document.write( "Question 1094775: A rectangular block has a square base. The length of each side of the base is (√3 - √2)m and volume of the block is (4√2 - 3√3) cubic metre. Find the height of the block in the form (a√2 + b√3) where a and b are constants. \n" ); document.write( "

Algebra.Com's Answer #709365 by KMST(5328) $\"\"$ $\"About$

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THE WORD PROBLEM PART:
\n" ); document.write( " $\"sqrt%283%29-sqrt%282%29\"$ = length of a side of the square base in meters.
\n" ); document.write( "

= surface area of the square base in square meters.
\n" ); document.write( "With that and the $\"4sqrt%282%29-3sqrt%283%29\"$ volume of the block in cubic meters,
\n" ); document.write( "knowing that volume = (area of the base)(height),
\n" ); document.write( "we can calculate the height in meters as
\n" ); document.write( " $\"%284sqrt%282%29-3sqrt%283%29%29%2F%285-2sqrt%283%29sqrt%282%29%29\"$ or $\"%284sqrt%282%29-3sqrt%283%29%29%2F%285-2sqrt%286%29%29\"$ .
\n" ); document.write( "
\n" ); document.write( "HOW TO SIMPLIFY THAT QUOTIENT:
\n" ); document.write( "To get rid of the irrationality in the denominator,
\n" ); document.write( "you need to do something you probably do almost often: rationalize.
\n" ); document.write( "In math, you do that by multiplying numerator and denominator times the irrational number that will result in a rational denominator.
\n" ); document.write( "In thid case, the irrational factor we need to multiply by is $\"%285%2B2sqrt%286%29%29\"$ ,
\n" ); document.write( "because

.
\n" ); document.write( "

\n" ); document.write( "With either of those equivalent expressions, the denominator is $\"1\"$ ,
\n" ); document.write( "and all you have to do is carefully calculate the numerator,
\n" ); document.write( "without making mistakes.
\n" ); document.write( "You could calculate it as
\n" ); document.write( " $\"%284sqrt%282%29-3sqrt%283%29%29%285%2B2sqrt%286%29%29\"$ $\"%22=%22\"$ $\"20sqrt%282%29%2B4sqrt%282%29%2A2sqrt%286%29-15sqrt%283%29-3sqrt%283%29%2A2sqrt%286%29\"$ $\"%22=%22\"$ $\"20sqrt%282%29%2B8sqrt%2812%29-15sqrt%283%29-6sqrt%2818%29\"$ $\"%22=%22\"$ $\"20sqrt%282%29%2B8sqrt%284%2A3%29-15sqrt%283%29-6sqrt%289%2A2%29\"$ $\"%22=%22\"$ $\"20sqrt%282%29%2B8%2A2sqrt%283%29-15sqrt%283%29-6%2A3sqrt%282%29\"$ $\"%22=%22\"$ $\"20sqrt%282%29%2B16sqrt%283%29-15sqrt%283%29-18sqrt%282%29\"$ $\"%22=%22\"$ $\"highlight%282sqrt%282%29%2Bsqrt%283%29%29\"$ .
\n" ); document.write( "Or you could calculate it as
\n" ); document.write( " $\"%284sqrt%282%29-3sqrt%283%29%29%285%2B2sqrt%283%29sqrt%282%29%29\"$ $\"%22=%22\"$
\n" ); document.write( "

$\"%22=%22\"$
\n" ); document.write( " $\"20sqrt%282%29%2B4%2A2%2A2sqrt%283%29-15sqrt%283%29-3%2A2%2A3sqrt%282%29\"$ $\"%22=%22\"$
\n" ); document.write( " $\"20sqrt%282%29%2B16sqrt%283%29-15sqrt%283%29-18sqrt%282%29\"$ $\"%22=%22\"$
\n" ); document.write( " $\"highlight%282sqrt%282%29%2Bsqrt%283%29%29\"$ .
\n" ); document.write( "
\n" ); document.write( "VERIFYING YOUR ANSWER:
\n" ); document.write( "You could also calculate approximate values for
\n" ); document.write( " $\"4sqrt%282%29-3sqrt%283%29\"$ $\"%22=%22\"$ $\"approximately\"$ $\"0.460702\"$ and
\n" ); document.write( " $\"5%2B2sqrt%286%29\"$ $\"%22=%22\"$ $\"approximately\"$ $\"9.898979\"$ ,
\n" ); document.write( "and then multiply them together to find the height in meters as
\n" ); document.write( " $\"approximately\"$ $\"0.460702%2A9.898979\"$ $\"%22=%22\"$ $\"approximately\"$ $\"4.560478\"$ .\r
\n" ); document.write( "\n" ); document.write( "That is not the answer \"in the form of $\"a%2Asqrt%282%29%2Bb%2Asqrt%283%29\"$ ,\"
\n" ); document.write( "but it is easy to calculate with a computer or calculator,
\n" ); document.write( "and a way to verify if you made a mistake.
\n" ); document.write( "
\n" ); document.write( "Another way to verify the answer
\n" ); document.write( "is to multiply area in square meters = times the answer you found:
\n" ); document.write( "

$\"%22=%22\"$
\n" ); document.write( "

\n" ); document.write( " $\"%22=%22\"$ $\"4sqrt%282%29-3sqrt%283%29\"$ .
\n" ); document.write( "
\n" ); document.write( "NOTE: I have not yet decided if assigning this problem is a way to give students practice with square roots, or a way to torture them. I suppose that assigning one or two such exercises would drive home
\n" ); document.write( "1. the need to understand the situation in word problems to figure out the needed calculations ,
\n" ); document.write( "2. the idea of \"rationalizing\" denominators by multiplying \"conjugate irrational numbers\" as a pair of irrational numbers like $\"7%2Bsqrt%285%29\"$ and $\"7-sqrt%285%29\"$ , and
\n" ); document.write( "3. the idea of simplifying roots as in $\"2sqrt%2818%29=2sqrt%289%2A2%29=2sqrt%289%29%2Asqrt%282%29=2%2A3%2Asqrt%282%29=6sqrt%282%29\"$ . \n" ); document.write( "

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