Physics/Dr. Lankar
Name: ________________ section: ______________: grade% (my spot!):
_______
If the length of a 3D shape (any: cube, cylinder, cone ...) is multiplied by 10, what happens to its surface area ? its volume ?
BACK
1) TAKE SOME GUESS
You are going to work with unit cubes. each side is 1cm. So each face has an area 0f ________ cm2 . The volume of the unit cube is ___________ cm3
(that's why it is called a unit cube ! da !). Before doing the activity, take a guess. If you multiply each side by 3 (so you have a cube 3 x 3 x 3 , such as
width = 3, length = 3 and height = 3), predict the area of each side ------------ (take an educated guess).
Predict the volume __________________ cm3
2) Check if you are right
A) Take one unit cube :
area of 1 face = ________ cm2
volume of the unit cube = ___________cm3
B) build a 3x3x3 cube.
Area of 1 face = _________cm2 (count the "tiles" covering the face)
Volume of the cube = _______________ cm3 (count the unit cubes that nake up the large 1).
So if you multiply one side by 3: (the side of the cube is 3 times larger, we say the linear dimension is multiplied by 3)
The surface area is multiplied by _________. Which is the ____________ of 3/ (square ? cube ? fourth power ? )
The volume is multiplied by ____________. Which is the _____________ of 3. (square ? cube ? fourth power ?)
3) what if your multiply the dimension by 10.
Consider the unit square below (the small unit squareof area =1cm2) . It was scaled up 10 times.
THat is each linear dimension (side) has been multiplied by 10.
The square is now a 10 x 10 square.
What is the area of the 10 times scaled up square ?
(count the "tiles").
Area = ________________ cm2
So the Area is multiplied by ____________ when the side is multiplied by 10. (is it 10? 100? 1000 ?)
This will be true for any geometric shape.
If you multiply a length by a factor A, the area is multiplied by ________. (is it A ? A2 ? A3 ? A4 ? )
That is , if the shape is scaled by A times, the area is multiplied by ______. same answer as before.
-
What about the volume ?Let's scale up 10 times the unit cube (volume is 1cm3) ( each side of the cube is multiplied by 10):
The cube is a 10 x 10 x 10 cube.
Use a ruler and a pencil to draw the 10 layers . On the top draw the grid 10 x 10 .
Now you can answer the question : If the cube is scaled up 10 times, the volume is multiplied by ____________ (is it 10 ? 100? 1000? )
This will be true for any shape. If you multiply a length by a factor A, the volume is multiplied by _______. (is it A? A2 ? A3 ? A4 ? )
That is, if the shape is scaled up A times, the volume is multiplied by _____________. same answer as before.
get it ? if the unit cube is scaled up 4 times , each side is multiplied by 4 , but the AREA of each face is multiplied by ________. ( ask me to check)
the VOLUME is multiplied by ___________________. (ask me to check)
If you scale up the unit cube 5 times, each side is multiplied by __________, the area is multiplied by __________ , the volume is multiplied by __________
If you scale up the unit cube 6 times, the side is multiplied by ___________, the area is multiplied by ___________, the volume is multiplied by __________
4) your turn
Let's say you have a cylinder. The radius of the top (and bottom) is 2 cm and the height is 4 cm.
(see picture of the small one)
Let's scale up the cylinder 4 times. That is the radius is multiplied by 4 and the height is multiplied by 4.
A) With a ruler check and show that the height has been multiplied by four
and that the radius has been also multiplied by four.
B) Predict: The surface area of the top will be multiplied by ________ (according to what you just learnt is it 4 ?42 ? 43 ? 44 ? )
The volume of the cylinder will be multiplied by ____________________. (according to what you just learnt is it 4? 42 ? 43 ? 44 ? )
C) Let's Check.
I) Find the area of the top surface area of the small cylinder (R1=2cm) S1 = _________ cm2 (area of a circle = 3.14 R2 , with R = 2cm)
Find the area of the top surface area of the large cylinder , fill the blanks:
R2 = 4R1 = ________ cm so S2 = _________cm2 (use the formula area of a circle 3.14 R22 )
FInd the ratio S2/S1 = ___________ so if you multiply the radius (1 linear dimension) by 4 , the surface area is multiplied by ________.
