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The Effect Of Multiplying A Dimension By A Factor Of K

13.4 The effect of multiplying a dimension by a factor of k

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 , and . Below we multiply one, two and three of its dimensions by a constant factor of and calculate the new volume and surface area.

Dimensions

Volume

Surface

Original dimensions4211cb1aecbfd4e7b88092e7219d1afd.png

Multiply one dimension by ed991ab7b8a7011fe8a0f3069ef135cb.png

Multiply two dimensions by 06d4e51ae19a4f423f40501d99418d0c.png

Multiply all three dimensions by 9eb11d5dfeaa9134ccf0862dc6afd2f5.png

Multiply all three dimensions by k4a729f6395763834825e446e637c5668.png

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

Consider a rectangular prism with a height of and base lengths of .1b6985afdae12a4ed2b5edea799b21d2.png

  1. Calculate the surface area and volume.

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

  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

Calculate the new volume and surface area

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

Express the new dimensions as a factor of the original dimensions

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

Prove that if the height of a rectangular prism with dimensions , and is multiplied by a constant value of , the volume will also increase by a factor .a0bc3c01d9e5c7252907de78b4f4a1e1.png

Calculate the original volume

We are given the original dimensions , and and so the original volume is .

Calculate the new volume

The new dimensions are , , and and so the new volume is:

Write the final answer

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

Worked example 20: Multiplying the dimensions of a cylinder by

Consider a cylinder with a radius of and a height of . Calculate the new volume and surface area (expressed in terms of and ) if the radius is multiplied by a constant factor of .845c8fa92ed742229a55db39749b9153.png

Calculate the original volume and surface area

Calculate the new volume and surface area

The new dimensions are and .

Exercise 13.6See solutions

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?d47b8fd833e895b3a32e01f7fdb0f876.png

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

Therefore, if the radius of a circle is a third of its original size, the area of the new circle will be 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?8953d2b7b32832ffeb15f79fc7f924f4.png

We can find the new area by noting that the area will change by a factor of 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:

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

So the new area of the cone will be 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:

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

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 .

The volume of a rectangular prism is given by . If we increase the length and breadth by a constant factor of 3 the volume is:

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 .

The volume of a rectangular prism is given by . If we increase the length, breadth and height by a constant factor of 3 the volume is:

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?10265d1a3b98d1569d553937d8d9afcb.png

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

Now we can calculate .

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 and a surface area of , find the new surface area and volume for a prism if all dimensions are increased by a constant factor of .

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

Therefore the volume is and the surface area is .

See solutions