Question

In a size transformation,the area of the image of an object of area 6m^2 is 37.5cm^2
Calculate:
1.the scale factor
2.the volume of an object in m^3 , if the volume of the image is 360 cm^3

Answer 1

?

Answer 2

The scale factor is the ratio of a length of the image divided by the corresponding length of the original figure.

We are not given the ratio of lengths, but we are given areas.
New_Area / Old_Area = (new_length / old_length )^2

Answer 3

lets say the two objects are right triangles.
object A has a hypotenuse of length x.
the image A' has hypotenuse of length y.
the scale factor is y / x

Answer 4

Getting it.
What about the volume?

Answer 5

we want to find y/x , the scale factor. but we are not given two corresponding lengths.
we are given areas. $$ \large \sf \frac{Area~ A'}{Area ~A }= \left( \frac{y}{x} \right)^2 $$

Answer 6

So how do we do it?

Answer 7

Volume of image is given as 360 cm^3

Answer 8

???

Answer 9

we can take the square root, which gives us 2.5
that is the scale factor

Answer 10

I'm getting confused.

Answer 11

i was explaining part 1)

Answer 12

It's 2.5 right?The scale factor.

Answer 13

yes

Answer 14

for part 2) are we supposed to use information from part 1) ?
the directions seem incomplete

Answer 15

the volume of an object in m^3 , if the volume of the image is 360 cm^2

Answer 16

volume in m^3 = 2.5*360

Answer 17

9 m^3?

Answer 18

the question is strange because units are different.
first make the units the same
the volume of an object in `m^3` , if the volume of the image is `360 cm^2`

Answer 19

you sure you copied the directions correctly

Answer 20

I have.We have to convert the units.

Answer 21

WAIT! Its 360 cm^3.I made a typo.

Answer 22

Sorry for the same.

Answer 23

ok but should we use the same scale factor from part 1 ?

Answer 24

part 1) and 2) are for the same transformation?

Answer 25

Yep.

Answer 26

we can use the equation
$$\large \frac{V '}{ V} = \left(\frac y x\right) ^3$$
we already know y/x = 2.5 $$\large \frac{360}{ V} = \left( 2.5 \right) ^3$$
solve for V, the original volume

Answer 27

15.625/360?

Answer 28

V = 360 / (2.5^3) cm^3
V = 23.04 cm^3

Answer 29

How??

Answer 30

23.04

Answer 31

right, in cm^3.
but the problem says to find volume in m^3?

Answer 32

we need to convert cm^3 to m^3

Answer 33

wait, we did part 1) wrong
i just saw that the units are not the same

Answer 34

In a size transformation,the area of the image of an object of area 6m^2 is 37.5cm^2
you cant compare different units. we need same units

Answer 35

600

Answer 36

google says
6 m^2 = 60000 cm^2
https://www.google.com/search?q=6+m^2+%3D+cm^2&ie=utf-8&oe=utf-8

Answer 37

Okay!
What next?

Answer 38

$$ \sf
\large \frac{A' }{A}= (scale~ factor)^2
\\~\\ \large \frac{60000 }{37.5}= (scale~ factor)^2
$$

Answer 39

40?

Answer 40

correct

Answer 41

wb part 2?

Answer 42

17.77 im getting

Answer 43

0.005625

Answer 44

I have to do this again

Answer 45

Getting lost now :(

Answer 46

the original object is 6 m^2 which is equal to 60,000 cm^2
the transformation is shrinking the object 37.5 cm^2

Answer 47

I misread the directions
Correction:

In a size transformation,the area of the image of an object of area 6m^2 is 37.5cm^2
original object is 6m^2 = 60000 cm^2
image is 37.5 cm^2 .

A ' / A = (scale factor)^2
(37.5 cm^2) / (60,000 cm^2) = (scale factor)^2
( 0.000625 ) = (scale factor )^2
scale factor = √ .000625
scale factor = .025 = 1/40

Calculate:
1.the scale factor : 1/40
2.the volume of an object in m^3 , if the volume of the image is 360 cm^3

V' / V = ( scale factor)^3
360 / V = (1/40)^3

360 / (1/40)^3 = V
V = 23040000 cm^3
change to m^3

Answer 48

This is what you should remember. Take notes of this.
Key:
L = a length of original object
L ' = corresponding length of image of object
A= area of original object
A' = area of image of object
V = volume of original object
V' = volume of image of object

The following equations are valid, assuming you have the same units.
$$\large{
\frac{ L'}{ L} = scale factor
\\~\\ \frac{ A'}{A} = (scale~ factor) ^2
\\~\\ \frac{V '}{ V} = (scale ~factor) ^3
}$$

Answer 49

we should word the problem again....

Answer 50

We have to parse the problem carefully.
`In a size transformation,the area of the image of an object of area 6m^2 is 37.5cm^2`
A' = 37.5 cm^2
A = 6 m^2 = 60 000 cm^2

We want to find L' / L , the scale factor.
We are given the areas. \[ \large{
\sf \frac{Area~ A'}{Area ~A }= \left( scale~ factor \right)^2
\\~\\ \frac{37.5}{60000 }= \left( scale~ factor \right)^2
}\]

Answer 51

Okay!

Answer 52

0.25

Answer 53

almost
0.025

Answer 54

what next?

Answer 55

Now that we have the scale factor, use this equation
\[\large{
\\~\\ \frac{V '}{ V} = (scale ~factor) ^3
}\]

Answer 56

360/V = 0.025^3

Answer 57

360 / V = 0.025^3
solve for V.
360 / V = 0.025^3 / 1
cross multiply
360 / (0.025^3) = V

Answer 58

23040000

Answer 59

right, the units of that is cm^3

Answer 60

now change to m^3

Answer 61

23.04

Answer 62

perfect.
23.04 m^3