The diagram shows a semi-circle with diameter F of lenget .
Angle radians and the shaded region has an aree of Aem",
Show that .
b Given that is inereasing at a rate of 0.05 radians per second, find the tate of
change of when radians.

Part a: Show that

1. Identify the components of the area :
   • The area of the sector formed by angle in a circle with radius is given by:
   • The radius of the semi-circle is half of the diameter :
   • Therefore, the area of the sector is:
2. Calculate the area of the triangle :
   • The area of triangle can be calculated using the formula:
   • The base and the height can be found using the sine function:
   • Thus, the area of triangle is:
3. Combine the areas:
   • The area of the shaded region is the area of the sector minus the area of the triangle:
4. Using the double angle identity:
   • We can express in terms of :
   • Therefore, we can rewrite the area as:
   • This simplifies to:

Part b: Find the rate of change of when

1. Differentiate with respect to time :
   • Using the chain rule:
2. Calculate :
   • Differentiate :
3. Evaluate at :
   • First, calculate :
   • Therefore:
4. Given radians/second:
   • Now, calculate :
5. Calculate the final result:
   • Thus:

Final Answers:

• Part a:
• Part b: The rate of change of when is .