Question

(a) The angle pitch of a roof is safest when measuring between 18° – 27°. According to these guidelines, is the roof pictured in the image safe?

(b) What is length of the roof line (segment PR)? Round answer to the nearest tenth of a foot and show all your work.

Answer 1

Find Q

• tanQ=Perpendicular/Base
• tanQ=4/15
• tanQ=0.26
• Q=tan^-¹(0.26)
• Q=14.6°

Not safe

Apply Pythagorean theorem

PR²=4²+15²=16+225=241

• PR=√241

Answer 2

a) not safe

b) 15.5 ft (nearest tenth)

Explanation:

Part (a)

Tan trig ratio

`\sf \tan(\theta)=(O)/(A)`

where:

• 
  `\theta` is the angle
• O is the side opposite the angle
• A is the side adjacent the angle

Given:

• 
  `\theta` = x
• O = PV = 4 ft
• A = RV = VQ = 15 ft

Substituting the given values into the formula and solving for x:

`\implies \sf \tan(x)=(4)/(15)`

`\implies \sf x=\tan^(-1)\left((4)/(15)\right)`

`\implies \sf x=14.9^(\circ)\:(nearest\:tenth)`

As 14.9° is not between 18° and 27°, the roof is not safe.

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Part (b)

Pythagoras’ Theorem

`\sf a^2+b^2=c^2`

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

Given:

• a = RV = VQ = 15 ft
• b = PV = 4 ft
• c = PR

Substituting the given values into the formula and solving for PR:

`\implies \sf 15^2+4^2=PR^2`

`\implies \sf 225+16=PR^2`

`\implies \sf PR^2=241`

`\implies \sf PR=√(241)`

`\implies \sf PR=15.5\:ft\:(nearest\:tenth)`