Respuesta :
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:
[tex]V\text{ = }\frac{lwh}{3}[/tex]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
[tex]l_2=\text{ }\frac{2}{3}l_1[/tex]The width is scaled by 5/3
[tex]w_2=\text{ }\frac{5}{3}w_1[/tex]The volume of the initial building is:
[tex]V_1=\text{ }\frac{l_1w_1h_1}{3}[/tex]The volume of the new building will be:
[tex]\begin{gathered} V_2=\text{ }\frac{l_2w_2h_2}{3} \\ V_2=\text{ }\frac{\frac{2l_1}{3}\times\frac{5w_1}{3}\times h_2}{3} \\ V_2=\frac{\frac{10l_1w_1}{9}\times h_2}{3} \\ V_2=\text{ }\frac{\frac{10l_1w_1h_2}{9}}{3} \\ V_2=\text{ }\frac{10l_1w_1h_2}{27} \end{gathered}[/tex]Divide V₂ by V₁
[tex]\begin{gathered} \frac{V_2}{V_1}=\text{ }\frac{10l_1w_1h_2}{27}\div\frac{l_1w_1h_1}{3} \\ \frac{V_2}{V_1}=\frac{10l_1w_1h_2}{27}\times\frac{3}{l_1w_1h_1} \\ \frac{V_2}{V_1}=\frac{30h_2}{27h_1} \\ \frac{V_2}{V_1}=\frac{10h_2}{9h_1} \end{gathered}[/tex]Since V₂ = V₁, V₂ / V₁ = 1
The equation above then becomes:
[tex]\begin{gathered} 1\text{ = }\frac{10h_2}{9h_1} \\ 9h_1=10h_2 \\ h_2=\text{ }\frac{9h_1}{10} \end{gathered}[/tex]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.
