For this second question below, please remember to show your work in full detail, but also to reply to your peers and ask clarifying questions of them. Enjoy!
Thursday, September 25, 2014
Math POTW #2
Wow, well done everyone on last week's POTW. I was especially happy with the fact it was only our first POTW of the year. Keep up the great work and continue to show all of your work and explain your thinking. I also encourage students to reply, comment, ask questions, and respond to one another's answers as well. Stu Dent's actual average was 84%, in the end. Interesting how a possible "outlier" like 39% can really affect one's overall average!
For this second question below, please remember to show your work in full detail, but also to reply to your peers and ask clarifying questions of them. Enjoy!
For this second question below, please remember to show your work in full detail, but also to reply to your peers and ask clarifying questions of them. Enjoy!
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the area of the shaded region is 2.6. to solve this question, I first found the area of the circle. the formula for that is pi x radius x radius. that is 3.14 x 9 (3x3) which equals 28.26. Because the portion of the circle given is a 90 degree angle, it is also quarter of the circle so I divided 28.26 by 4 and got 7.065 which is the area of that quarter. then I found the area of triangle AOB. the formula for that is half base x height. because this is a right angled triangle the side of the triangle can be used as the height. so, the height is 3 cm and the base is half of 3, 1.5 cm. 3x1.5 is 4.5 cm. I subtracted 4.5 cm from 7.065 and got 2.565 or 2.6 which is the area of the shaded region.
ReplyDeleteThat is quite a thorough answer Aparna. good job. How did you know the formula for calculating the area of a circle? The area of a triangle?
DeleteAparna, next time remember to put the units or measurements such as cm square.
DeleteFirst, I found the area of the circle with the formula of piRsquare. That's 3.14 times R (radius) square. This gives me 28.26. Knowing that 4 90 degree angles make up a whole circle, I divided 28.26 to get 7.065. Each quarter is worth 7.065 square centimetres. Then, I decided to find the area of the portion within the outline of the lines AO, OB and BA, then subtract the area from 7.065 to find the area of the shaded portion. The formula to find the area of a triangle would be height times base divided by two. Two triangles make up a square, and the formula to find the area of a square is height times base. Half of that, since a triangle is half of a square, would be height times base divided by 2. The height of the triangle can also be the side length, which is 3 cm, as stated in the statement. Then i needed the base length, which was 3 cm. The height must be vertical from the base, and the 3 cm line connects to the other through the 90 degree angle. 3x3 divided by 2 = 9/2 = 4.5 The area of the triangle is 4.5 cm square. To find the are of the shade portion, I subtracted the area of the triangle from the total area of the quarter of the circle, 7.065-4.5=2.565, which rounds to 2.6. The area of the shaded region is 2.6 cm square.
ReplyDeleteThe answer should be around 2.6. TO put everything together, it would be helpful to find the area of the whole circle (formula=radius² x pi (3.1415)). So radius squared (3x3=9) x 3.1415 = 28.2735 rounded to 28.3. Visually I can see that the AOB triangle plus the shaded part is exactly one quarter of the circle. Because of this, I can divide the area of the circle by 4 (28.3/4=7.075 rounded to 7.1) so now we have the area of the triangle + the shaded part. Now we just need to subtract the area of the triangle to get our final answer (formula for the area of a triangle is (base x height) / 2). We know that 2 sides of the triangle are 3cm because it is isosceles, because of the right angle, we can tell 2 things. Firstly, the right angle tells us the other 2 angles are 45 degrees because a triangles angles always add up to 180 degrees. Because 45 degrees is perfectly diagonal of a square it spans the same length and spans half the height. There are 2 3cm sides with 45 degree angles so now we know the base is 6cm(3+3) and the height is 1.5 (3 / 2) and so now we can get the triangles area. ((1.5 x 6) / 2 =4.5) so if we subtract 4.5 from 7.1 (7.1 - 4.5= 2.6) we get our answer.
ReplyDeleteCalista Wrote:
ReplyDeleteTo get the answer, I found the area of the full circle first:
A= π x r^2
= π x 3^2
= π x (3 x 3)
= 3.14... x 9
= 28.27 cm^2
Next, I divided the area of the circle by four:
28.27 ÷ 4 = 7.0675
Then, I found the area of the triangle ABO:
A = b x h ÷ 2
= 3 x 3 ÷ 2
= 9 ÷ 2
= 4.5
Last but not least, I subtracted the area of the triangle AOB from 7.065:
7.065 - 4.5 = 2.565 cm^2
2.565 cm^2 rounds up to 2.6
Therefore, the area of the shaded region is 2.565 cm^2
Sivan wrote:
ReplyDeleteFirst, I had to find the area of the whole circle, to do this I needed to write a formula
( 3squared) this meant that I had to take 3.14 x (3x3)= 28.26. I then knew that because this was the area of the whole circle, that I needed to divide the whole circle into 4 sectors (28.26 4 =7.065 ). Know that I knew the area of 14of the circle, I needed to find the area of the triangle ABO. To find the area of the triangle, I multiplied height x half base (3 x (3 2) 1.5=4.5. I then, subtracted the area of the triangle from the sector and got 2.565 2.6 which meant that the area of the shaded part was about 2.6 cm squared
Circle Area: π(pi) x 9 cm (3 cm to the power of 2) = 28.3 cm squared
ReplyDeleteTriangle Area: 3 cm x 3 cm = 9 cm squared, 9 cm squared ÷ 2 = 4.5 cm squared
Area of Shaded + Triangle: 28.3 cm (Circle area) ÷ 4 = 7.1 cm squared
Area of Shaded: 7.1 cm squared - 4.5 cm squared (Triangle Area) = 2.6 cm squared
The shaded area is 2.6 cm squared
The area of the circle is calculated with this formula: 3.14 (pi) x (3x3)=28.27 squared
ReplyDeleteArea of triangle: 3cm x 3cm (the two sides, not base) =9cm squared ÷ 2= 4.5.
To find the area of the shaded part and the triangle (both are necessary currently)you have to use the area of the circle and then divide that by 4 (since we need to find 1\4) which equals 7.1 cm
then by removing the area of the triangle (4.5 cm) from 7.1 cm the answer becomes 2.6 cm.
So the answer to this question is 2.6cm
The area of the circle is calculated with this formula: 3.14 (pi) x (3x3)=28.27 squared
ReplyDeleteArea of triangle: 3cm x 3cm (the two sides, not base) =9cm squared ÷ 2= 4.5.
To find the area of the shaded part and the triangle (both are necessary currently)you have to use the area of the circle and then divide that by 4 (since we need to find 1\4) which equals 7.1 cm
then by removing the area of the triangle (4.5 cm) from 7.1 cm the answer becomes 2.6 cm.
So the answer to this question is 2.6cm
The shaded area is 2.6. I found this by first finding the area of the circle. the formula for that is radius x radius(radius 2) then x3.14, the radius was 3, so 3x3 is 9, 9x3.14=28.26. This is the area of the circle. The full circle is 360 degrees, so 90 degrees(the slice) is exactly 1/4 of that, therefore i divided 28.26 by 4 and got 7.065, that is the slice. Next i found the area of the triangle, whcih was heightxbase divided by 2. we only got one side, but just by looking at the anglesand sides, you can tell that it is an isosilece traingle, meaning that one aother side is 3. 3x3 divided by 2 is 4.5. that is the traingles area. now all i had to do was subtract 4.5 from 7.065, this gave me 2.565 or 2.6 whcih was the answer
ReplyDelete