A geologist is mapping geological formations and discovers a sedimentary layer that grows at a rate proportional to its thickness. If the layer grows from 50 meters to 200 meters over 200 years, assuming exponential growth, find the annual growth rate. Use the formula \( N(t) = N_0 \times e^{rt} \). Calculate \( r \).

["Title: Geologist Uncovers Exponential Growth in Sedimentary Layer: Calculating the Annual Rate", "Meta Description: A geologist mapping sedimentary formations discovers a layer growing exponentially. Using the exponential growth model, we calculate the annual growth rate when thickness expands from 50 m to 200 m over 200 years.", "---", "### When a Sedimentary Layer Grows Exponentially\nIn the field of geology, understanding the rates at which rock layers form provides critical insights into Earth’s history and sedimentation processes. A recent discovery by a dedicated geologist reveals a sedimentary layer undergoing exponential growth—thicker over time, with thickness increasing proportionally to its current size.", "This pattern follows a mathematical model commonly used in natural sciences: exponential growth. The formula describing this phenomenon is:", "[\nN(t) = N_0 \ imes e^{rt}\n]", "Where:\n- ( N(t) ) = thickness at time ( t ) (in meters)\n- ( N_0 ) = initial thickness (50 meters)\n- ( r ) = annual growth rate (what we aim to find)\n- ( t ) = time in years\n- ( e ) = base of the natural logarithm (~2.718)", "### From 50 Metres to 200 Metres in 200 Years", "The data tells us:\n- Initial thickness ( N_0 = 50 ) meters\n- Thickness after 200 years: ( N(200) = 200 ) meters\n- Time interval: ( t = 200 ) years", "Substitute into the exponential growth formula:", "[\n200 = 50 \ imes e^{r \ imes 200}\n]", "Divide both sides by 50:", "[\n4 = e^{200r}\n]", "To solve for ( r ), take the natural logarithm of both sides:", "[\n\ln(4) = 200r\n]", "Calculate ( \ln(4) ):", "[\n\ln(4) = \ln(2^2) = 2\ln(2) \approx 2 \ imes 0.6931 = 1.3862\n]", "Now solve for ( r ):", "[\nr = \frac{1.3862}{200} \approx 0.006931\n]", "Expressing ( r ) as a percentage (annual growth rate):", "[\nr \approx 0.693%\n]", "### Conclusion\nThe sedimentary layer grows at an annual exponential growth rate of approximately 0.693%. This discovery not only sheds light on dynamic geological processes but also illustrates how precise exponential modeling enables geologists to decode Earth’s layered history with remarkable accuracy.", "Understanding such growth rates helps interpret sediment accumulation, assess groundwater storage potential, and reconstruct paleoenvironments—proving that every layer tells a story, rooted in time—and now, in science.", "---\nKeywords: geologist, sedimentary layer, exponential growth, exponential model, ( N(t) = N_0 e^{rt} ), annual growth rate, geological formations, natural sciences, Earth history, sedimentation rate, annual growth rate calculation."]