Final answer:
To find the values of A and B such that the graph of the exponential function f(t)=AeBt goes through the points (0,8) and (5,4), we can substitute the coordinates of each point into the equation and solve for A and B. After substituting the coordinates and solving for A and B, we find A ≈ 3.59 and B ≈ -0.1397.
Step-by-step explanation:
To find the values of A and B such that the graph of the exponential function f(t)=AeBt goes through the points (0,8) and (5,4), we can substitute the coordinates of each point into the equation and solve for A and B.
Step 1: Substitute (0,8) into the equation: 8 = AeB(0). This simplifies to 8 = A.
Step 2: Substitute (5,4) into the equation: 4 = AeB(5). Now substitute A = 8 from Step 1 and solve for B: 4 = 8e5B. Dividing both sides by 8 gives 0.5 = e5B.
Step 3: Take the natural logarithm of both sides to solve for B: ln(0.5) = ln(e5B). Using the property ln(ex) = x, this simplifies to ln(0.5) = 5B. Divide both sides by 5 to find B: B = ln(0.5) / 5 ≈ -0.1397.
Step 4: Substitute B = -0.1397 back into the equation 4 = 8e5B to find A: 4 = 8e5(-0.1397). Dividing both sides by 8 gives 0.5 = e-0.6985. Taking the natural logarithm of both sides gives ln(0.5) = -0.6985. Solving for A, we find A ≈ 3.59.
So, A ≈ 3.59 and B ≈ -0.1397.