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TUGASAN 2
Building a Snow Statue - posted February 26, 2001
When the snow stopped falling, Kelly, Miguel, and Neha rushed outside to build a
snow statue. Kelly made one large snowball for the head. Miguel made a larger snowball for the middle section. Neha made a giant snowball for the bottom section.
As they were getting ready to put the snow statue together, the children noticed that
the circumference of the smallest section was about 2/3 the circumference of the
middle section. The middle section was about 3/4 the circumference of the largest
section. When they measured the smallest snowball, they found that it had a
circumference of about 80 centimeters.
How tall will the snow statue be when the sections are piled one on top of the other?
Assume that the snow does not compact.
Bonus: The group would like to make another snow statue with a total height of at
least two meters. If all other conditions are kept the same, what circumference will
they need for the smallest section?
Note: Please be sure to use the appropriate units of measurement, and state your final
answer in a complete sentence.
Comments
I know that Kelly, Miguel, and Neha had a great time building their snow statue. I
hope that you enjoyed trying to solve this problem!
A common error this week was calculator dependency. Remember that these kids are
out playing in the snow. How exact do you think their measurements were? My calculator states that the answer when I divide 360 by pi is 114.591559026, but I
can't accept that as the final height of the statue! Understanding this concept is very
difficult, so don't be discouraged.
Take another look at the information in the problem. Count how many times the word
"about" is used. That is your first clue that your final answer can not be exact. The
circumference of the smallest snowball is about 80 cm. How exact is that number? Is it
rounded to the nearest whole number or to the nearest ten? We really don't know for
sure. If the number had been stated as 80.0, we would have known that it was rounded to the nearest tenth. This tells us that the circumference of the smallest
snowball is not rounded to the nearest tenth or hundredth or thousandth. The very best
we can do with our final answer is to round to a whole number. My answer would be
about 115 cm. On the other hand, if you assume that 80 cm has been rounded to the
tens place, your final answer would need to be rounded to the tens place and you
would say that the height of the statue is about 110 cm.
Because our final answer can be no more accurate than a whole number, the value we
choose to use for pi does not matter. The two most commonly acceptable values of pi
are 3.14 and 22/7. Both of these values, as well as the pi key on your calculator, will
lead to the same final answer.
As you read the solutions below, you will find that 115 and 114 were both accepted
as correct final answers. Read the explanations to see why both answers are acceptable.
Teachers interested in reading comments from other teachers and joining a discussion
about this problem should visit the pow-teach discussion group and look for the discussion about Building a Snow Statue.
Keep Problem Solving
Judy Ann Brown
Middle School Project Coordinator
Highlighted solutions:
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From:
The diameter of the smallest snowball is bout 25.478 cm.
To find the diameter of the middle snowball, I set the following
equation.
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From:
Next we have to do L. So we have to divide 120 by 3(the amount of
fourths) so you can find out one fourth and add it on getting you L.
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From:
The circumference of the largest section is 160. I could have added
all the circumferences up but that would give me the total
circumference; I wanted the height. I drew the snowman, and drew a
line through the middle of him. That line is what I needed to find.
The line divided the snowman in half and that meant I could find the
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Comments I know that Kelly, Miguel, and Neha had a great time building their snow statue. I hope that you enjoyed trying to solve this problem! A common error this week was calculator dependency. Remember that these kids are out playing in the snow. How exact do you think their measurements were? My calculator states that the answer when I divide 360 by pi is 114.591559026, but I can't accept that as the final height of the statue! Understanding this concept is very difficult, so don't be discouraged. Take another look at the information in the problem. Count how many times the word "about" is used. That is your first clue that your final answer can not be exact. The circumference of the smallest snowball is about 80 cm. How exact is that number? Is it rounded to the nearest whole number or to the nearest ten? We really don't know for sure. If the number had been stated as 80.0, we would have known that it was rounded to the nearest tenth. This tells us that the circumference of the smallest snowball is not rounded to the nearest tenth or hundredth or thousandth. The very best we can do with our final answer is to round to a whole number. My answer would be about 115 cm. On the other hand, if you assume that 80 cm has been rounded to the tens place, your final answer would need to be rounded to the tens place and you would say that the height of the statue is about 110 cm. Because our final answer can be no more accurate than a whole number, the value we choose to use for pi does not matter. The two most commonly acceptable values of pi are 3.14 and 22/7. Both of these values, as well as the pi key on your calculator, will lead to the same final answer. As you read the solutions below, you will find that 115 and 114 were both accepted as correct final answers. Read the explanations to see why both answers are acceptable. Teachers interested in reading comments from other teachers and joining a discussion about this problem should visit the pow-teach discussion group and look for the discussion about Building a Snow Statue. Keep Problem Solving Judy Ann Brown Middle School Project Coordinator |
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From:
Now, we have to get the diameter of each snowball in order to get the
height of the snowman. The formula used to calculate circumference is
diameter*pie, and we can apply this formula on this situation.
