Final answer:
To simplify the expression involving exponential and logarithmic terms, apply logarithmic properties. Evaluate the trigonometric function by substituting π/3 for t in the given expression for x(t).
Step-by-step explanation:
The question requires simplifying an expression and then evaluating a trigonometric function.
To simplify the expression (e^2x+1/2(ln10-ln5)), you can apply the properties of logarithms since (ln10 - ln5) is equivalent to ln(10/5) or simply ln2. The expression becomes e^2x + 1/2(ln2). The simplification of the logarithmic part relies on knowing that log properties allow us to write the difference of logs as the log of a quotient.
For finding (x(π/3)), where x(t) = -1/2cos3t+9/2, you need to substitute π/3 for t and solve. Calculating -1/2cos(π) gives us -1/2 since the cosine of π is -1. Therefore, x(π/3) = -1/2 * (-1) + 9/2 = 1/2 + 9/2 = 5.