Welcome to the Math POTW. The POTW is posted to challenge you and enrich your understanding and interest in math. The POTW tends to be just at, and sometimes beyond, specific curriculum expectations for Gr. 7/8 and is provided by the prestigious University of Waterloo's CEMC Program. You must try both questions. Full solutions will be posted with the POTW's next posting.
POTW #1 Question Gr. 7:
POTW# 1 Question Grade 8:
Triangle ABC is a 3-4-5 triangle, meaning it's a right angle triangle. Then you calculate the area of the triangle which is (BxH)/2 or (12x16)/2 which is equal to 96cm2.
ReplyDeleteSince BC is the diameter, half of it is the radius. Half of BC is 10. The formula for the area of a circle is Pi x R^2 or in our case Pi x 10^2 which is 100pi. 100 x pi is 314.16cm2(rounded to hundredths) and subtracted 96 cm2 it is equal to 218.16cm2 .
The shaded area is 218.18cm2.
Yay, Aiden first ever post for 2019/2020 school year! Make sure you check your answers when the next posting is up.
DeleteGrade 8 POTW:
ReplyDeleteIn order to find the area of the unshaded region of the circle, we need to find the area of both the circle and the triangle, and find the difference between said values.
To find the area of the triangle, we use the formula (b x h) / 2 = A. The base length of the triangle, in this case, is 12 cm, and the height is 16 cm. Now, to input these values into our formula:
A = (b x h) / 2
A = (12 x 16) / 2
A = 192 / 2
A = 96.
Therefore, the area of the triangle is 96 cm2.
Now, to find the area of the circle. The formula for the area of a circle is: 𝝅r^2. The radius of the circle is half the length of line segment BC (the diameter), therefore the radius of the circle is 10cm. Now, to input these digits into our formula:
A = 𝝅r^2
A = 𝝅(10^2)
A = 𝝅 x 100
A = 314.16
Therefore, the area of the circle is 314.16 cm2.
Now, to find the area of the unshaded region, we simply find the difference between the area of the circle (314.16cm2) and the area of the triangle (96cm2):
314.16 - 96 = 218.16cm2
The area of the unshaded region of the circle is 218.16cm2.
Grade 7: First I would list the numbers here where η is Riky's age.
ReplyDelete180, 181, 183, 187, 188, 190, 193, 195, 196, η
There are a even number of numbers here in this data set so I would simply consider the two numbers separately. The pairs of numbers that are possible for the median are:
(188, 190), (187,188), (η,188), (188, η). Since we know that the median changed by +0.5 cm, then the difference between the pairs of numbers from the old median is 0.5 less than the pair of new median. (Before and after the mistake correction). I need to find 2 pairs of numbers above that can be the old and new medians.(187,188) and (η,188) where η is 188. (188, 190) and (188, η) can also work where η is 191. I have established 2 answers which still need to satisfy the mean. Let's try η = 188
180, 181, 183, 187, 188, 188, 190, 193, 195, 196, The mean for this data set is 188.1 which is the new mean. The old mean is 188.1 - 0.5 = 187.6 So the old η is 183 which satisfies the requirements.
Old data set 180, 181, 183, 183, 187, 188, 190, 193, 195, 196,
New data set 180, 181, 183, 187, 188, 188, 190, 193, 195, 196,
Now when η is 191. The new mean would be 188.4 so the old mean must be 187.9 The old η will must be 186
Old data set 180, 181, 183, 186, 187, 188, 190, 193, 195, 196,
New data set 180, 181, 183, 187, 188, 190, 191, 193, 195, 196,
Now in this case, we look at the median pairs (188, 190) and (188, η) but it appears that (188,190) is not the old median for this set thus η = 191 is not possible.
Therefore, the only possible height for Riky is 183cm
This was Edward's answer I don't know if it says unknown on all screens
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ReplyDeleteThis comment has been removed by the author.
ReplyDeleteSo the area of a triangle is (bxh)/2.
ReplyDeleteIn this case, it is (12x16)/2=96
SO the area of the triangle is 96cm^2.
Since the diameter is bc, which is 20, which is 2 times the radius, 10 is the radius of the circle.
The area of a circle is πr^2.
In this case it is π100.
π100 is equivalent to 314.16(rounded to the hundreths)
So 314.16-96=218.16
The area of the unshaded region is 218.16cm^2.
Grade 7 POTW
ReplyDelete(I swear I didn’t copy I SWEAR) so with the ten ppl on the team, the median must be a pair of numbers then calculated w/ mean. This pair must be X (riky’s height) and 188. This is because any other eligible pair would not include X. The only numbers that could be X must be 188, 189, or 190 because otherwise they wouldn’t raise the median by 0.5. Proof:
(188 + 188) / 2 = 188. (***Old number + 188) / 2 = 187.5 . ***can be anything 187 or lower for now
(189 + 188) / 2 = 188.5. (****Old number + 188) / 2 = 188 . **** must be 188
(190 + 188) / 2 = 189. (*****Old number + 188) / 2 = 188.5 . ***** must be 189
So for now, our options for X are 188, 189, and 190.
