Question

TRANSAMERICA BUILDINGThe TransAmerica building in San Francisco is built of concrete and is shaped like a square-based pyramid. The building is periodically power-washed using one gallon of cleaning solution for every 250 square meters of surface. As the new building manager, you need to order the cleaning supplies for this large task. The problem is that you do not know the height of each triangular face of the building; you only know the vertical height of the building from the base to the top vertex.Your Task: Determine the amount of cleaning solution needed to wash the TransAmerica building if an edge of the square base is meters and the height of the building is meters. Include a sketch in your solution.

Answer 1

we know that

The surface area of a square pyramid is given by the formula

`SA=B+LA`

where

B is the area of the square base

LA is the lateral area (area of the four triangular faces)

step 1

Find out the area of the square base B

`\begin{gathered} B=96^2 \\ B=9,216\text{ m}^2 \end{gathered}`

step 2

Find out the area of one triangular face

the area of one triangular face is given by the formula

`A=(1)/(2)*b*h`

where

b=96 m

h=?

Applying the Pythagorean Theorem, find out the value of h

`h^2=H^2+((b)/(2))^2`

where

H=220 m (height of the building)

b/2=96/2=48 m

substitute

`\begin{gathered} h^2=220^2+(48)^2 \\ h^2=50,704 \\ h=225.18\text{ m} \end{gathered}`

substitute

`\begin{gathered} A=(1)/(2)*96*225.18 \\ A=10,808.64\text{ m}^2 \end{gathered}`

Multiply by 4 to obtain the Lateral Area

`\begin{gathered} LA=4*10,808.64 \\ LA=43,234.56\text{ m}^2 \end{gathered}`

The surface area of the building is equal to

`\begin{gathered} SA=9,216+43,234.56 \\ SA=52,450.56\text{ m}^2 \end{gathered}`

step 3

we know that

one gallon of cleaning solution -----> for every 250 square meters of surface

Applying proportion

1/250=x/52,450.56

solve for x

x=52,450.56/250

x=210 gallons

The answer is 210 gallons

see the figure below to better understand the calculation to obtain the value of

h (the height of each triangular face)