Answer:
Step-by-step explanation:
Step-by-step explanation: As shown in the attached figure, the prism-shaped roof has equilateral triangular bases, one of which is ΔABC. We need to create an equation that models the height of one of the roof's triangular bases in terms of its sides. Let ii be AD.
See the figure attached herewith, ΔABC forms an equilateral triangle, in which AD is the height. So, D will be the mid-point of BC and ∠ADB = ∠ADC = 90°.
Now, in ΔADB, we have
AD^2=AB^2-BD^2
AD^2=AB^2-(1/2AB^2)^2
AD=√3/4AB^2
we can find the height of any one of the roof's triangular bases.
2.1. Check picture 1. Let the one side of the triangle be a, drop one perpendicular, CD. Then triangle ADB is a right triangle, with hypothenuse a and one side equal to 1/2a. By the Pythagorean theorem, as shown in the picture, the height is √3/2a
2. if a=25 ft, then the height is √3/2a=√3/2*25=1.732/2*25=21.7(ft)
3. consider picture 2. Let the length of the roof be l feet.
one side of the prism (the roof) is a rectangle with dimensions a and l, so the area of one side is a*l
the lateral Area of the roof is 3a*l
the area of the equilateral surfaces is 2*(1/2*a*√3/2a)=√3/2a^2
so the total area of the roof is
4. The total area was the 2 triangular surfaces + the 3 equal lateral rectangular surfaces. Now instead of 3 lateral triangular surfaces, we have 2.
So the total area found previously will be decreased by al
5. so the area now is √3/2a^2 + 2al
6. now a=25 and l=2a=50
Area= √3/2a^2+2al=√3/2*25^2+2*25*50=25^2(√3/2+4)=625*4.866
=3041.3 (ft squared)
