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
10.1 Step 1: Substitute t=3 into the given equation for M(t). The equation for the number of molecules is M(t) = -t^3 + 3t^2 + 72t. M(3) = -(3)^3 + 3(3)^2 + 72(3) Step 2: Calculate the value. M(3) = -27 + 3(9) + 216 M(3) = -27 + 27 + 216 M(3) = 216 The number of molecules of the drug in the bloodstream 3 hours after the drug was taken is 216 molecules. 10.2 Step 1: Find the first derivative of M(t) to determine the rate of change. M(t) = -t^3 + 3t^2 + 72t M'(t) = (d)/(dt)(-t^3 + 3t^2 + 72t) M'(t) = -3t^2 + 6t + 72 Step 2: Substitute t=2 hours into the rate equation M'(t). M'(2) = -3(2)^2 + 6(2) + 72 Step 3: Calculate the rate. M'(2) = -3(4) + 12 + 72 M'(2) = -12 + 12 + 72 M'(2) = 72 The rate at which the number of molecules of the drug is changing at exactly 2 hours is 72 molecules/hour. 10.3 Step 1: To find when the rate of change is a maximum, we need to find the derivative of the rate function, M'(t), and set it to zero. This is the second derivative of M(t). M'(t) = -3t^2 + 6t + 72 M''(t) = (d)/(dt)(-3t^2 + 6t + 72) M''(t) = -6t + 6 Step 2: Set M''(t) to zero and solve for t. -6t + 6 = 0 -6t = -6 t = 1 Step 3: Verify that this value corresponds to a maximum. The third derivative M'''(t) = -6, which is less than 0, confirming it's a maximum. The value t=1 is within the domain 0 < t < 10. The rate at which the number of molecules of the drug in the bloodstream is changing will be a maximum after 1 hour. What's next? Send 'em! 📸