The area for a semi circle in Cartesian coordinate system with the flat edge against the x-axis is:A=∫0r2−x2√∫−rrdxdy
Bounding limits for x are from -r to r. And for y, this comes from the equation for a circle:x2+y2=r2
Solve for y:y=±r2−x2−−−−−−√
But since we are only dealing with a semi circle, we can always look at the positive answer.
To convert to a polar coordinate system, we used some equations you have probably seen:x=rcosθy=rsinθdxdy=rdrdθ
The substitute to convert to polar coordinate system:A=∫0π∫0rrdrdθSolve this for the area.A=πr2/2
To find the x coordinate of the centroid:x=1/A∫∫xdxdyorx=1/A∫0π∫0rrcosθ(rdrdθ)Solve and you will find x = 0 which intuitively makes sense.
Similar for the y coordinate of the centroid:y=1/A∫∫ydxdyory=1/A∫0π∫0rrsinθ(rdrdθ)Solve and you will find y=4r/3π
This means the centroid of a semicircle is at: (0,4r/3π)
Answer:
60
Step-by-step explanation:
did this before so easy
Answer:
see explanation
Step-by-step explanation:
6 × 24 ← replace 24 by 4 + 20
= 6(4 + 20) ← multiply each term in the parenthesis by 6
= 24 + 120
= 124
Break it down into 2-Dimensional shapes. Then add the areas together.
From the picture you can see;
front & back rectangles are 2*(4 x 8) = 64 m²
2 side rectangles are 2*(4 x 12) = 56 m²
2 triangular front & back pieces are (1/2)*8*3 = 12 m²
2 roof rectangles are 2*(5 x 12) = 120 m²
total Surface area = 64 m² + 56 m² + 12 m² + 120 m²
= 252 m²
For the volume; break it up into 3-dimenssional shapes and add the volumes together.
From the picture you can see;
Rectangular box volume is 4 x 8 x 12 = 384 m³
Triangular roof volume is area of front triangle multiplied through the length. (1/2)*8*3* 12 = 144 m³
Total volume = 384 m³ + 144 m³
= 528 m³
