The length of a rectangle exceeds three times its width by 9 if the perimeter is 66 find the width of the rectangle?
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<h3>Answer: 12</h3>
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Explanation:
The two integers multiply to 36, so,
ab = 36
which solves to
a = 36/b
Then we want to add the numbers such that we get the smallest possible result.
a+b = (36/b)+b
So we want (36/b)+b to be as small as possible.
Let's say we replace b with x and we consider this function
f(x) = (36/x) + x
The goal is to find when f(x) is smallest, ie, we want to minimize the function.
If we were to graph out the function, we get the curve shown below.
To make things easier, we'll only focus on positive values of x.
The lowest part of the curve is what we're after. Using the "minimum" function/feature on the graphing calculator, we would then find the lowest point occurs at (6,12). This point is considered a local minimum because it's the lowest point in that given neighborhood of x values.
So the input x = 6 leads to the smallest output f(x) = 12.
This in turn means b = 6 is going to pair with a = 36/b = 36/6 = 6.
In short, a = 6 and b = 6.
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As a check,
a*b = 6*6 = 36
a+b = 6+6 = 12
We can make a table of various values to help confirm that 12 is the smallest sum.
Side note: If you're not allowed to use a graphing calculator, then you'll need to use calculus.