A hydrologist uses a groundwater model where water level drops exponentially: h(t) = h₀ × e^(−0.02t). If the initial level is 50 meters, what is the drop in water level after 25 years? - Coaching Toolbox

Understanding the Context

Where:

• h(t) = groundwater level at time t (in meters)
  
• h₀ = initial water level
  
• e = base of natural logarithms (~2.718)
  
• t = time in years
  
• The coefficient −0.02 represents the annual decay rate
  
• This model assumes a consistent decline, reflecting real-world conditions like over-extraction or drought

The Exponential Decline Model Explained

The equation h(t) = h₀ × e^(−0.02t) mathematically captures how groundwater depth diminishes gradually but predictably. The negative exponent indicates decay toward zero, representing stored water being withdrawn or replenishment failing.

Why exponential? Because natural aquifer systems often respond to stress—such as pumping—with losses that scale over time, not linearly. This model allows scientists to forecast future water levels accurately and plan accordingly.

Key Insights

What Happens After 25 Years?

Let’s apply this model to a real scenario. Suppose a groundwater reservoir starts at h₀ = 50 meters. Using the formula:

h(25) = 50 × e^(−0.02 × 25)
h(25) = 50 × e^(−0.5)

Approximately, e^(−0.5) ≈ 0.6065

So,
h(25) ≈ 50 × 0.6065 ≈ 30.325 meters

Final Thoughts

Calculating the Drop in Water Level

Initial level: 50 meters
Level after 25 years: ~30.325 meters
Drop = 50 – 30.325 = ~19.675 meters

That’s a significant decline—nearly 40% of the original level—highlighting the urgency of sustainable water policies.

Why This Matters in Hydrology

An exponential decline suggests the aquifer is being depleted faster during prolonged pumping, increasing risks like land subsidence, saltwater intrusion, and water shortages. Hydrologists use such models to simulate extraction impacts and guide groundwater management strategies.

Real-World Applications