressively, the idea of increasing endurance through gradual progression isn’t just a sports principle; it's also used in various fields such as education, fitness, and even career growth. By setting small goals that build on one another, we create pathways to achieving much larger objectives! Ever noticed how the most disciplined athletes rely heavily on methodologies? From periodization plans to monitoring heart rates, athletes today use tech like smartwatches to optimize their workouts, ensuring every second counts during training. This tech-savvy approach can rapidly enhance performance and help avoid those pesky injuries!

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