Sum of 9 teammates is 1693
Mean formula: (1693 + X) / 10 = ((1693 + old height) / 10) + 0.5
Application:
(1693 + 190) / 10 = 188.3 ≠ [((1693 + 189) / 10) + 0.5 = 188.2 + 0.5 = 188.7] This cannot be X
(1693 + 189) / 10 = 188.2 ≠ [((1693 + 188) / 10) + 0.5 = 188.1 + 0.5 = 188.6] This cannot be X
The last available height is 188, however, as previously stated, it’s old height can be anything 187 or lower. But sike, that only applies for median. In order for this to apply for the mean rule, the old height must be 183. (I found this by jus5 doing that same formula 4 times descending the old height by 1 each time from 187, but I didn’t really feel like doing that in a comment so I just said it.)
(1693 + 188) / 10 = 188.1 = ((1693 + 183) / 10) + 0.5 = 187.6 + 0.5 = 188.1 This must be X
Therefore, Riky’s height is 188 cm but the coach originally measured it as 183 cm.
Old data: 180, 181, 183, 183, 187, 188, 190, 193, 195, 196
New data: 180, 181, 183, 187, 188, 188, 190, 193, 195, 196
That was probably really confusing, and I might’ve done something wrong idk
(We had to both right?) Grade 8 POTW
ReplyDeleteSo this one I had to look at aiden’s answer for to help me, but only for the 3-4-5 triangle part I swear.
So apparently a 3:4:5 ratio triangle means it’s a right angle, so height is 16, base is 12.
12 x 16 = 192
192 / 2 = 96 ———> area of triangle is 96cm squared.
Are of circle is πr squared, and diameter is 20cm so radius must be 10.
π10 squared = π100.
3.14 is π rounded to the hundredth, and 3.14 x 100 is 314
Circle area: 314 cm squared
314 - 96 = 218 cm squared ... (do I just put a “2” instead??)
The unshaded area of the circle is 218 cm squared.
... I really hope that was supposed to be mandatory
Kevin
ReplyDeleteIf you use the formula for calculating a triangles area B*H/2 = A so I would be the distance form AC which is 12 cm as the base and A-B which is 16 as the height it would be 16*12 = 192 / 2 = 96. So the area of the triangle is 96cm^2. Since the area of a circle is πr^2 and the diameter of the circle is BC which is 20 cm and the radius is half of the diameter which would be 10cm. Since 10^2 is 100 and π * 100 is 314.16 however this is with the shaded area included. So I took 314.16 - 96 = 218.16. So the are of the unshaded area is 218.16^2
To find the area of the unshaded region of the circle, I will first find the area of the triangle. But since I do not know the height of the triangle, and even though the triangle may look like a right triangle, I can’t be sure, I must find another way to calculate the area. This is why I decided to use Huron’s formula, which is √S(S-AB)(S-AC)(S-BC). (S = Semi Perimeter)
ReplyDeleteS = AB+AC+BC/2
S = 16+12+20/2
S = 48/2
S = 24
A = √S(S-AB)(S-AC)(S-BC)
A = √24(24-16)(24-12)(24-20)
A = √24(8)(12)(4)
A = √9216
A = 96
The area of the triangle is 96 cm squared.
Next, I must find the area of the circle, which is pi x radius squared.
A = πr^2
A = 3.14159265359 x 10 squared
A = 3.14159265359 x 100
A = 314.16
The area of the circle is 314.16 cm squared.
Finally, to find the area of the unshaded region of the circle, I will subtract the area of the triangle from the area of the circle to find the difference.
314.16 cm squared - 96 cm squared = 218.16 cm squared
The area of the unshaded region of the circle is 218.16 cm squared.
So this was absolutely brutal, but:
ReplyDeleteStarting off, I decided that it was necessary to figure out the area of the triangle. By multiplying the base times height (which was borrowed from Aiden, thanks) and dividing it by two ((12 * 16) / 2), I got 96 as a result.
So from here I had to find the total area of the circle.
The most useful information currently provided is the diameter, being 20 cm.
Using the method I found, I divided the diameter in half (10cm) and squared it (100cm), multiplied it by pi. The end result was ~314 cm squared.
Finally, I subtracted the area of the triangle (314 - 96), and got 218.
In conclusion, the area of the the unshaded parts of the circle are 218 cm squared.
Grade 7:
ReplyDelete180, 181, 183, 187, 188, 190, 193, 195, 196, ?
There are a even number so median would be either:
188, 190 187,188 ?,188 or 188 , ?.
The median changed by +0.5 cm
(187,188) and (?,188) where ? is 188. or (188, 190) and (188, ?) where ? is 191.
188
180, 181, 183, 187, 188, 188, 190, 193, 195, 196,
The mean for this data set is 188.1 (New)
The old mean is 188.1 - 0.5 = 187.6
So the old η is 183 which satisfies the requirements.
Therefore, Riky is 183cm
Grade 7:
ReplyDelete80, 181, 183, 187, 188, 190, 193, 195, 196, ?
There are a even number of numbers here in this data
possible for median are:
(188, 190), (187,188), (?,188), (188, ?).
median changed by +0.5 cm
187,188) and (?,188) where ? is 188 or(188, 190) and (188, ?) where ? is 191.
180, 181, 183, 187, 188, 188, 190, 193, 195, 196, The mean for this data set is 188.1 which is the new mean. The old mean is 188.1 - 0.5 = 187.6 So the old η is 183 which satisfies the requirements.
Therefore,Riky is 183cm