David Gwyn J. answered • 10/30/20

Highly Experienced Tutor (Oxbridge graduate and former tech CEO)

Area of triangle = 1/2 x height x base

in this case, 728 = 1/2 hb => hb = 1456

but we are also told that h = 8b - 8 (height is 8 less than 8 times the base)

hence we can substitute to eliminate h, and get** (8b - 8)(b) = 1456**

At this point we can EITHER use trial-and-error (guess!)...

try b = 10 for 72 x 10 = 720 too small

try b = 15 for 112 x 15 = too big

try b = 14 for 104 x 14 = 1456 just right :-) and it has given us the height too.

OR factorize...

(8b - 8)(b) = 1456

=> 8b^{2} - 8b - 1456 = 0

=> 8 (b^{2} - b - 182) = 0

=> 8 (b + 13)(b - 14)

Hence either b + 13 = 0 => b = -13 (negative base not possible for triangle)

or b - 14 = 0 => b = 14

If b = 14 substitute to get h in h = 8b - 8 = 8(14) - 8 = 104

Hence, **base = 14 feet and height = 104 feet **(or vice versa)

*Double-check: 1/2 x 14 x 104 = 7 x 104 = 700 + 28 = 728 which is what we need.*