A robotics engineer programs a drone to ascend 3 meters in the first second, then 95% of the previous second’s height each subsequent second. What is the total height, in meters, the drone reaches after 10 seconds, rounded to the nearest tenth?
["Title: How a Robotics Engineer Programmed a Drone to Ascend Using a Geometric Sequence — Total Height After 10 Seconds", "In the world of robotics, precision and predictive motion control are essential. One fascinating application involves programming drones to follow specific flight profiles using mathematical sequences. Consider this scenario: a robotics engineer designs a drone that ascends in a controlled pattern — rising 3 meters in the first second, then 95% of the prior second’s height in each subsequent second.", "This motion follows a geometric sequence, presenting an excellent real-world example of exponential decay in vertical ascent. Understanding how to compute the total distance covered over time is key for optimizing drone performance, especially in applications like delivery, surveillance, and environmental monitoring.", "### The Pattern: A Geometric Sequence", "Let’s break down the drone’s vertical gain:", "- Second 1: $ 3 $ meters\n- Second 2: $ 3 \ imes 0.95 = 2.85 $ meters\n- Second 3: $ 3 \ imes (0.95)^2 = 2.7075 $ meters\n- ... and so on", "This forms a geometric sequence where:\n- The first term $ a = 3 $\n- The common ratio $ r = 0.95 $\n- The number of terms $ n = 10 $", "The total height after 10 seconds is the sum of the first 10 terms of this geometric sequence, given by the formula:", "$$\nS_n = a \cdot \frac{1 - r^n}{1 - r}\n$$", "Substituting the values:", "$$\nS_{10} = 3 \cdot \frac{1 - (0.95)^{10}}{1 - 0.95}\n$$", "First, calculate $ (0.95)^{10} $. Using a calculator:", "$$\n(0.95)^{10} \approx 0.598736\n$$", "Now plug into the formula:", "$$\nS_{10} = 3 \cdot \frac{1 - 0.598736}{0.05} = 3 \cdot \frac{0.401264}{0.05} = 3 \cdot 8.02528 = 24.07584\n$$", "Rounding to the nearest tenth:", "$$\n\boxed{24.1}\n$$", "### Conclusion", "Through precise engineering and mathematical modeling, the robotics engineer successfully programmed a drone to ascend using a geometric pattern. After 10 seconds, the total height reached is approximately 24.1 meters. This elegant application of sequences in drone motion control highlights how robotics combines physics, programming, and mathematics to achieve controlled aerial performance—critical for advancing autonomous flight systems.", "Whether for educational insight or real-world drone design, understanding such geometric progression enables engineers to predict and optimize dynamic behavior with confidence."]
