POTW #6 Grade 8 Question:
POTW #6 Grade 7 Question:
POTW #5 Grade 8 Answer:
POTW #5 Grade 7 Answer:
Thursday, October 13, 2016
POTW #6 - Time Flies!
Wow, it's already POTW #6! Great work so far gang and keep it up. Please see the latest POTW questions below. Don't forget to check your previous POTW answers below as well! Enjoy.
POTW #6 Grade 8 Question:
POTW #6 Grade 7 Question:
POTW #5 Grade 8 Answer:
POTW #5 Grade 7 Answer:
POTW #6 Grade 8 Question:
POTW #6 Grade 7 Question:
POTW #5 Grade 8 Answer:
POTW #5 Grade 7 Answer:
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Grade 8 POTW:
ReplyDeleteFirst, I use formulas to figure out all the areas.
To figure out the area of all the B areas, I use the full area of the semi-circle, and subtract it by the area of the smaller semi-circle that contains the smallest semi-circle and the whole area of R. To figure out R, I use the area of the middle semi-circle and subtract by the smallest one. Here are my calculations:
The radius of the middle semi-circle is 3/2, which is 1.5. The radius of the small circle is 1/2, which is 0.5, and the radius of the largest semi-circle is 5/2, which is 2.5.
Considering the area formula for circles, which is pi*radius squared, I can use it for each semicircle, dividing the area by 2 for each semi-circle.
The area of the largest semi-circle is 3.125 pi, the middle one is 1.125 pi, and the smallest is 0.125 pi.
So, the area of sections that contain b is 3.125pi - 1.125pi = 2pi, and the area of the sections that contain r is 1.125 pi - 0.125 pi = pi.
Therefore, the ratio of the two areas is 2pi to 1pi, but I can cancel it out so I get 2:1.
-Alan
Grade 7 POTW:
ReplyDeleteThe formula of the big semi-circle would be the diameter multiplied by pi. This would get me 100pi. But, since it is a semicircle I divide by 2 and get 50pi.
For each small semi-circle, the diameter is 100/5 since there are 5 sections. So, the diameter for each small semi-circle is 20, but I divide by 2 since it isn't a circle and it would be 10pi.
Since there are 5 of the small circles, I would multiply 10pi by 5, giving me 50pi.
Therefore, the two people running would tie on going to reach the finish line.
They will get to the point at the same time. To do this we have to find out how much they travel. To find the circumference of a circle is radius times 2 pi. Then since it's a semi circle you divide that by 2. 2 50 (100)*pi/2=157.079632679. Then for the bottom path, you have to find the circumference for each circle. 2 10(20)*pi/2=31.4159265359. Then you multiply that by 5 to get 157.079632679. Since it's the same amount of distance to travel, they'll get their at the same time.
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ReplyDeleteTo solve the grade seven POTW, I had to find out the circumference of each "circle" because they were travelling in a semi-circular path.
ReplyDeleteThe top path, which Bev rides on, has a diameter of 100m because A to F is 100 m. Because he is travelling only half of the "circle", I did 100*pi/2, which procured the result of 157.079632679. Then, for the path Mike took, there were 5 different circular paths. The straight distance (diameter) between each point is 20m because A-F is 100 m. So, having the diameter, I calculated that each path is 31.4159265359, because 20*pi/2. I divided it by two because he is only traveling half the circle. I then did 31.4159265359*5 because he traveled on five of those paths. The total was 157.079632679. This means that if they travel at the same speed, they will get to Point F from Point A at the same time, because it is the same distance.
About the Grade 8 POTW...
ReplyDeleteI screwed up on both the other two answers, the one about 1:2 and 2:1. The ratio is actually 1/3:2/5. I got this from dividing 2pi into 5 pieces since there are 5 pieces labeled B. I also got 1/3 from dividing pi into 3 pieces, giving me 1/3 : 2/5.
Alan
P.S. Hopefully I didn't screw up this time.
Grade 8: There are 3 red segments. There are 5 blue segments. They are all congruent. Therefore, the ratio from red to blue is 3 to 5. The amount of pieces matches up to the meters perfectly!
ReplyDeleteGrade 7: The dashed path is 314M so I would say it is longer.
So first I am going to figure out the diameter which I concluded to be 20 because A to F in a straight line is 100 and the semi circle diameters are going along the straight line from A to F and there are 5 semicircles by dividing 100 / 5 I got 20 m. Now since 20 m is the diameter, I need to multiply 20 x 3.14 (pi). That gives me 62.8 which is the perimeter of the circle. Now I divide it by 2 to get the perimeter of the semicircle and that gave me 31.4 which is the perimeter of the circles. Now I just need to multiply 31.4 by five to get 157 m for the bottom lane. to be continued coz i dont have time right now.
ReplyDeleteTo figure out this question, you need to find the circumference of upper and lower semi-circles. To find the circumference of a semi-circle, you have to do (Ï€*d)/2
ReplyDeleteFor the upper path, the diameter of that semi-circle is 100. So you would do π*100/2. The circumference would be 157 rounded. That would be the entire circumference and would take 157 meters to reach the end.
For the lower path, I need to calculate one of the small semi-circle. I know that the diameter from the previous path was 100. Since there are 5 equal semi-circles, you need to divide 100 by 5. That means that the diameter for each semi circle is 20 meters. Now we need to go back to the formula to find the perimeter. We would do π*20/2, which is 31 rounded to the nearest ones place value. But its not done yet. We have to multiply 31 by 5 because there are 5 semi-circles. That leaves us with 155 meters to reach the end.
