Part (a)
It's not stated, but I'm assuming that RV = VQ
If so, then we use the tangent ratio to connect the opposite side PV = 7 and adjacent side RV = 16
Focus on triangle RPV
tan(angle) = opposite/adjacent
tan(R) = PV/RV
tan(x) = 7/16
x = arctan(7/16) ...... same as inverse tangent
x = 23.6293777 ...... which is approximate
Rounding to the nearest degree, the angle x is roughly 24 degrees.
Since 18 < 24 < 27, this means the roof pitch follows the safety guidelines.
Answer: Yes the roof is safe
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Part (b)
Again we only focus on triangle RPV
We have two pathways we can take. The first option is to use the pythagorean theorem
a^2+b^2 = c^2
c = sqrt(a^2 + b^2)
PR = sqrt( (RV)^2 + (PV)^2 )
PR = sqrt( (16)^2 + (7)^2 )
PR = 17.464249 .... approximate
PR = 17.5
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The second path you could take is to use the angle x we found in part (a). Use either the cosine or sine ratios like so
sin(angle) = opposite/hypotenuse
sin(x) = PV/PR
PR = PV/sin(x)
PR = 7/sin(23.6293777)
PR = 17.464249
PR = 17.5
The pathway for cosine will look almost identical to this, but you'll be computing PR = RV/cos(x) instead.
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Answer: 17.5 feet
