Question - Calculating Volumes and Areas
Solution:
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.
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