A hydrologist models the discharge of a river as a function of rainfall: \( Q = 50R^{0.7} \), where \( Q \) is discharge in m³/s and \( R \) is rainfall in cm. If rainfall increases from 30 cm to 40 cm, by how much does discharge increase? Round to the nearest whole number.

["Title: How Hydrologists Model River Discharge: A Mathematical Approach to Rainfall Impact", "Understanding how rainfall influences river flow is essential in hydrology, flood forecasting, and water resource management. One widely used model relates river discharge ( Q ) to rainfall ( R ) through a power law:\n[ Q = 50R^{0.7} ]\nwhere ( Q ) is measured in cubic meters per second (m³/s) and ( R ) in centimeters of rainfall.", "In this article, we explore a key application of this model by calculating how discharge increases when rainfall rises from 30 cm to 40 cm. By analyzing the mathematical relationship, hydrologists gain precise insight into watershed behavior and flood risks.", "### The Discharge Model Explained\nThe equation ( Q = 50R^{0.7} ) captures the nonlinear response of river discharge to rainfall intensity. A small increase in rainfall produces a larger increase in discharge due to the exponent ( 0.7 > 1 ), meaning the relationship is superlinear but moderated.", "Let’s compute the discharge at both rainfall levels.", "### Step 1: Calculate Discharge at 30 cm Rainfall\n[ Q_{30} = 50 \ imes (30)^{0.7} ]\nFirst, compute ( 30^{0.7} ):\n( 30^{0.7} \approx 12.32 ) (using a calculator)\nThen:\n[ Q_{30} = 50 \ imes 12.32 = 616 , \ ext{m}^3/\ ext{s} ]", "### Step 2: Calculate Discharge at 40 cm Rainfall\n[ Q_{40} = 50 \ imes (40)^{0.7} ]\nCompute ( 40^{0.7} ):\n( 40^{0.7} \approx 14.98 )\nThen:\n[ Q_{40} = 50 \ imes 14.98 = 749 , \ ext{m}^3/\ ext{s} ]", "### Step 3: Find the Increase in Discharge\n[ \Delta Q = Q_{40} - Q_{30} = 749 - 616 = 133 , \ ext{m}^3/\ ext{s} ]\nRounded to the nearest whole number, the discharge increases by 133 m³/s.", "### Why This Matters in Hydrology\nThis model helps hydrologists predict how rivers respond dynamically to storm events. A 10 cm rise in rainfall does not just double discharge—instead, due to the exponent, discharge rises significantly more (by 133 m³/s from 30 to 40 cm). Such insights inform infrastructure planning, flood warnings, and environmental flow assessments.", "### Conclusion\nBy applying the mathematical model ( Q = 50R^{0.7} ), we’ve quantified the impact of increasing rainfall from 30 cm to 40 cm. The discharge rises by approximately 133 m³/s, highlighting the sensitive relationship between rainfall intensity and river flow. Understanding this link enables better management of water systems and improved resilience to extreme weather.", "---", "Keywords: hydrologist, river discharge, rainfall model, Q = 50R^0.7, flood forecasting, watershed modeling, superlinear precipitation-runoff relationship, water resources."]