The present value of a building in the downtown area is given by the function
p(t) = 300,000e^-0.09t+(t)/2 f or 0 _< t _< 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 optimal present value of the building, we need to find the value of t for which the slope of the tangent line to the graph is 0. Here's how you can do it:
1. Use a graphing utility: Plot the function p(t) = 300,000e^(-0.09t) + (t)/2 for the given domain 0 ≤ t ≤ 10. You can use software like Microsoft Excel or online graphing tools like Desmos or Wolfram Alpha.
2. Once you have the graph plotted, examine it visually to identify the point where the slope of the tangent line is 0. This point will represent the optimal present value of the building.
3. To find the specific value of t, you can use the derivative of the function p(t). Taking the derivative of p(t) will give you the slope of the tangent line at any given point.
The derivative of p(t) can be calculated as follows:
p'(t) = -300,000 * 0.09 * e^(-0.09t) + 1/2
4. Set the derivative equal to 0 and solve for t:
-300,000 * 0.09 * e^(-0.09t) + 1/2 = 0
5. Solve the equation to find the value of t that satisfies it. You can use numerical methods or techniques such as factoring or the quadratic formula to solve for t.
Once you find the value of t, substitute it back into the original function p(t) to obtain the optimal present value of the building, P(t).