Math Forum - Problem of the Week

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Math Forum - Problem of the Week

Submissions for Funny Ducks

Student / Short Answer / Long Answer
Student 1 / The area bounded by the ducks is 615 sq ft / The Pond has an area of 17.5ft x 20.5 ft. Since the ducks stay exactly
3 feet from the pond on their walks, that means the corners are
rounded. Each corner is 1/4 of a circle with a radius of 3. The area
bounded by their path is: (17.5 x 20.5) + 2(17.5 x 3) + 2(20.5 x 3.5)
+ 3^2PI, 358.75ft^2 + 105ft^2 + 123ft^2 + 28.27ft^2 which equals
615.02ft^2 which rounds to 615ft^2
Student 2 / The area of the walk and pond that the ducks use is 615 ft^2. / The area of the pond:
A = L x W
Ap = (20.5)(17.5)
Ap = 358.75 ft^2
______
__|__Aw___|__
| | |
| | |
As| Ap | |20.5 ft
| | |
__|______|_ |
|______|3ft
17.5 ft
As= The area of the path along the length= 3ft x 20.5ft = 61.5ft^2
Since there are two identical areas, one on each side of the pond
2As = 123ft^2
Aw= The area of the path along the width = 3ft x 17.5ft= 52.5 ft^2
2Aw = 105ft^2
The area of each corner is 1/4 of a circle.
Ac= The area of the 4 corners =( pi) r^2 = (pi)3^2= 28.27 ft^2
Total area is the sum of 2As +2Aw +Ac+Ap
358.75ft^2 + 123 ft^2 + 105 ft^2 + 28.27ft^2 =615.02 ft^2
Student 3 / The area bound by the path is 615 square feet (including the pond's area) or 256 square feet (excluding the pond's area). / 20 feet6inches=20.5 feet
17 feet6inches=17.5 feet
The ducks walk at 3 feet from the pond all the time.This means that
the path is parallel to the pond's margins (at 3 feet distance) and
has the form of a quarter of a circle (with the radius of 3 feet) in
its corners.The area bound by the path (without the pond's area)is
made from 2 rectangles (20.5 feet * 3 feet), 2 rectangles (17.5 feet *
3 feet) and 4 quarters of a circle (with a 3 feet radius):
2*(20.5*3)+2*(17.5*3)+pi*3^2=123+105+28=256 square feet
(I rounded pi*9=3.14*9=28.26 to 28 square feet)
The total area bound by the path (including the pond's area) is:
256+20.5*17.5=256+358.75=614.75 square feet
Rounding the avove number we have 615 square feet.
The answer is:
* the total area (including the pond's area) is 615 square feet
* the "ground" area (excluding the pond's area) is 256 square feet.
Student 4 / I found the answer for the sidewalk's final area to be 147 square feet and the pond's final area to be 523 square feet. / This POW is about a family of ducks that chose to live at a lily pond
25.5 feet long and 20.5 feet wide. Whenever they walked, they stayed
3 feet away from the pond. The problem is to find the area of the
sidewalk and pond.
My soliution for the area of the pond is 522.75 square feet, which
I rounded up to 523 square feet. I did it as the following:
20.5x25.5
For the solution of the sidewalk, I frist added 3 to each measurement
as follows:
(20.5+3)x(25.5+3)
Then I subtracted the area of the pond before rounding, in which I got
147 square feet. I got my answer by doing the following:
669.75-522.75
I then checked my answer by redoing the entire problem. There is
really no way you can check this problem other than to check the
equations. I got the same answer every time.
Finally, I did not receive any help or had any influences on
finding the answer to this problem other than the teacher who told
everyone to do the problem. :oP
Student 5 / My solution for the area of the pathway is 1,476 feet and the area of the pond and pathway together is 5,781 feet. / I did this by finding the area of ther pond and then the area of the
path and pond together. Then i subtracted the area od the pond from
the area of the path and pond together to find the area of the path
only. You can check this by making sure the path added to the pond
equals the total of the pond and path together.
Student 6 / the total area of the pond and path is 615.05^ft. / first I found the lengths of the sides which were 17.5 and 20.5 ft
you had to add six ft to the length and width so the new lengths were
23.5 and 26.5. To find the area of those you have to multiply them
23.5 * 26.5 which equals 622.75^ft
next I have to find the area of the rounded corners. there are 4
rounded corners which equals one circle.the area for circles is A =
pi * r^. A = 3.14 * 3^. A = 28.3^ft.
the total area is 28.3 +622.75=651.05^ft.
then you have to subtract the square corners which is 36^ft.
