The yearbook staff receive 75 submissions for yearbook articles. Paul will accept 2/5 of all submissions. Currently, Paul plans
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So hmm check the picture below, that's the silo, notice the top of the cylinder is inside with grain... and the bottom, well, that's the (base no included)
so
![\bf \textit{now, let's check the critical points}\\\\ \cfrac{dc}{dr}=-\cfrac{28000}{r^2}+\cfrac{16\pi r}{3}\implies \cfrac{dc}{dr}=\cfrac{16\pi r^3-84000}{3r^2} \\\\\\ 0=\cfrac{16\pi r^3-84000}{3r^2}\implies 84000=16\pi r^3\implies 5250=\pi r^3 \\\\\\ \boxed{\sqrt[3]{\cfrac{5250}{\pi }}=r}](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7Bnow%2C%20let%27s%20check%20the%20critical%20points%7D%5C%5C%5C%5C%0A%5Ccfrac%7Bdc%7D%7Bdr%7D%3D-%5Ccfrac%7B28000%7D%7Br%5E2%7D%2B%5Ccfrac%7B16%5Cpi%20r%7D%7B3%7D%5Cimplies%20%5Ccfrac%7Bdc%7D%7Bdr%7D%3D%5Ccfrac%7B16%5Cpi%20r%5E3-84000%7D%7B3r%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0A0%3D%5Ccfrac%7B16%5Cpi%20r%5E3-84000%7D%7B3r%5E2%7D%5Cimplies%2084000%3D16%5Cpi%20r%5E3%5Cimplies%205250%3D%5Cpi%20r%5E3%0A%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7B%5Csqrt%5B3%5D%7B%5Ccfrac%7B5250%7D%7B%5Cpi%20%7D%7D%3Dr%7D)
now, I'd like to point out, we also get critical points from the denominator being zero, so 3r²=0, gives us r=0, which is a critical point, however, with a radius of 0, well, there's no volume, so, the is not feasible for the phenomena, so is not applicable, though is a valid critical point
critical points from the denominator being zero, are usually asymptotic or "cusps", the function reaches an extrema, however the tangent is not horizontal and thus is not differentiable at that point, even though is an extrema.
so
now, I'd like to point out, we also get critical points from the denominator being zero, so 3r²=0, gives us r=0, which is a critical point, however, with a radius of 0, well, there's no volume, so, the is not feasible for the phenomena, so is not applicable, though is a valid critical point
critical points from the denominator being zero, are usually asymptotic or "cusps", the function reaches an extrema, however the tangent is not horizontal and thus is not differentiable at that point, even though is an extrema.
