Question

A roof support with equally spaced vertical pieces is shown in below figure. Find the total length of the vertical pieces if the shortest one is 10.0 in .long

Answer 1

The diagram points at a horizontal distance of 10.0 inches.  But it says vertical support is 10.0 inches.  I take it as the length (height) of shortest vertical support. See my diagram. There are totally 14 vertical supports.

Angle made by slanting support with horizontal = 90⁰ - 84.8⁰ = 5.2⁰

tan 5.2⁰ = 10.0" / a ,   where a is the distance from O to the shortest support.

a = 10.0" / tan 5.2⁰ = 109.88" 

tan 5.2⁰ = 10.0" / a = h / 224.0"

h = 224.0" * tan 5.2⁰ = 20.385 "  = height of longest vertical support.

Now the heights of the vertical supports are in arithmetic progression, as their lengths increase by fixed quantity = tan 5.2⁰ * horizontal spacing between them.

So sum of the heights of vertical supports = sum of arithmetic series
          = number of terms * ( first term + last term) / 2 
         = Number of vertical supports *
                    (height of smallest support + height of tallest support) / 2

        = 14 * ( 10" + 20.385")/2 = 212.695"