Therefore, Mike will get to point F first.
Grade 7 POTW:
ReplyDeleteTo solve this question, I will need to find the circumference of the semi-circles that make up the paths.
The formula to do this is (Ï€ x diameter)/2
Bev's Path:
(Ï€ x diameter)/2
=(Ï€ x 100)/2
=(314.16)/2
=157.079632679
The top path is 157.079632679 meters long.
Mikes path (one semi-circle):
(Ï€ x diameter)/2
=(Ï€ x 20)/2
=(62.83)/2
=31.4159265359
Each tiny loop on the bottom path is 31.4159265359
All the loops on the bottom path = 31.4159265359 x 5. That results in 157.079632679
The bottom path is 157.079632679 meters long.
Both the paths are the same distance so Mike and Bev will reach point F at the same time.
I got about 1:2, as when dividing blue over red, I got 1.999......, rounded to two. I did my work in my math workbook.
ReplyDeleteGrade 8 POTW:
ReplyDeleteHere's how I solved it:
First I decided what I would do. What I planned to do was find the areas of the semi circles and the subtract from one and another to get the arch and divide by however many segments there were.
So first I found the area of the big semi circle by doing the formula for finding the area of a circle, and then dividing by two.
So the radius of the circle ( if another semicircle of the same size was added) is 2.5 as if the diameter is 5 then half of that is the radius:
(2.5 * pi)^2 / 2 ( I used 3.141592653 as pi)
7.853981633^2 / 2
61.68502749 / 2
30.84251374m squared
We'll label this semi circle B
Then I found the area for the smaller semi circles
Semicircle with the diameter of 3m: will be called R
R: ( 1.5 * pi) ^2 / 2
4.71238898^2 / 2
22.2066099 / 2 - 11.10330495m squared
The smallest semicircle will be called C:
(0.5 * pi)^2 / 2
1.570796327^2 / 2
2.4674011 / 2 = 1.23370055m squared
Next to find the area of the blue segments, I took semicircle A area and subtracted semicircle B area from it.
30.84251374 - 11.10330495 = 19.73920879m squared
Then I divided that by 5 to get the area for one segment:
19.73920879 / 5 = 3.947842758m squared
Next I got the area of the red segments by subtracting the area of semicircle C from Semicircle B:
11.10330495 - 1.23370055 = 9.8696044m squared
Next I divided it by three to find the area of one segment due to there being three segments in total:
9.8696044 / 3 = 3.289868133m squared
There for the ratio of the area of the blue to red segments is:
3.947841758m squared : 3.289868133m square
or if you round,
4m squared : 3 m squared
Here is how I went about solving the Grade 7 POTW:
ReplyDeleteThe circumference of a circle is found by multiplying its diameter by π. So to find the circumference of a semi-circle, you would have to divide the semi-circle by two. The length of the upper path is equal to half the circumference of a circle with diameter 100 m. So the equation to find the the length of the upper path would be π * 100 / 2 = 50π m. This would bring us to an answer of around 157.1, and would give us the length of Bev's path. Each of the semi-circles along the lower path have the same diameter. Therefore, the diameter of each of these semi-circles is 100 / 5 (the number of semi-circles) which then gives us a quotient of 20 m. This means that the length of the lower line is equal to half the circumference of five circles, each with a diameter of 20 m. Now, in order to find the length of the lower line we do π multiplied by the diameter, divided by 2. In other words, we do (π * 20) / 2. This would give us 62.83 divided by 2, which equals 31.4159265359. This is the length for one semi-circle. So we now do (31.4159265359 * 5) = 157.079632679. That would be Mike's length/distance. So since both paths are the same distance, Bev and Mike will reach point F at the same time.
First i started looking at A-F to figure out Bev's path, which was 100 meters long. That means that half of it would be 50 which is the radius. Then i tried to find out the perimeter of the circle so i did 2*50*Ï€ but, that was the full perimeter i only needed half. Half of 100 would be 50, which is how longs Bev's path was. Now i moved onto mike.
ReplyDeleteThen again, since A-F equals 100m, one of the 5 little sections would be 20m. Half of that would be 10 which is the radius. Then again i found out the perimeter of the smaller circle which was 20. But then since i was looking at only half once again, its would actually still stay at 10. But thats for only 1 of the sections so i did 10*5 and got 50. That means that Mikes path was also 50m long.
This means that both Mike and Bev's paths were the same length and they would finish at the same time. (I didn't multiply by π for this question but it ends up the same so yeah)
Great answer!
DeleteI can not remember if I posted my answer with an explanation or not but I did my work on hard copy thus getting the answer of 2:1
ReplyDeleteUnfortunately, my answer got deleted when I tried to preview it, so I decided to just right the gist of my process.
ReplyDeleteTo calculate Bev's path, I multiplied the diameter of the semi-circle (100m) by half of pi (since there is a semi-circle, not a full one) and got 157m.
To calculate Mike's, I used the same process, but only for 1 semi circle, and divided the diameter by 5 (number of circles). My product was 31.4, and multiplied by 5 semi-circles, totalled 157m.
Since both of my answers were the same, Bev and Mike will both finish at the exact same time.
Both of them would reach the end at the same time. I did this question on paper.
ReplyDelete