Question

A building in the shape of a rectangular pyramid is used to store construction stones. The building has a volume of 21,000 ft3. A new building is being designed. It will also be in the shape of a rectangular pyramid, but compared to the existing building, the length of the base will be scaled by a factor of 2/3 and the width will be scaled by a factor of 5/3. The height of a new building will be scaled such that the volume of the new building will be the same as the existing building. What scale factor will be used for the height of the new building?

Answer 1

Answer:

The scale factor used for the snew building = 9/10

Explanations:

Let the volume of the old building be V₁

Let the volume of the new building be V₂

The volume of the old bulding, V₁ = 21000 ft cube

The volume of the new building = The volume of the old building

V₁ = V₂ = 21000

The volume of a rectangular pyramid is given as:

`V\text{ = }(lwh)/(3)`

Where l is the length of the base

w is the width

h is the height of the pyramid

The length of the base is scaled by 2/3

`l_2=\text{ }(2)/(3)l_1`

The width is scaled by 5/3

`w_2=\text{ }(5)/(3)w_1`

The volume of the initial building is:

`V_1=\text{ }(l_1w_1h_1)/(3)`

The volume of the new building will be:

`\begin{gathered} V_2=\text{ }(l_2w_2h_2)/(3) \\ V_2=\text{ }((2l_1)/(3)*(5w_1)/(3)* h_2)/(3) \\ V_2=((10l_1w_1)/(9)* h_2)/(3) \\ V_2=\text{ }((10l_1w_1h_2)/(9))/(3) \\ V_2=\text{ }(10l_1w_1h_2)/(27) \end{gathered}`

Divide V₂ by V₁

`\begin{gathered} (V_2)/(V_1)=\text{ }(10l_1w_1h_2)/(27)/(l_1w_1h_1)/(3) \\ (V_2)/(V_1)=(10l_1w_1h_2)/(27)*(3)/(l_1w_1h_1) \\ (V_2)/(V_1)=(30h_2)/(27h_1) \\ (V_2)/(V_1)=(10h_2)/(9h_1) \end{gathered}`

Since V₂ = V₁, V₂ / V₁ = 1

The equation above then becomes:

`\begin{gathered} 1\text{ = }(10h_2)/(9h_1) \\ 9h_1=10h_2 \\ h_2=\text{ }(9h_1)/(10) \end{gathered}`

The new building will be scaled by a factor of 9/10 to make the volume of the new building the same as the existing building.