The present value of a building in the downtown area is given by the function
P(t) = 300,000e^-0.09t+¡Ìt/2 f or 0 ¡Ü t ¡Ü 10
Find the optimal present value of the building. (Hint: Use a graphing utility to graph the function, P(t), and find the value of t0 that gives a point on the graph, (t0, P(t0)), where the slope of the tangent line is 0.)
To find the optimum present value of the building, we need to find the point on the graph of the function P(t) where the slope of the tangent line is 0.
Step 1: Graph the function P(t) = 300,000e^(-0.09t+√t/2) for the given range, 0 ≤ t ≤ 10, using a graphing utility.
Step 2: Look for the point on the graph where the slope of the tangent line is 0. This point will represent the optimal present value of the building.
To find the slope of the tangent line at a particular point on the graph, we can take the derivative of the function P(t) with respect to t.
The derivative of P(t) = 300,000e^(-0.09t+√t/2) can be found by applying the power rule and the chain rule:
P'(t) = (-0.09)(300,000)(e^(-0.09t+√t/2)) + (1/2)(300,000)(e^(-0.09t+√t/2))(1/√t)
This can also be simplified as:
P'(t) = -27,000e^(-0.09t+√t/2) + 150,000e^(-0.09t+√t/2)/√t
Step 3: Set the derivative equal to 0 and solve for t to find the critical points where the slope is 0:
-27,000e^(-0.09t+√t/2) + 150,000e^(-0.09t+√t/2)/√t = 0
Step 4: Use a numerical method, such as Newton's method or a graphing calculator, to approximate the values of t that satisfy the equation from the previous step. This will give us the values of t where the slope of the tangent line is 0.
Step 5: Substitute the values of t obtained in Step 4 back into the original function P(t) to find the corresponding optimal present values of the building.
Note: The specific numerical values for t and the optimal present value of the building will depend on the graph of the function P(t) and the calculated values.