II) Find the volume of the small cylinder. (H1 = 4cm) V1 = _________cm3 (volume = area x height =S1 x H1 = 3.14 R2 x H1 with R1 = 2cm )
Find the volume of the large cylinder. height is H2 = 4H1 = _______ cm
V2 = ______________ cm3 (use volume = 3.14 R22 x H2 with R2 = 8cm )
Find the ratio V2/V1 = __________ so if you multiplied 1 linear dimension by 4 (multiply the radius by 4 and the height by 4)
the volume is multiplied by ________________. (4? 42 = 16? 43 =64 ? 44 ?. So if you scale up a cylinder by 4 , the volume is multiplied by _________.
D) Try that :
I) A sphere has a radius of R1 = 5cm. Its surface area is S1 = ______ cm2 (4 pi R2 with pi = 3.14)
Its volume V1 = _____________ cm3 (4/3 pi R3)
You scale up the linear dimension by 5 . That means R2 = _________ cm.
Your new sphere has now a surface area S2 = ______ cm2 (multiply by 5 ? 52=25 ? 53=125 ? )and a volume V2 = ________ cm3 (multiply by 5 ? 25? 125?)
II) Here is a cone (using what you learnt)
The area of the base S1 = 5 cm2
The volume of the cone V1 = 40 cm3
You scale up the cone by 6. that means the height is multiplied by ____ and the radius is multiplied by ____.
The new surface S2 = ______ cm2 (multiply by 6 ? 62=36? 63=108 ? )
The new volume V2 = ______ cm3 (multiply by 6? 36 ? 108? )
II) A person is 1.70m tall. If you are a magician and scale the guy up 10 times, he will be ______ times taller. (linear dimension, length)
The volume of the guy will be multiplied by ___________. (10 ? 100 ? 1000? )That means that his weight will be multiplied by ________. (same answer)
(weight is proportional to the volume, suppose the density the same ). So he will be ________ heavier. (same answer)
The surface of the cross sections (area) of his bones (see drawing below) will be multiplied by ____________. (10 ? 100 ? 1000 ?)
The bones support your weight. So if you multiply the area of the bones by ______ it means that the bones will be ______ times
stronger. Do you see a problem here ? you are _____ heavier but your bones will be only _______ stronger ?
________________________________________________________________________
GOING FURTHER
1) I hope you got the idea. If you have a cube, or any other 3D shape and you make it larger such as each of its side is mutliplied by the same factor.
Let's take a factor of 2. The length is multiplied by ________, the width is multiplied by ______, the height is multiplied by __________2.
But the area is multiplied by _______. THat means the new area will be ______ times larger than the previous one.
The volume is multiplied by _____________. Or the new volume is _________ times larger than the previous one.
2) 1 m = 100 cm , the factor is _________ (in scientific notation ) ( a factor is by what you multiply )
but 1m2 (area) = 1mx1m = ___________ cm2 , the factor is now ________ (in scientific notation)
1 m3 (volume) = 1mx 1m x1m = _____________ cm3, the factor is now ________ (scientific notation)
3)
1 cm = 10mm , the factor is _________
1cm2 = _________mm2, the factor is ____________
1cm3 = _____________mm3, the factor is ____________
4) What is your height ? my height is ___________m (you can check lab4)
What is your weight ? my weight is __________ pounds.
IF a demon makes you 4 times taller but is nice enough to keep all proportions the same. (same proportion between head and arms ..)
you new height is ____________ m and you arm will be also multiplied by a factor of ________ and your leg will be ______ times longer
and your head will be _____ times longer.
However, your volume will be multiplied by ____________ !!! (not 4). The volume is like the weight if we consider you are made of the same
stuff everywhere in the body. So your new weight is now __________ pounds !!
The length of your leg bone, the fenur, is multiplied by ________ (like any new length). The surface area of the cross-section of the
bone will be multiplied by ______________ ! (its a surface area, so not 4).
Do your think your bones (or its surface area) can support your new weight ?
Why ?
5) Once you get the idea, you can answer al kinds of questions.
From lab2 you know that the Sun is about 10 times larger than Jupiter. (that its radius is 10 tomes larger).
but the surface area of the Sun is _____ times larger than Jupiter's, and its mass (volume) is ________ times larger than jupiter.
on a scale , if you have 1 sun on one plate , you need _________ jupiters on the other to balance out.
