Calculating Volumes and Areas
Let's solve both parts of this question one by one:
### Part a)
To calculate the volume of the pyramid with a square base:
The formula for the volume of a pyramid is \( \frac{1}{3} \times \text{base area} \times \text{height} \).
Given:
- The height (\(h\)) of the pyramid is 10 cm.
- The side (\(s\)) of the square base is 5 cm.
First, calculate the base area:
- Base area = \( s^2 = 5 \text{ cm} \times 5 \text{ cm} = 25 \text{ cm}^2 \).
Next, calculate the volume of the pyramid:
- Volume = \( \frac{1}{3} \times \text{base area} \times h = \frac{1}{3} \times 25 \text{ cm}^2 \times 10 \text{ cm} \).
- Volume = \( \frac{1}{3} \times 25 \times 10 \text{ cm}^3 = \frac{250}{3} \text{ cm}^3 \approx 83.33 \text{ cm}^3 \).
To convert cubic centimeters to liters, remember that 1 liter equals 1000 cubic centimeters:
- Volume in liters = \( \frac{83.33}{1000} \text{ liters} \approx 0.083 \text{ liters} \).
So the volume of the pyramid is approximately 0.083 liters (to 2 decimal places: 0.08 liters).
### Part b)
For the rectangular cover plate with rounded corners to form sectors:
Given:
- The dimensions of the rectangular plate are 500 mm by 300 mm.
- Each corner is rounded to form a sector with a radius of 25 mm.
Firstly, calculate the area of the original rectangle:
- Area of rectangle = Length × Width = 500 mm × 300 mm = 150000 mm².
Next, you have to subtract the area of the four sectors. Each sector is a quarter of a circle with a radius of 25 mm.
The area of one full circle with radius \(r\) is \( \pi r^2 \). Therefore, the area of one quarter-circle (or sector) would be \( \frac{1}{4} \pi r^2 \).
- Area of one sector = \( \frac{1}{4} \pi (25 \text{ mm})^2 \).
- Area of one sector = \( \frac{1}{4} \pi \times 625 \text{ mm}^2 \).
- Area of one sector = \( 156.25 \pi \text{ mm}^2 \).
Since there are four such sectors:
- Total area of the four sectors = \( 4 \times 156.25 \pi \text{ mm}^2 \).
- Total area of the four sectors = \( 625 \pi \text{ mm}^2 \).
Subtracting this from the area of the rectangle gives the final area:
- Final area of the plate = Area of rectangle - Total area of the four sectors.
- Final area of the plate = 150000 mm² - 625π mm².
- Final area of the plate = \( 150000 \text{ mm}^2 - 1963.4954 \text{ mm}^2 \) (using \( \pi \approx 3.1416 \)).
- Final area of the plate ≈ 147036.5 mm² (rounded to one decimal place).
So, the final area of the rectangular cover plate, after rounding off the corners to form sectors, is approximately 147036.5 mm².
