For the Grade 7s, your question involves not only knowledge of measurement, but also algebra. Do you see why?
Both questions also relate to our next driving task in our Patterning and Algebra unit.
The Grade 7s must also TRY to complete both POTWs this week and the Grade 8s MUST complete both POTWs this week. Enjoy!
Grade 8 POTW Enrichment!
Mr. Milette and his wife had a blast in San Antonio, Texas last week (even though the Raptors posted their worse loss of the season against the Spurs). The weather there was so nice that they stopped for some ice cream while on a bike ride visiting the UNESCO Mission sites, like The Alamo. Mr. Milette thought it was the perfect opportunity to ask his students a measurement question. He wants to know the volume of both ice cream cones, together. The radius of the half-sphere of ice cream is 5cm and the height of the cone is 12cm. Use the image below to help visualize this and draw it out as well. Again, what is the volume of both ice cream cones together? (It will actually be more like an estimate in the end, because as you can see the ice cream and cone are not perfect half-spheres/cones).
Grade 7 POTW Challenge!
Mr. Milette and his nephew made an epic snow castle over the winter break. Mr. Milette, with his teacher rulebook in-hand, reminded his nephew not to throw snow at people and that snowballs can do a lot of damage. His nephew did not seem convinced so he showed him this parent fail video. His precocious nephew next wondered what the volume of each brick at the top of their castle was. What is the volume of each of these snow castle bricks? An image of one brick example is shown below. If their castle had 11 of these bricks, what is the total volume of them?
Grade 8 Question:
ReplyDeleteFirst of all, we need to find the formula of finding the volume of a cone. The formula is πr2h/3. Essentially, this formula is the same as 3.14*radius²*height/3. Now, we need to identify what the radius and the height of the cone is. The radius of this ice cream cone is 5cm, while the height of this ice cream cone is 12cm. Now, we need to follow the formula of the volume of a cone.
First, we would do r² first, or 5². This gives us 25.
Next, we would do π*25, which is also 3.14*25. This gives us 78.54 when rounded to the nearest hundredths place value.
Then, we would do 78.54*h, or 78.54*12. This gives us a product of 942.48.
Finally, we would take that number and divide is by 3, or 942.48/3. This leaves us with 314.16. However, we are not done yet. The question is asking us to find the total volume of 2 cones.
So we would have to do 314.16*2, which is 628.32.
Therefore, the volume of the 2 ice cream cones together is 628.32cm³.
(I'm not sure if this is correct)
The Dawn of a new age, and the dawn of a new POTW! This is how I figured out the Grade 8 POTW(Because why not do Grade 8 first? XD):
ReplyDeleteFirst of all, I need the formula of the volume of a cone and the formula for the volume of a sphere. The formula for the cone is 1/3 of pi squared times the height. The formula for the sphere is 4/3 pi times the radius cubed. However, since the ice cream is only a half sphere, I would have to divide my total volume of the sphere by 2. Now, onto the calculations!
Radius of sphere: 5cm. Height of cone: 12 cm. Note: Radius of sphere is also radius of cone since the sphere is filled into the cone.
Cone Volume: pi*25*12/3=Pi*25*4=100 pi. Pi is about 3.14, so I multiply it and get 314cm cubed.
Sphere Volume: 4/3*pi*5*5*5(5 cubed)=4/3*125 pi=166.6(6 goes on infinitely)pi. Multiplying by Pi(3.14), I get 523.3(roughly)cm cubed. I also need to divide by 2 to get the half sphere, the volume is 261.6(goes on infinitely) cm cubed.
The total volume would be 261.6+ 314=575.6 cm cubed for one ice cream cone. I can round this to get approximately 576 cm cubed. To figure out the volume for 2 ice cream cones, I simply multiply by 2 and get a total of approximately 1152cm cubed.
I'm not sure if all my calculations are correct, but I think my answer should be somewhat close to the answer.
