13.4 The effect of multiplying a dimension by a factor of k (EMA7T)

When one or more of the dimensions of a prism or cylinder is multiplied by a constant, the surface area and volume will change. The new surface area and volume can be calculated by using the formulae from the preceding section.

It is possible to see a relationship between the change in dimensions and the resulting change in surface area and volume. These relationships make it simpler to calculate the new volume or surface area of an object when its dimensions are scaled up or down.

Consider a rectangular prism of dimensions ll, bb and hh. Below we multiply one, two and three of its dimensions by a constant factor of 55 and calculate the new volume and surface area.

Dimensions

Volume

Surface

Original dimensions

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V=l×b×h=lbhV=l×b×h=lbhA=2[(l×h)+(l×b)+(b×h)]=2(lh+lb+bh)A=2[(l×h)+(l×b)+(b×h)]=2(lh+lb+bh)

Multiply one dimension by 55

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V1=l×b×5h=5(lbh)=5VV1=l×b×5h=5(lbh)=5VA1=2[(l×5h)+(l×b)+(b×5h)]=2(5lh+lb+5bh)A1=2[(l×5h)+(l×b)+(b×5h)]=2(5lh+lb+5bh)

Multiply two dimensions by 55

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V2=5l×b×5h=5.5(lbh)=52×VV2=5l×b×5h=5.5(lbh)=52×VA2=2[(5l×5h)+(5l×b)+(b×5h)]=2×5(5lh+lb+bh)A2=2[(5l×5h)+(5l×b)+(b×5h)]=2×5(5lh+lb+bh)

Multiply all three dimensions by 55

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V3=5l×5b×5h=53(lbh)=53VV3=5l×5b×5h=53(lbh)=53VA3=2[(5l×5h)+(5l×5b)+(5b×5h)]=2(52lh+52lb+52bh)=52×2(lh+lb+bh)=52AA3=2[(5l×5h)+(5l×5b)+(5b×5h)]=2(52lh+52lb+52bh)=52×2(lh+lb+bh)=52A

Multiply all three dimensions by k

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Vk=kl×kb×kh=k3(lbh)=k3VVk=kl×kb×kh=k3(lbh)=k3VAk=2[(kl×kh)+(kl×kb)+(kb×kh)]=k2×2(lh+lb+bh)=k2AAk=2[(kl×kh)+(kl×kb)+(kb×kh)]=k2×2(lh+lb+bh)=k2A

Worked example 18: Calculating the new dimensions of a rectangular prism

Consider a rectangular prism with a height of 44 cmcm and base lengths of 33 cmcm.

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  1. Calculate the surface area and volume.

  2. Calculate the new surface area (AnAn) and volume (VnVn) if the base lengths are multiplied by a constant factor of 33.

  3. Express the new surface area and volume as a factor of the original surface area and volume.

Calculate the original volume and surface area

V=l×b×h=3×3×4=36 cm3A=2[(l×h)+(l×b)+(b×h)]=2[(3×4)+(3×3)+(3×4)]=66 cm2
VA=l×b×h=3×3×4=36 cm3=2[(l×h)+(l×b)+(b×h)]=2[(3×4)+(3×3)+(3×4)]=66 cm2

Calculate the new volume and surface area

Two of the dimensions are multiplied by a factor of 3.

Vn=3l×3b×h=3(3)×3(3)×4=324 cm3An=2[(3l×h)+(3l×3b)+(3b×h)]=2[(3(3)×4)+(3(3)×3(3))+(3(3)×4)]=306 cm2
VnAn=3l×3b×h=3(3)×3(3)×4=324 cm3=2[(3l×h)+(3l×3b)+(3b×h)]=2[(3(3)×4)+(3(3)×3(3))+(3(3)×4)]=306 cm2

Express the new dimensions as a factor of the original dimensions

V=36Vn=324VnV=32436=9Vn=9V=32VA=66An=306AnA=30666An=30666A=5111A
VVnVnVVnAAnAnAAn=36=324=32436=9=9V=32V=66=306=30666=30666A=5111A

Worked example 19: Multiplying the dimensions of a rectangular prism by kk

Prove that if the height of a rectangular prism with dimensions ll, bb and hh is multiplied by a constant value of kk, the volume will also increase by a factor kk.

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Calculate the original volume

We are given the original dimensions ll, bb and hh and so the original volume is V=l×b×hV=l×b×h.

