Answer is: option3

-2.5 units/sec

Solution:

We are given the function:

\[ y = 2e^{\cos x} \]

Both \( x \) and \( y \) vary with time \( t \), and we are given that:

\[ \frac{dy}{dt} = 5 \]

We need to find \( \frac{dx}{dt} \) when \( x = \frac{\pi}{2} \).

Using Implicit Differentiation:

\[ \frac{d}{dt} (y) = \frac{d}{dt} (2e^{\cos x}) \]

Using the Chain Rule:

\[ \frac{dy}{dt} = 2e^{\cos x} \cdot (-\sin x) \cdot \frac{dx}{dt} \] \[ 5 = -2e^{\cos x} \sin x \cdot \frac{dx}{dt} \]

Substituting \( x = \frac{\pi}{2} \):

• \(\cos \frac{\pi}{2} = 0\), so \( e^{\cos x} = e^0 = 1 \).
• \(\sin \frac{\pi}{2} = 1 \).

Thus, our equation simplifies to:

\[ 5 = -2(1)(1) \cdot \frac{dx}{dt} \] \[ 5 = -2 \frac{dx}{dt} \]

Solving for \( \frac{dx}{dt} \):

\[ \frac{dx}{dt} = \frac{5}{-2} = -2.5 \]

Final Answer: \(-2.5\) units/sec