651.05-36=615.05^ft
Student 7 / The answer we got was that the area of the pond and the sidewalk was 368 feet. / First we found the area of the pond. To find the area you multiply the
length by the width. The area of the pond is equal to 358.75 feet.
Then we added three inches to the length and the width. That would be
the area of the land between the duck path and the pond. Then we
found the area of both by multiplying new length, which was 20'9", and
by the width which is 17'9". The area we found was about 368 feet.
Then we figured that the total area minus the ponds area would give us
the area of the sidewalk. 368.3125'- 358.75'= 9.5625'. That is the
area of the sidewalk only. We then checked our answer by adding the
area of the pond with the area of the sidewalk to get the area of the
pond and the sidewalk. 358.75' + 9.5625'= 368.3125'.
Student 8 / The area of the entire region is 615.02 feet squared and the area of the pathway is 256.27 feet squared. / I found the area of the lilly pond which was 358.75. I then
found the area of the strips surrounding the lilly pond that the ducks
wlaked on: 52.5, 61.5, 52.5, and 61.5. I then knew that the corners
would have to be rounded for the ducks to stay exactly 3 feet from the
pond and that each corner was one quarter of a circle. The 4 corners
added up to one whole circle. The radius of the circle was 3 feet and
to find the area I multiplied 3.14 by 3 squared and came up with
28.27. I then added all of these numbers together and got 615.02. To
get the area of the pahtway you would add 52.5, 61.5, 52.5, 61.5, and
28.27 and you would come up with 256.27.
I can verify this because if you add 256.75 and 358.27 it equals
615.02.
Student 9 / The area of the region is 623 feet / I got this by adding six feet to the width and six feet to the length
of the pond since there is three extra feet on each end totaling
twelve. Then I multiplyed the numbers (23.5ftx26.5) to get 623 feet
Student 10 / The area of the outside path was 264ft2 while the arae of the pnd was 358.75 feet2 and the total pond + Path was 716.75 feet2 / see any changes at bottom of sheet. Done Chronologicaly.
first, i drew a diagram of both squares- the pond as one and the
dotted line
that they walked on as the 2nd one. Then i figured out the area of
the inside
pond by Multiplying 17.5 by 20.5
Then, i was confused about the numbers for the 2nd square, so i
counted the
squares from left the right and got two numbers- 23.5 and 26.5 ,
which I then
multiplyed toghter to get 622.75 ft2.
Okay, so now I have the inner and outer circle done, i just have to
find the
path, so i used my diagram again and SUBTRACTED the outer square, by
the inner
square.
This equation was 622.75 - 358.75 and anwser was 264 ft for the
path.
I confirmed my anwser by adding the path + the inner square to get
the outer
square , and by subtracting the path - outer square to get the inner
one.
This was helpful and confirmed my anwser.
DOHH!!!
But i had frogot somthing though, they always stay 3 ft away form the
square, so when they turn corners, they go in a cricel, rather then a
square like form.
so, using the area for a circle, i did Pie Radius squared, and got
6.42. Then i found the total area of all the corners, which was (3 x
3 x 4= 36 ft), then subtracted that from the outer walkway (264 -
36) which equals 228, then i added the total circle area and Voila!
228 + 29.57 = 257.57 which is the area of the outer path!
So, for the total area, i would add 257.57 to the origial 266.75 and
for the total area got .. 524.34ft2!
Student 11 / 23 feet 6 inches long and 20 feet 6 inches widw with a total of 47 feet long and 41 feet wide. / The way I came to this answer is I added 3 feet to each side then double the two sides.
Student 12 / My solution is that the path around the pond is 264 ft. squared. The total rectangle including the path is 622.75 feet squared. / How I solved this problem was I figured out the demensions of the
lilly pond by multiplying the width by the length. Next I added 6 ft.
to both the length and the width. I did this because the ducks won't
walk any closer or farther than 3 ft. from the pond. From each point
on the pond it needs to go out three feet farther, this equalls six
feet total on each side. Now I multiplied the width by the length to
find the area. With this number I subtracted the area of just the pond
from the area of the total rectangle including the path. This told me
how large the path was. To verify my answer I added the pond's area
plus the sidewalk's area. This answer gives you the total of the large
rectangle.
