Set \( 50 \times 2^{t/3} \geq 800,000 \)

["# Solve the Inequality ( 50 \ imes 2^{t/3} \geq 800{,}000 ): Step-by-Step Solution", "Understanding how to solve exponential inequalities is essential in fields like finance, physics, and computer science. In this article, we’ll break down how to solve the inequality:", "[\n50 \ imes 2^{t/3} \geq 800{,}000\n]", "This equation helps model real-world scenarios such as population growth, radioactive decay, or compound interest, where values grow exponentially over time.", "---", "## Step 1: Isolate the exponential expression", "Start by dividing both sides of the inequality by 50 to simplify:", "[\n2^{t/3} \geq \frac{800{,}000}{50} = 16{,}000\n]", "Now the inequality becomes:", "[\n2^{t/3} \geq 16{,}000\n]", "---", "## Step 2: Express 16,000 as a power of 2", "To make solving easier, rewrite 16,000 in terms of powers of 2.", "Recall that:", "[\n2^{14} = 16{,}384 \quad \ ext{(since } 2^{10} = 1024, \quad 2^{14} = 2^{10} \ imes 2^4 = 1024 \ imes 16 = 16{,}384\ ext{)}\n]", "Since ( 16{,}000 ) is very close to ( 16{,}384 ), we approximate:", "[\n16{,}000 \approx 2^{14}\n]", "Keep in mind this is an estimation; for precise solving, logarithms are preferred, but this gives a close starting point.", "---", "## Step 3: Apply logarithms to eliminate the exponent", "Take the base-2 logarithm of both sides:", "[\n\log_2(2^{t/3}) \geq \log_2(16{,}000)\n]", "Using logarithmic identity ( \log_b(b^x) = x ):", "[\n\frac{t}{3} \geq \log_2(16{,}000)\n]", "Now compute ( \log_2(16{,}000) ). Using change of base:", "[\n\log_2(16{,}000) = \frac{\log_{10}(16{,}000)}{\log_{10}(2)} \approx \frac{4.204}{0.3010} \approx 13.97\n]", "---", "## Step 4: Solve for ( t )", "Multiply both sides by 3:", "[\nt \geq 3 \ imes 13.97 = 41.91\n]", "---", "## Final Answer", "[\n\boxed{t \geq 41.91}\n]", "This means that the inequality ( 50 \ imes 2^{t/3} \geq 800{,}000 ) holds true for all ( t ) values greater than or equal to approximately 41.91 time units.", "---", "## What does this mean practically?", "This solution tells you the minimum time required for an exponentially growing quantity starting at 50 and doubling every 3 units to reach at least 800,000. For example, in a biological growth model, after about 42 time units, a population growing exponentially at this rate surpasses 800,000.", "---", "## Tips for Solving Similar Inequalities", "- Always isolate the exponential term.\n- Express constants as powers of the base when possible.\n- Use logarithms to bring down exponents.\n- Estimate logarithms when exact values aren’t needed.\n- Always include units and interpret results in context.", "---", "# Summary", "Solving exponential inequalities like ( 50 \ imes 2^{t/3} \geq 800{,}000 ) involves isolating the exponent, using logarithms, and interpreting the result carefully. With preventive estimation and precise computation, such problems become manageable and powerful tools in modeling real-world phenomena."]