A science policy analyst is comparing energy output from three renewable sources: solar, wind, and hydro. Solar generates 120 MWh on day one, increasing by 10% daily. Wind generates 150 MWh daily with no growth. Hydroelectric is 90 MWh daily. After how many full days will solar’s daily output exceed both wind and hydro combined?

["When Will Solar Power Surpass Wind and Hydro Combined? A Science Policy Analyst Compares Renewable Energy Growth", "In the global transition toward clean energy, understanding the growth dynamics of renewable sources is essential for effective policy planning. A science policy analyst recently compared the projected daily energy output of three major renewable sources—solar, wind, and hydroelectric—over time. The goal: determine after how many full days solar power will exceed the combined daily output of wind and hydroelectric generation.", "---", "### Daily Energy Output Breakdown", "- Solar: Starts at 120 MWh on day one, increasing by 10% each day.\n- Wind: Constant at 150 MWh every day.\n- Hydroelectric: Steady at 90 MWh per day.", "Combined wind + hydro output:\n[\n150 + 90 = 240 \ ext{ MWh}\n]", "Solar’s output each day follows a geometric progression:\n[\nS_n = 120 \ imes (1.1)^{n-1}\n]\nwhere ( n ) is the day number (starting from day 1).", "---", "### When Does Solar Exceed Combined Wind & Hydro?", "We want the smallest integer ( n ) such that:\n[\n120 \ imes (1.1)^{n-1} > 240\n]", "Divide both sides by 120:\n[\n(1.1)^{n-1} > 2\n]", "Take the logarithm of both sides (base 10 or natural log works—we use base 10):\n[\n\log((1.1)^{n-1}) > \log(2)\n]\n[\n(n - 1) \cdot \log(1.1) > \log(2)\n]", "Using approximate values:\n[\n\log(1.1) \approx 0.04139, \quad \log(2) \approx 0.3010\n]", "Solve for ( n - 1 ):\n[\nn - 1 > \frac{0.3010}{0.04139} \approx 7.27\n]\n[\nn > 8.27\n]", "Thus, the smallest integer ( n ) satisfying this is ( n = 9 ).", "---", "### Verification", "- Day 8: ( 120 \ imes (1.1)^7 \approx 120 \ imes 1.9487 \approx 233.84 ) MWh (< 240)\n- Day 9: ( 120 \ imes (1.1)^8 \approx 120 \ imes 2.1436 \approx 257.23 ) MWh (> 240)", "---", "### Policy Implications", "This analysis reveals that while solar energy grows steadily—benefiting from technological improvements and economies of scale—its daily output only surpasses combined wind and hydro after nine full days. This insight supports policy measures such as:", "- Incentivizing storage solutions to manage solar intermittency post-day 8.\n- Accelerating infrastructure deployment to maximize wind and hydro contributions during the transition.\n- Long-term planning emphasizing modular, scalable solar installations that gain momentum after day 8.", "---", "Conclusion\nThough solar begins with lower output, its exponential growth allows it to eventually surpass the steady but unchanged totals of wind and hydro. Understanding this timeline enables policymakers and energy planners to anticipate energy shifts, allocate resources efficiently, and design targeted interventions to ensure reliable, resilient renewable energy systems."]