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Volume: 48 cm3
Volume: 384 cm3
Volume: 6 cm3
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Preimage | Image |
---|---|
2 | 2(2)=4 |
6 | 2(6)=12 |
4 | 2(4)=8 |
Surface Area (cm2) | Volume (cm3) | |
---|---|---|
Formula | S=Ph+2B | V=Bh |
Substitution | S=(32)(8)+2(48) | V=(48)(8) |
Calculation | S=352 | V=384 |
Preimage | Image |
---|---|
2 | 21(2)=1 |
6 | 21(6)=3 |
4 | 21(4)=2 |
Surface Area (cm2) | Volume, (cm3) | |
---|---|---|
Formula | S=Ph+2B | V=Bh |
Substitution | S=(8)(2)+2(3) | V=(3)(2) |
Calculation | S=22 | V=6 |
Let's make a table showing the surface areas and volumes of the given prism and the two images.
Preimage | Image B | Image C | |
---|---|---|---|
Surface Area | 88 cm2 | 352 cm2 | 22 cm2 |
Volume | 48 cm3 | 384 cm3 | 6 cm3 |
We see that a dilation with a scale factor of 2 makes the surface area 4 times more. Also, a dilation with a scale factor of 21 makes the surface area 41 of what it used to be.
Preimage | Image B | Image C | |
---|---|---|---|
Surface Area | 88 cm2 | 352 cm2=4(88 cm2) | 22 cm2=41(88 cm2) |
Volume | 48 cm3 | 384 cm3 | 6 cm3 |
Now we can compare the volumes.
Preimage | Image B | Image C | |
---|---|---|---|
Surface Area | 88 cm2 | 352 cm2 | 22 cm2 |
Volume | 48 cm3 | 384 cm3=8(48 cm3) | 6 cm3=81(48 cm3) |
A dilation with a scale factor of 2 makes the volume 8 times more. A dilation with a scale factor of 21 makes the volume 81 of what it used to be.
Preimage | Image B | Image C | |
---|---|---|---|
Surface Area | 88 cm2 | 352 cm2=4(88 cm2) | 22 cm2=41(88 cm2) |
Volume | 48 cm3 | 384 cm3=8(48 cm3) | 6 cm3=81(48 cm3) |
See that with a dilation of 2 the surface area was multiplied by 4=22, and the volume was multiplied by 8=23. With a dilation of 21 the surface area was multiplied by 41=(21)2, and the volume was multiplied by 81=(21)3.
Preimage | Image B | Image C | |
---|---|---|---|
Surface Area | 88 cm2 | 352 cm2=22(88 cm2) | 22 cm2=(21)2(88 cm2) |
Volume | 48 cm3 | 384 cm3=23(48 cm3) | 6 cm3=(21)3(48 cm3) |
We can conclude that if r is the scale factor of a dilation of a prism, the surface area of the preimage would be multiplied by r2 and the volume of the preimage would be multiplied by r3.