You have about the same ratio between Earth and Jupiter.
1) TAKE SOME GUESS
You are going to work with unit cubes. each side is 1cm. So each face has an area 0f ________ cm2 . The volume of the unit cube is ___________ cm3
(that's why it is called a unit cube ! da !). Before doing the activity, take a guess. If you multiply each side by 3 (so you have a cube 3 x 3 x 3 , such as
width = 3, length = 3 and height = 3), predict the area of each side ------------ (take an educated guess).
Predict the volume __________________ cm3
2) Check if you are right
A) Take one unit cube :
area of 1 face = ________ cm2
volume of the unit cube = ___________cm3
B) build a 3x3x3 cube.
Area of 1 face = _________cm2 (count the "tiles" covering the face)
Volume of the cube = _______________ cm3 (count the unit cubes that nake up the large 1).
So if you multiply one side by 3: (the side of the cube is 3 times larger, we say the linear dimension is multiplied by 3)
The surface area is multiplied by _________. Which is the ____________ of 3/ (square ? cube ? fourth power ? )
The volume is multiplied by ____________. Which is the _____________ of 3. (square ? cube ? fourth power ?)
3) what if your multiply the dimension by 10.
Consider the unit square below (the small unit squareof area =1cm2) . It was scaled up 10 times.
THat is each linear dimension (side) has been multiplied by 10.
The square is now a 10 x 10 square.
What is the area of the 10 times scaled up square ?
(count the "tiles").
Area = ________________ cm2
So the Area is multiplied by ____________ when the side is multiplied by 10. (is it 10? 100? 1000 ?)
This will be true for any geometric shape.
If you multiply a length by a factor A, the area is multiplied by ________. (is it A ? A2 ? A3 ? A4 ? )
That is , if the shape is scaled by A times, the area is multiplied by ______. same answer as before.
-
What about the volume ?Let's scale up 10 times the unit cube (volume is 1cm3) ( each side of the cube is multiplied by 10):
The cube is a 10 x 10 x 10 cube.
Use a ruler and a pencil to draw the 10 layers . On the top draw the grid 10 x 10 .
Now you can answer the question : If the cube is scaled up 10 times, the volume is multiplied by ____________ (is it 10 ? 100? 1000? )
This will be true for any shape. If you multiply a length by a factor A, the volume is multiplied by _______. (is it A? A2 ? A3 ? A4 ? )
That is, if the shape is scaled up A times, the volume is multiplied by _____________. same answer as before.
get it ? if the unit cube is scaled up 4 times , each side is multiplied by 4 , but the AREA of each face is multiplied by ________. ( ask me to check)
the VOLUME is multiplied by ___________________. (ask me to check)
If you scale up the unit cube 5 times, each side is multiplied by __________, the area is multiplied by __________ , the volume is multiplied by __________
If you scale up the unit cube 6 times, the side is multiplied by ___________, the area is multiplied by ___________, the volume is multiplied by __________
4) your turn
Let's say you have a cylinder. The radius of the top (and bottom) is 2 cm and the height is 4 cm.
(see picture of the small one)
Let's scale up the cylinder 4 times. That is the radius is multiplied by 4 and the height is multiplied by 4.
A) With a ruler check and show that the height has been multiplied by four
and that the radius has been also multiplied by four.
B) Predict: The surface area of the top will be multiplied by ________ (according to what you just learnt is it 4 ?42 ? 43 ? 44 ? )
The volume of the cylinder will be multiplied by ____________________. (according to what you just learnt is it 4? 42 ? 43 ? 44 ? )
C) Let's Check.
I) Find the area of the top surface area of the small cylinder (R1=2cm) S1 = _________ cm2 (area of a circle = 3.14 R2 , with R = 2cm)
Find the area of the top surface area of the large cylinder , fill the blanks:
R2 = 4R1 = ________ cm so S2 = _________cm2 (use the formula area of a circle 3.14 R22 )
FInd the ratio S2/S1 = ___________ so if you multiply the radius (1 linear dimension) by 4 , the surface area is multiplied by ________.
