This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
The first problem is incomplete as it does not provide enough information (e.g., initial amount, growth rate, or amount at 10 hours) to calculate the amount present after three hours. Here is the solution for the second problem: Step 1: Identify the given information and the exponential growth formula. The exponential growth formula is P(t) = P_0 e^kt, where P(t) is the amount at time t, P_0 is the initial amount, k is the growth constant, and t is the time. Given: Initial amount P_0 = 200 grams at t=0 (6 am). Amount at 9 am is P(3) = 500 grams. The time elapsed from 6 am to 9 am is 3 hours. We need to find the amount at noon, which is t=6 hours from 6 am. Step 2: Calculate the growth constant k. Substitute the known values into the formula: 500 = 200 e^k · 3 Divide both sides by 200: (500)/(200) = e^3k 2.5 = e^3k Take the natural logarithm of both sides: (2.5) = (e^3k) (2.5) = 3k Solve for k: k = ((2.5))/(3) Step 3: Calculate the amount present at noon. Noon is 6 hours after 6 am (t=6). Use the calculated k and the initial amount P_0 = 200: P(6) = 200 e^(((2.5))/(3)) · 6 P(6) = 200 e^2 (2.5) Using the logarithm property a (b) = (b^a): P(6) = 200 e^((2.5)^2) Using the property e^(x) = x: P(6) = 200 · (2.5)^2 P(6) = 200 · 6.25 P(6) = 1250 The amount of bacterial culture present at noon will be 1250 grams. Send me the next one 📸