Question

1. Prove that a conical tent of given capacity will require the least amount of canvas when the height is root 2times the radius of the base.

Answer 1

Capacity is equal, that means volume is constant.
To Prove that- The Surface Area will be minimum if the height is √2 times the radius
Proof- v=volume
                 v1=v2
       v1=1/3$\pi$r²h 
     so, SA1= $\pi$rl
      L1=√h²+r²
SA1=$\pi$r√h²+r²


Now v2 in which h is √2 times tthe radius
  v2=1/3$\pi$r²√2r    (h=√2r)
     SA2=$\pi$rl
L2= √r²+(√2r)²
L2=√3r²=r√3
SA2= $\pi$r*r√3
SA2=$\pi$r²√3
Now we wil compare the volume
v1=v2
1/3$\pi$r²h =1/3$\pi$r²√2r  
Hence we will get the answer!