document.write( "Question 266652: Suppose the volume must be 50in^3, What are the values for r and h that will minimize the amount of sheet metal required to obtain this volume. I know volume is v= (pi)r^2h. I also know the surface area is sa= 2(pi)rh+2(pi)r^2.
\n" ); document.write( "I have been working on this all day. Any help would be greatly appreciated. Thanks \n" ); document.write( "

Algebra.Com's Answer #195898 by drk(1908) $\"\"$ $\"About$

You can put this solution on YOUR website!
lets start with
\n" ); document.write( "(i) $\"v+=+%28pi%29r%5E2%2Ah\"$
\n" ); document.write( "we know v = 50, so we get
\n" ); document.write( "(ii) $\"50=+%28pi%29r%5E2%2Ah\"$
\n" ); document.write( "If we solve for h, we get
\n" ); document.write( "(iii) $\"h+=+50%2F%28pi%2Ar%5E2%29\"$
\n" ); document.write( "now we look at surface area as
\n" ); document.write( "(iv) $\"sa+=+2%28pi%29%2Ar%2Ah%2B2%28pi%29%2Ar%5E2\"$
\n" ); document.write( "substitute (iii) into (iv) to get
\n" ); document.write( "(v) $\"sa+=+2%2Api%2Ar%2A%2850%2F%28pi%2Ar%5E2%29%29+%2B+2%2Api%2Ar%5E2\"$
\n" ); document.write( "now everything is in terms of r.
\n" ); document.write( "getting a common denominator as pir^2, we get
\n" ); document.write( "(vi) $\"sa+=+%28100%2Api%2Ar+%2B+2%2Api%5E2%2Ar%5E4%29%2F%28pi%2Ar%5E2%29\"$
\n" ); document.write( "now, factoring, we get
\n" ); document.write( "(vii) $\"sa+=+%282%2Api%2Ar%2850%2Bpi%2Ar%5E3%29%29%2F%28pi%2Ar%5E2%29\"$
\n" ); document.write( "reducing we get
\n" ); document.write( "(viii) $\"sa+=+2%2850%2Bpi%2Ar%5E3%29%2Fr\"$
\n" ); document.write( "It turns out that r is minimum at 2.
\n" ); document.write( "This means that h = 50/4pi or h ~ 3.97887
\n" ); document.write( "This gives us a minimum sa at
\n" ); document.write( "sa = 75.1326 \n" ); document.write( "

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