II) Find the volume of the small cylinder. (H1 = 4cm) V1 = _________cm3 (volume = area x height =S1 x H1 = 3.14 R2 x H1 with R1 = 2cm )
Find the volume of the large cylinder. height is H2 = 4H1 = _______ cm
V2 = ______________ cm3 (use volume = 3.14 R22 x H2 with R2 = 8cm )
Find the ratio V2/V1 = __________ so if you multiplied 1 linear dimension by 4 (multiply the radius by 4 and the height by 4)
the volume is multiplied by ________________. (4? 42 = 16? 43 =64 ? 44 ?. So if you scale up a cylinder by 4 , the volume is multiplied by _________.
D) Try that :
I) A sphere has a radius of R1 = 5cm. Its surface area is S1 = ______ cm2 (4 pi R2 with pi = 3.14)
Its volume V1 = _____________ cm3 (4/3 pi R3)
You scale up the linear dimension by 5 . That means R2 = _________ cm.
Your new sphere has now a surface area S2 = ______ cm2 (multiply by 5 ? 52=25 ? 53=125 ? )and a volume V2 = ________ cm3 (multiply by 5 ? 25? 125?)
II) Here is a cone (using what you learnt)
The area of the base S1 = 5 cm2
The volume of the cone V1 = 40 cm3
You scale up the cone by 6. that means the height is multiplied by ____ and the radius is multiplied by ____.
The new surface S2 = ______ cm2 (multiply by 6 ? 62=36? 63=108 ? )
The new volume V2 = ______ cm3 (multiply by 6? 36 ? 108? )
II) A person is 1.70m tall. If you are a magician and scale the guy up 10 times, he will be ______ times taller. (linear dimension, length)
The volume of the guy will be multiplied by ___________. (10 ? 100 ? 1000? )That means that his weight will be multiplied by ________. (same answer)
(weight is proportional to the volume, suppose the density the same ). So he will be ________ heavier. (same answer)
The surface of the cross sections (area) of his bones (see drawing below) will be multiplied by ____________. (10 ? 100 ? 1000 ?)
The bones support your weight. So if you multiply the area of the bones by ______ it means that the bones will be ______ times
stronger. Do you see a problem here ? you are _____ heavier but your bones will be only _______ stronger ?
________________________________________________________________________
GOING FURTHER
1) I hope you got the idea. If you have a cube, or any other 3D shape and you make it larger such as each of its side is mutliplied by the same factor.
Let's take a factor of 2. The length is multiplied by ________, the width is multiplied by ______, the height is multiplied by __________2.
But the area is multiplied by _______. THat means the new area will be ______ times larger than the previous one.
The volume is multiplied by _____________. Or the new volume is _________ times larger than the previous one.
2) 1 m = 100 cm , the factor is _________ (in scientific notation ) ( a factor is by what you multiply )
but 1m2 (area) = 1mx1m = ___________ cm2 , the factor is now ________ (in scientific notation)
1 m3 (volume) = 1mx 1m x1m = _____________ cm3, the factor is now ________ (scientific notation)
3)
1 cm = 10mm , the factor is _________
1cm2 = _________mm2, the factor is ____________
1cm3 = _____________mm3, the factor is ____________
4) What is your height ? my height is ___________m (you can check lab4)
What is your weight ? my weight is __________ pounds.
IF a demon makes you 4 times taller but is nice enough to keep all proportions the same. (same proportion between head and arms ..)
you new height is ____________ m and you arm will be also multiplied by a factor of ________ and your leg will be ______ times longer
and your head will be _____ times longer.
However, your volume will be multiplied by ____________ !!! (not 4). The volume is like the weight if we consider you are made of the same
stuff everywhere in the body. So your new weight is now __________ pounds !!
The length of your leg bone, the fenur, is multiplied by ________ (like any new length). The surface area of the cross-section of the
bone will be multiplied by ______________ ! (its a surface area, so not 4).
Do your think your bones (or its surface area) can support your new weight ?
Why ?
5) Once you get the idea, you can answer al kinds of questions.
From lab2 you know that the Sun is about 10 times larger than Jupiter. (that its radius is 10 tomes larger).
but the surface area of the Sun is _____ times larger than Jupiter's, and its mass (volume) is ________ times larger than jupiter.
on a scale , if you have 1 sun on one plate , you need _________ jupiters on the other to balance out.
You have about the same ratio between Earth and Jupiter.
pub google
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