--Alan--
Grade 8 POTW:
ReplyDeleteAfter thoroughly reading the question, I understood that I was supposed to find the volume of an object which doesn't have a formula, so then I made my work easier by splitting the object into different and simpler objects. Here, It was an ice-cream cone with ice-cream on top of it shaped as a hemisphere. So I split the shapes to a cone and a hemisphere. After that, I found the volume of both figures and added them up. Here is my full work:
r = 5 cm
π = 3.14 cm
h = 12 cm
Area of the object = (πr^2 x h/3) + (2/3πr^3)
= (3.14 x 5^2 x 12/3) + (2/3 x 3.14 x 5^3)
= (3.14 x 25 x 4) + (2/3 x 3.14 x 125)
= (3.14 x 100) + (2/3 x 392.5)
= 314 + 261.66
= 575.66 cm^3
Therefore, after all calculations, The volume of 1 ice-cream cone is 575.66 cm^3. But the question had asked for the area of 2 ice-cream cones, so then I multiplied the answer by 2:
= 575.66 x 2
= 1151.32 cm^3
Therefore, The volume of 2 Ice-cream cones is 1151.32 cm^3.
[ I HOPE THIS IS RIGHT :^) ]
This is how I went about solving the Grade 7 POTW:
ReplyDeleteI looked at the length and saw that it was equal to 10 cm. And after looking at the missing space, it was quite obvious that it was going to be of a different length just by the look of it. So I used the width (3 cm) to fill up the sides and make them all line up. This means that there would be 3 cm on each side and
10-3-3 = 4. That means that the middle length is equal to 4 cm. That length has been assigned a value of x so that means the other x value has to be 4. Since we know that the width is 3 cm we can do (4*4*3) and get a total of 48 cm^3 (I don't know why (command + <) didn't work this time). Since we have that empty space out of the way, we can take the original dimensions and multiply them. We find that (10*3*8) is equal to 240. Then we use the empty space of 48 cm^3 and subtract it from 240. After doing calculations, (240 - 48) gives us a difference of 192. Now, to find the volume of 11 of these bricks we simply do (192*11) to get our total of 2112 cm ^3.
- Ameya
Are we allowed to assume the side lengths or is the figure not drawn to scale?
Delete-Alan
This comment has been removed by the author.
DeleteOMG I totally screwed up! I noticed today when I was reviewing this POTW with the Gr. 7s that some info IS missing. The basic question is: "solve for x if the total volume of the shape is 165 cubic cm". So, you may now more easily solve it.
DeleteOMG I totally screwed up! I noticed today when I was reviewing this POTW with the Gr. 7s that some info IS missing. The basic question is: "solve for x if the total volume of the shape is 165 cubic cm". So, you may now more easily solve it.
ReplyDeleteI completed both POTWs in my math notebook and came up with the answer x=5 for the grade 7 POTW and the answer that both ice cream cones together had a volume of 1152cm-cubed.
ReplyDeleteFor the grade 7 potw I decided first to just get the total volume of all the bricks by multiplying 165cm3 by 11 to get 1'815 so that's the total volume of the 11 bricks. Then as for x, if 165 is the volume left you do what is listed below.
ReplyDelete240-165=3X2
75=3X2
25=X2
5=X
This is because we first have to find the amount that isn't there. 240-165=75. Then 75=3X2 because it's X*X*3 (volume). Then to get rid of the 3 you divide both sides by 3. 75/3=25 so 25=X2. Then for the square if you get the square root of both sides it becomes 5=X.
Kevin, make sure that you add the unit cubed for the volume of the 11 bricks. :)
Delete-Alan
Grade 7 POTW:
ReplyDelete1815 cm^3 would be the total volume of 11 of these shapes
75 cm^3 would be the portion cut off. Not sure i did this correct,
math in book
Michelle, you also need to find the value of x.
DeleteGrade 7 POTW:
ReplyDeleteNow with the additional information, now I can solve the problem. To find x, I need to find the total volume of the missing space.
To figure that out, you first would find the volume of the whole rectangle, pretending there is still space is the missing spot. You would do 10*3*8. That gives us 240 cubic cm.
Now, we would subtract the volume of a whole rectangle by the actual volume of the shape. This way, we can find the volume of the missing space, leading to finding what x is. The actual volume of this shape is 165 cubic cm.
So we would do 240-165, leaving us with 75. So the volume of the missing space is 75 cubic cm.
Now, we can take that number and find x. To do that, we would work backwards. The formula for finding the volume is L*W*H.