To find the diameter of the smallest snowball, I set the following
equation.
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From: Marissa Miller, age 13 School: National Cathedral School-Girls, Washington, DC I think that the snowman that the group is building will be about 1 meter and 15 cms tall. First, you have to think about how you are going to find out the snowman's height. Thinking about it in a two-d form makes it easier. If you mentally stack the circles on top of eachother, you realize that you are going to need to find out the diameter. But first you have to get each one's circumference. Here's what you know: S(small circle)- 2/3M M(MEdium circle)- 3/4 L L(Large circle) - L (duh) Then you find out that S = 80 cm So you punch in the values. S= 80 For M you have to find out what one third is and then add it on to eighty. So you divide 80 by 2(the number of thirds in 80) and you get 40. Now you add those two up : + 40 80 -------- 120 so M = 120 cm Next we have to do L. So we have to divide 120 by 3(the amount of fourths) so you can find out one fourth and add it on getting you L. 120 / 3 = 40 So you add those up: + 40 120 ------ 160 To get the circumference with the diameter you just multiply diameter * pi but now you just switch it around and divide. You could divide by pi for each and then add them all up, but it's more accurate to add them all up and then divide by pi.So 80 +120 160 --------- 360 And then 360/ pi = about 114.591559 cm so about 1 meter and 15 cm |
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From: Emily Cupelo, age 14 School: Christian Brothers Academy, Syracuse, New York My funal answer to this weeks PoW is 115 centimeters. (Bonus explained below) First, for this problem. I wrote down what I knew about the Snow Statue: The smallest is 80 cm; 2/3 of the middle section. The middle section is 3/4 of the largest section. I started to try and find the circumference of the middle section. The problem stated that the smallest section was 2/3 of the middle section. (The other information would be useless because I don't know the circumference of the middle section yet.) From this information, I could set up a porportion to get the circumference:( x = middle section) 80 = 2 ---- --- x 3 I solved the porportion. 80 = 2 ------ ---- x 3 2x =240 --- --- 2 2 x = 120 The circumference of the middle section is 120 cm. From this information I could then find the circumference of the largest section. The information stated that the middle section is 3/4 of the largest section. I could set up another porportion to find the circumference of the largest section:( x = largest section) 120 = 3 ------- ---- x 4 3x = 480 ---- ---- 3 3 x = 160 The circumference of the largest section is 160. I could have added all the circumferences up but that would give me the total circumference; I wanted the height. I drew the snowman, and drew a line through the middle of him. That line is what I needed to find. The line divided the snowman in half and that meant I could find the diameters and add them all up. So, now I need to find all of their diameters. The circumference of a circle is pi * diameter. The diameter is the circumference / pi. That meant the height of the snowman is 80/ pi + 120/pi + 160/pi. I could solve this equation to get my answer: 80/pi + 120/pi + 160/pi = height 360/pi = height height= 114.59 cm. (rounded to 115 cm.) The height of the Snow Statue is 115 centimeters. BONUS: TO get the bonus, I tried to come up with an equation. It could be the smallest sec. + middle section + largest section = 200 cm (2 meters) The smallest section would be s and to make it easy I made all the variables in terms of s. Middle section = 3/2 * s Largest section = 4/3 * middle section = 4/3 * 3/2 * s = 12/6 * s = 2s So the equation for the total height would be s + 3/2s(middle secton) + 2s (largest section) = 200 I solved for s. (s + 3/2s + 2s)/ pi = 200 (3s + 3/2s)/ pi = 200 (9/2s)/pi = 200 9s/2pi = 200 Multiply both sides by 2pi, you get 9s = 400*pi s = 400pi/9 s = 139.63cm (rounded to 140) The circumference that the smallest snowball would have to be 140 centimeters in order for the snow statue to be 2 meters (or 200cm) high. |