Prove that 2? = 133° 34’, when the volume of concrete is minimum. Make any assumption...
Prove that 2? = 133° 34’, when the volume of concrete is minimum. Make any assumption where necessary.
Homework Answers
Consider an arch of
- Thickness 't'
- Angle - 2 ?
- Radius - 'r'
- Total length of the arch - 'L'
- Area of the arch - A = r * Constant = kr
V= A *r * 2?
= k r *r *2?
=k *r2 *2?
Now ,
Sin ? = Perpendicular / Base
Perpendicular = (Total Lemgth of the arch /2) = L/2
Base = Radius = r
Sin ? = (L/2) / r
Sin? = L/2r
2r = L/Sin ?
r = L/2Sin?
Now ,
Volume = k * (L /2Sin ?)2 *2? =2? kL 2 / 4 Sin 2 ?
Differentiating both sides
dV/d? = d/d? (2?k L2 /4 Sin 2 ? )
For minimum volume of concrete dV/d? = 0
0 =( kL2 / 2 ) d/d? (?/Sin2 ? )
0 = (Sin 2 ? - 2?Sin?Cos?) / Sin 4 ? )
Sin 2 ?-2?Sin?Cos? = 0
Sin 2 ?=2?Sin?Cos?
Tan ? = 2?
133°34' = 2?
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