document.write( "Question 1179691: The area of the region under the curve of the function f(x) = 5x + 7 on the interval [1, b] is 88 square units, where b > 1.\r
\n" ); document.write( "\n" ); document.write( "the value of b is ? \n" ); document.write( "
Algebra.Com's Answer #809293 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "This problem can be solved easily without calculus. The given function is a straight line, so the area under the curve on the interval [1,b] is a trapezoid. The area of a trapezoid is

\n" ); document.write( "\"%28%28a%2Bb%29%2F2%29%28h%29\"

\n" ); document.write( "where h is the height and a and b are the lengths of the two bases.

\n" ); document.write( "For this problem, the height is (b-1); the two bases are f(1)=12 and f(b)=5b+7.

\n" ); document.write( "\"%28%28a%2Bb%29%2F2%29%28h%29=%28%285b%2B19%29%2F2%29%28b-1%29=88\"
\n" ); document.write( "\"%285b%2B19%29%28b-1%29=176\"
\n" ); document.write( "\"5b%5E2%2B14b-19=176\"
\n" ); document.write( "\"5b%5E2%2B14b-195=0\"

\n" ); document.write( "Note this is the same equation the other tutor ends up with using calculus.

\n" ); document.write( "She then goes on, as she always does, to magically separate the middle term 14b into two parts, producing a quadratic with four terms that can be factored by grouping. That's fine for showing the solution -- but it does nothing to teach the student how the factoring is done.

\n" ); document.write( "So let me take a bit of time here to talk about ways to factor a quadratic like this. There are many methods that are taught; most of them that I have seen consist of well defined steps that mysteriously lead to the right factorization.

\n" ); document.write( "I prefer one that calls on the student to use his powers of reasoning to obtain the answer.

\n" ); document.write( "In this case we want a factorization of the form

\n" ); document.write( "\"5b%5E2%2B14b-195=%28pb%2Bq%29%28rb%2Bs%29\"

\n" ); document.write( "Since the b^2 term comes from (pb)(rb), and the coefficient of b^2 in the quadratic is 5, clearly p and r have to be 5 and 1, in some order. We can assume p=5 and r=1, giving us

\n" ); document.write( "\"5b%5E2%2B14b-195=%285b%2Bq%29%281b%2Bs%29\"

\n" ); document.write( "Now look at the constant term in the quadratic: -195. That constant comes from q times s: \"qs=-195\"

\n" ); document.write( "We can see that q and s are of opposite sign; and we can see that one of them contains 5 as a factor.

\n" ); document.write( "Here is the where the logical reasoning makes what looks like a big problem much easier. If q contained a factor of 5, then the two terms in the binomial (pb+q) would have a common factor of 5. That would make the quadratic have a common factor of 5; but it does not. So q cannot contain a factor of 5; so s must contain a factor of 5.

\n" ); document.write( "-195 is 5(-39); at this point we can play with the signs of the two binomial factors and the possible values of q and s to find the combination that gives us the correct middle term.

\n" ); document.write( "With a little playing around like that, we find

\n" ); document.write( "\"5b%5E2%2B14b-195=%285b%2B39%29%281b-5%29\"

\n" ); document.write( "That gives us possible values of -39/5 or 5 for b; the statement of the problem asking us to find the area under the curve on the interval [1,b] means we choose the positive value for b.

\n" ); document.write( "So b=5, and now we have all we need to find the area as the area of a trapezoid; its height is b-1=4; the bases are f(1)=12 and f(5)=32; the area is

\n" ); document.write( "\"%28%28a%2Bb%29%2F2%29%28h%29+=+%28%2812%2B32%29%2F2%29%284%29+=+22%2A4+=+88\"

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