Solve \( 500 \cdot e^{0.4t} > 5000 \) → \( e^{0.4t} > 10 \) → \( 0.4t > \ln(10) \approx 2.3026 \) → \( t > 2.3026 / 0.4 = 5.7565 \).
["How to Solve the Inequality ( 500 \cdot e^{0.4t} > 5000 ) Step-by-Step | Exponential Growth Explained", "Understanding how to solve exponential inequalities is essential for fields like finance, biology, engineering, and physics. In this article, we’ll break down the step-by-step solution to the inequality:", "[\n500 \cdot e^{0.4t} > 5000\n]", "and explain the mathematical reasoning behind each transformation.", "---", "### Step 1: Isolate the Exponential Term", "To solve for ( t ), begin by dividing both sides of the inequality by 500 to simplify:", "[\ne^{0.4t} > \frac{5000}{500}\n]", "[\ne^{0.4t} > 10\n]", "This reduces the problem to determining when the exponential function exceeds a constant.", "---", "### Step 2: Apply Natural Logarithm to Both Sides", "Since the exponential function ( e^{x} ) is strictly increasing, we can apply the natural logarithm (ln) without flipping the inequality:", "[\n\ln\left(e^{0.4t}\right) > \ln(10)\n]", "Using the logarithmic identity ( \ln(e^x) = x ), this simplifies to:", "[\n0.4t > \ln(10)\n]", "---", "### Step 3: Solve for ( t )", "Now divide both sides by 0.4 to isolate ( t ):", "[\nt > \frac{\ln(10)}{0.4}\n]", "Using the approximation ( \ln(10) \approx 2.3026 ), we compute:", "[\nt > \frac{2.3026}{0.4} = 5.7565\n]", "---", "### Final Result", "Thus, the solution to the inequality ( 500 \cdot e^{0.4t} > 5000 ) is:", "[\nt > 5.7565\n]", "This means that ( t ) must be greater than approximately 5.76 units (timestamp, population growth, reaction time, etc.) for the original expression to exceed 5000.", "---", "### Why This Matters: Real-World Applications", "Exponential inequalities model phenomena involving growth or decay—such as compound interest, bacterial population growth, or radioactive decay. Knowing how to solve them empowers you to predict critical thresholds and optimize decision-making.", "---", "### Summary", "To solve ( 500 \cdot e^{0.4t} > 5000 ):\n1. Divide both sides by 500 → ( e^{0.4t} > 10 )\n2. Apply natural logarithm → ( 0.4t > \ln(10) )\n3. Isolate ( t ) → ( t > \frac{\ln(10)}{0.4} \approx 5.7565 )", "Mastering this method equips you with a powerful tool for tackling real-world exponential challenges.", "---", "Keywords: exponential inequality, solve ( e^{0.4t} > 10 ), natural logarithm, solve exponential inequality, real-world applications of e^t, t > 5.7565"]
