Let's break down the problem:

We have a rectangular piece of paper with dimensions 24 cm (length) and 16 cm (width), with a semicircle cut out from one side (the shorter side). The objective is to find:

1. The perimeter of the paper after the semicircle is cut out.
2. The area of the paper after the semicircle is cut out.

Part a) Finding the Perimeter

• The perimeter of the rectangle without the cutout would normally be: $\text{Perimeter of rectangle} = 2(\text{length} + \text{width}) = 2(24 + 16) = 80 \, \text{cm}$

• Since a semicircle is cut out, we need to subtract the width of the rectangle that matches the diameter of the semicircle and add the circumference of the semicircle.

• The diameter of the semicircle is 16 cm (the width of the rectangle), so the radius $r$ is: $r = \frac{16}{2} = 8 \, \text{cm}$

• The circumference of a full circle is $2\pi r$, so the circumference of the semicircle is: $\text{Circumference of semicircle} = \pi r = 3.14 \times 8 = 25.12 \, \text{cm}$

• The modified perimeter will be: $\text{Perimeter} = 80 \, \text{cm (original)} - 16 \, \text{cm (width)} + 25.12 \, \text{cm (semicircle circumference)} = 89.12 \, \text{cm}$

Part b) Finding the Area

• The area of the rectangle is: $\text{Area of rectangle} = \text{length} \times \text{width} = 24 \times 16 = 384 \, \text{cm}^2$

• The area of the semicircle is half the area of a full circle: $\text{Area of semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times 3.14 \times 8^2 = \frac{1}{2} \times 3.14 \times 64 = 100.48 \, \text{cm}^2$

• The modified area will be: $\text{Area} = 384 \, \text{cm}^2 - 100.48 \, \text{cm}^2 = 283.52 \, \text{cm}^2$

Final Answers:

• Perimeter = 89.12 cm
• Area = 283.52 cm²

Let me know if you would like any further clarification or additional details.

Here are 5 related questions you might want to explore:

1. How would the perimeter change if the semicircle had a different radius?
2. What is the perimeter if the shape had a full circle instead of a semicircle cut out?
3. How do the area and perimeter change if the dimensions of the rectangle increase proportionally?
4. What if the cutout was an equilateral triangle instead of a semicircle?
5. How would you calculate the perimeter and area if an ellipse was cut out instead of a semicircle?

Tip: Remember, when dealing with circles, always ensure you use the correct approximation for $\pi$ depending on the precision required (e.g., 3.14, or $\pi$ in exact terms).