Student 13 / The area bounded by the path of the ducks to the nearest square foot is 615 sq. ft. / Since the ducks always waddle 3 ft from the pond this path forms a
quadrilateral with rounded corners, since at the corner the duck stay
3 ft from a point which is part of a circle. So along the 20.5 ft
side 3 ft from it forms a 3 x 20.5 rectangle whose area is 61.5 sq.
ft. and you have two of them, one on each of the opposite sides. You
also have two 3 x 17.5 rectangles along the remaining opposite side,
these each have an area of 52.5 sq. ft. At each corner you have 1/4
of a circle with a radius of 3 ft, so all four corners form a circle
of radius 3 ft, using a value of pi=3.14 the area of this circle = pi
* 3^2 = 28.26 sq. ft. So the total area is 2* 61.5 + 2* 52.5 + 28.26
= 256.26 sq. ft. or approx. 256 sq. ft. Now add the area of the
pond, 17.5 x 20.5 = 358.75 sq. ft., 256.26+358.75 = 615.01 sq. ft. or
about 615 sq. ft.
I will try to be more careful in the future
Mike
Student 14 / The area of the entire region bound by the funny ducks path is 482 square feet. / L = length of pond = 20.5 feet.
W = width of pond = 17.5 feet.
L+3 feet = length of ducks path = 23.5 feet.
W+3 feet = width of ducks path = 20.5 feet.
Area of ducks path = (L+3)* (W+3)
= 23.5*20.5
= 481.75 square feet
= 482 square feet (rounded to nearest square foot)
Student 15 / 753 Square Feet. / Ok. First, I figured out what it would be without the 3 foot thing
around it. It was 27*17=459. Then, I figured out what it would be
on the 17 ft. ends if 3 ft. was added. It was 17*3=51 and doubled it
since there is 2 sides with 17 (102). Then I figured out the 26 ft.
sides. It was 26*3=78. Then I doubled that (156). Then I took
3*3=9 because of the corners and times that times 4 (36). Then I
added it all together. (459+102+156+36=753) And that is where my
answer came from.
Student 16 / I got 615 square feet. / I drew a picture to help me solve the problem this week. I drew a
rectangle for the pond and another bigger rectangle around it with
rounded corners to be the area where the ducks can walk. Then I drew
lines from the inside rectangle’s corners to the outer rectangle. Next
I labeled the distances – 20ft 6in for the length, 17ft 6in for the
width, and 3ft for distance between the two rectangles. I changed the
numbers into 20.5ft and 17.5ft. Then I multiplied those two numbers
and got 358.75ft squared. Next I did 20.5ft times 3ft and got 61.5ft
squared. Then I multiplied 61.5ft squared by 2 because there are two
of those rectangles. I got 123ft squared. Next I did 17.5ft times 3ft
and got 52.5ft squared. Then I multiplied 52.5 by two because there
are two of those rectangles. I got 105. To find a circle’s area you
have to multiply pie by radius squared. So I did 3ft (the radius)
squared which gave me 9ft and then 9ft multiplied by pie. That gave me
28.274ft squared. I did not have to divide by four to get a corner
because there are 4 corners, which makes one whole circle. Next I
added 123ft squared and 105ft squared and got 228ft squared. 228ft
squared plus 358.75ft squared is 586.75ft squared. Then I added 586.75
to 28.274ft squared to find the total area. That gave me 615.024ft
squared. I rounded it to the nearest square foot. I got 615 square
feet.
Student 17 / 736 Sq. Ft. / I added 3 twice to 17, and 3 twice to 26. Then I took 23 (17+6=23)
and 32 (26+6=32). 26*32=736.
Student 18 / The area of the rectangle is 623 square feet. / Add 3 feet two times to each dimension because the ducks stay that
far away on both sides. So the dimensions of the rectangle are 20
feet 6 inches plus 3 feet plus 3 feet (or 26 feet 6 inches) and 17
feet 6 inches plus 3 feet plus 3 feet (or 23 feet 6 inches). The
area is then 26.5 feet times 23.5 feet or 622.75 square feet (rounded
to 623 square feet).
Student 19 / The total area enclosed by the ducks' walking is about 615 feet. (notice, if this is not your answer, take a look at the way I approached this problem. You will see a difference) / Solution is in web page, but here's the text:
To do this problem I first defined it. You'll notice the corners are
bent. This is because if I went straight out the ducks would be 3 *
sqr(2) feet away from the pool. So, I actually have four quarter
circles on the edges. Figuring this in my total area is:
Area = 20.5' * 17.5' + 2 * 3 * 17.5' + 2 * 3 * 20.5' + pie * (3²)'
A = 358.75' + 105' + 123' + 9 pie'