Calculate the new volume

The new dimensions are ll, bb, and khkh and so the new volume is:

Vn=l×b×(kh)=k(lbh)=kV
Vn=l×b×(kh)=k(lbh)=kV

Write the final answer

If the height of a rectangular prism is multiplied by a constant kk, then the volume also increases by a factor of kk.

Worked example 20: Multiplying the dimensions of a cylinder by kk

Consider a cylinder with a radius of rr and a height of hh. Calculate the new volume and surface area (expressed in terms of rr and hh) if the radius is multiplied by a constant factor of kk.

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Calculate the original volume and surface area

V=πr2×hA=πr2+2πrh
VA=πr2×h=πr2+2πrh

Calculate the new volume and surface area

The new dimensions are krkr and hh.

Vn=π(kr)2×h=πk2r2×h=k2×πr2h=k2VAn=π(kr)2+2π(kr)h=πk2r2+2πkrh=k2(πr2)+k(2πrh)
VnAn=π(kr)2×h=πk2r2×h=k2×πr2h=k2V=π(kr)2+2π(kr)h=πk2r2+2πkrh=k2(πr2)+k(2πrh)

Exercise 13.6

If the length of the radius of a circle is a third of its original size, what will the area of the new circle be?

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The area of the original circle is: πr2πr2. Now we reduce the radius by a third. In other words we multiply rr by one third. The new area is:

Anew=π(13r)2=19πr2=19A
Anew=π(13r)2=19πr2=19A

Therefore, if the radius of a circle is a third of its original size, the area of the new circle will be 1919 the original area.

If the length of the base's radius and height of a cone is doubled, what will the surface area of the new cone be?

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We can find the new area by noting that the area will change by a factor of kk when we change the dimensions of the cone. In this case we are changing two dimensions of the cone and so the new area will be: Anew=k2AAnew=k2A

The value of kk comes from the word “doubled” in the question: the value of kk is 2.

So the new area of the cone will be Anew=4×AAnew=4×A if we double the height and the base's radius of the cone.

Therefore the surface area of the new cone will be 4 times the original surface area.

If the height of a prism is doubled, how much will its volume increase by?

We do not know if we have a rectangular prism or a triangular prism. However we do know that the volume of a prism is given by:

V=area of base×height of prism
V=area of base×height of prism

Now we are changing just one dimension of the prism: the height. Therefore the new volume is given by:

Vnew=area of base×2(height of prism)=2V
Vnew=area of base×2(height of prism)=2V

Therefore the volume of the prism doubles if the height is doubled.

Describe the change in the volume of a rectangular prism if the:

length and breadth increase by a constant factor of 33.

The volume of a rectangular prism is given by V=l×b×hV=l×b×h. If we increase the length and breadth by a constant factor of 3 the volume is:

Vnew=3(l)×3(b)×h=9V
Vnew=3(l)×3(b)×h=9V

Therefore the volume of the prism increases by a factor of 9 when the length and breadth are increased by a constant factor of 3.

length, breadth and height are multiplied by a constant factor of 33.

The volume of a rectangular prism is given by V=l×b×hV=l×b×h. If we increase the length, breadth and height by a constant factor of 3 the volume is:

Vnew=3(l)×3(b)×3(h)=27V
Vnew=3(l)×3(b)×3(h)=27V

Therefore the volume of the prism increases by a factor of 27 when the length, breadth and height are increased by a constant factor of 3.

If the length of each side of a triangular prism is quadrupled, what will the volume of the new triangular prism be?

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When multiplied by a factor of kk the volume of a shape will increase by k3k3. We are told that the dimensions are quadrupled. This means that each dimension is multiplied by 4. Therefore k=4k=4.

Now we can calculate k3k3.

k3=(4)3=64
k3=(4)3=64

Therefore, if each side of a triangular prism is quadrupled, the volume of the new triangular prism will be 64 times the original shape's volume.

Given a prism with a volume of 493493 cm3cm3 and a surface area of 6 0076 007 cm2cm2, find the new surface area and volume for a prism if all dimensions are increased by a constant factor of 44.

We are increasing all the dimensions by 4 and so the volume will increase by 4343. The surface area will increase by 4242.

V=493×43=31 552 cm3Surface area=6 007×42=96 112 cm2
VSurface area=493×43=31 552 cm3=6 007×42=96 112 cm2

Therefore the volume is 31 552 cm3 and the surface area is 96 112 cm2.