So first we would divide 75 by 3, which would give us 25. Then we would square root 25, leaving us with 5.
Therefore the length of x is 5 cm.
Now we need to find the volume of their castle, to do that, we would multiply the volume of 1 block by the number of blocks. So we would do 165*11. That gives us 1815.
Therefore, the volume of the entire castle is 1815 cubic cm.
Here is how I solved the Grade 8 POTW:
ReplyDeleteSo first I wrote the Formula for a Semi-sphere and a cone:
Semi-Sphere Volume Formula: (4/3pir^3)/2
Cone Volume Formula: pir^2h/3
Then I plugged in 5 for the radius and found the volume of the semi sphere:
(1.333 * 3.141 * ( 5 * 5 * 5))/2
( 1.333 * 3.141 * (25 * 5)) /2
( 1.333 * 3.141 * 125) / 2
( 4.186953 * 125 ) /2
523.269125 / 2
Volume of the semi circle: 261.6845625cm^3 ( or 261.6cm^3 if you round)
Volume of a cone:
3.141 * ( 5 * 5) * (12/3)
3.141 * 25 * (12/3)
3.141 * 25 * 4
78.525 * 4
Volume of the Cone: 314.1cm^3
Volume of the whole ice-cream cone: 314.1 + 261.6 = 575.7cm^3
Volume of TWO ice-cream cones: 575.7cm^3 * 2 = 1151.4cm ^3
The Volume of the two ice-cream cones Mr. Milette and his wife have is ABOUT 1151.4cm^3
This is how i solved the Grade 7 POTW.
ReplyDeleteFirst, I solved for X. I knew that if there wasn't the gap, the volume would be 240 cm3 (10*3*8). The volume of the shape with the gap is 165 cm3. 240-165=75, so the volume of the gap is 75cm3. I knew that the depth was 3, and 75*3 is 25, and 5*5 is 25. So, X is 5, 5*5*3=75. If there was 11 of the 165cm3 bricks, the total volume would be 1815cm3 because 11*165=1815.
Grade 7 POTW:
ReplyDeleteHow I did is was to first calculate x, the first part of the question. To figure this out I needed the formula of the rectangular prism, minus the gap in the middle. This would be the volume of one brick, which is 165 cm cubed. So, my equation is:
3*8*10-3*x squared = 165. This would simplify to 240-3x squared = 165.
To figure out x, I have to make x by itself. First of all, I use 240-165=3x squared. This means that 3x squared is equal to 75. To get x by itself, I have to divide 75 by 3, then square root it. 75/3=25. The square root of 25 is 5, so x=5.
First part done! Now to the second part.
Since I already know the volume of one brick, all I have to do is use that volume and multiply it by 11 to get the answer. 165*11= 1815, so the total volume of the 11 blocks would be 1815 cubic centimeters.
-Alan-
For the grade 8 POTW, I got around 1151 cm3, and for the grade 7 question I got an answer of 240 with the gap for 1 of the bricks, then found the area of the gap and multiplied it (165) by 11, thus receiving 1815 cm3.
ReplyDeleteGrade 7 POTW: The shape with no gap is 240 cm cubed. The shape with the gap is 165 centimeters cubed. From there out with process of elimination, I found X has to be 5cm for both of the sides. if X times X times 3 was 75, the difference, I found the right response.
ReplyDeleteGrade 8 POTW: Using the formula with pi being rounded to 3.14, I discovered the semi-sphere would have a volume of 129.634 cm cubed. The volume of the cone is 26.1202 cm cubed. Adding these together times 2 would be: 311.5084 cm cubed. That is the total volume of the 2 ice cream cones.
Here is how I went about solving the Grade 7 POTW:
ReplyDeletea) In order to solve for the volume of all 11 bricks, we would do (165 * 11) which gives us 1815 cm^3. That is why, the volume of all the bricks is 1815 cm^3.
b) To solve for "x", we need to just do (10*3*8) and pretend as though that gap was never even there. So the product would be 240 cm^3. Since we know that the volume with the gap is 165 cm^3, we subtract 165 from 240 and get 75. Then we divide 75 by 3 and get 25. Finally, we take the square root of 25 and that is 5. Therefore, "x" is equal to 5.
To solve this question I assumed that each scoop was exactly a semi-circle. To solve the semi-circles combined I just used the formula to calculate the volume of a sphere (4/3pi *r^3). Then I calculated a cone and just doubled it (pi*r^2*h/3)*2. Thus I got the totals of 523.6cm^3 and 628.32cm^3. Added together it would be a total of 1151.92cm^3
ReplyDeleteGrade 7 POTW:
ReplyDeleteI know that 165 cubic cm is the volume of the structure, so the volume of all the bricks would be 165 cubic cm * 11= 1815 cubic cm. When I solved for x I took the total volume by 8*3*10 which is 240 cubic cm. Then, I subtracted 240 cubic cm by 165 cubic cm which equals to 75 cubic cm. Then:
75/3=25
25 squared=5 cm
Therefore, X is equal to 5 cm.
I did this on paper. Sry i meantioned it a bit late
ReplyDeleteGrade 7 POTW
ReplyDeleteFirst, I decided to calculate the volume of all 11 bricks. Since each brick's volume is 165cm cubed, and there are 11 bricks, I just had to multiply the number of bricks by the volume of one.
165cm cubedx11= 1815cm cubed.
Therefore, the volume of all the bricks combined is 1815cm cubed.
Next, I had to solve for the "x" value. First, I calculated the area of the whole prism, assuming there is no gap.
8cmx3cmx10cm= 240cm cubed.
Since we know that the brick with the gap is 165 cm cubed, I subtracted the total volume from 165 cm cubed, to get the volume of the gap.
240cm cubed
-165cm cubed
-------
75cm cubed
I had to divided 75 by 3 (getting the number back in the 2nd dimension), and got 25cm squared. Since the gap is a square (both sides are of equal length), I have to find the square root of 25 to get the value of x, which is 5.
Therefore, the value of x is 5.
Grade 8 POTW
ReplyDeleteIn order to solve this question, I first calculated the volume of a hemisphere, using the formula
Volume= 4/3piradiuscubed/2. Since the radius of the ice cream is 5cm, the volume of the hemisphere is 261.8cm cubed Therefore, the volume of one ice cream scoop is 261.8cm cubed.
Next, I had to calculate the volume of the cone, using the formula Volume=piradiussquaredxheight/3. Since the radius of the cone is 5cm, and the height is 12cm, the volume is 314.16. Therefore, the volume of one ice cream cone is 314.16cm cubed.
In order to figure out the volume of 1 ice cream cone, I added the volume of the ice cream and cone together, to get 575.96cm cubed. Since there are 2 ice cream cones, I multiplied the volume of 1 cone by 2, to get 1151.92 cm cubed. Therefore, the volume of 2 ice cream cones combined is 1151.92 cm cubed.
Grade 7 POTW
ReplyDeleteThe total volume of all 11 blocks is 1815 cm cubed. (165 * 11 = 1815)
To find what x was, I sort of did it in a more complicated way since my mom helped me with this:
240 - x² ⋅ 3 = 165
Then subtract 240 from both sides.. You get:
-x² ⋅ 3 = -75 After this, divide both sides by 3. You get:
-x² = -25 And to make both numbers positive we multiply by -1 because a negative multiplied by a negative is a positive. This becomes:
x² = 25 Then the opposite an exponent is the square root so you find the square root of x² and 25. AND VOILA!
x is equal to 5.
Grade 7 POTW (sorry for the late submission)
ReplyDeleteTo find x, I need to find the volume of the space missing from the rectangular prism. To do this, I need to find the volume of the rectangular prism and subtract the volume of the figure from it.
volume of rectangular prism = b*h*d
V = 10 * 3 * 8
V = 240
240 - 165 = 75 cubic cm
The volume of the area missing is 75 cm.
I know that one dimension of the missing area is 3 cm and the other 2 are x. 75 / 3 = 25. 2x must equal 25. The square root of 25 is 5. X is equal to 5.
The value of x is 5 cm.
To make sure my answer was right, I put it into the formula to solve for the volume of the figure.
V = 10*3*8 - 5*5*3
V = 240 - 75
V = 165
As you can see, it works.
x is equal to 5 cm.