clara works for a marketing company and is designing packing for a new product. the
product can come in various sizes. clara has determined that the size of the packaging can be
found using the function, p(b) = (b)(2b + 1)(b + 5), where b is the shortest measurement of the
product. after some research, clara determined that packaging with 20, 500 cm^3 will be the
most appealing to customers.
what are the dimensions of the package?

  20 cm by 25 cm by 41 cm

Step-by-step explanation:

You are being asked to solve the cubic ...

  b(2b +1)(b +5) = 20500

I find the easiest way to do that is to use a graphing calculator. By subtracting 20500 to get ...

  b(2b +1)(b +5) -20500 = 0

we can look for a zero of the expression on the left. Many graphing calculators are good at showing those to high precision. Here, we find the only real zero of this equation is at b = 20.

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Filling in b=20 for the three product dimensions, we find the package size is ...

  20 cm by 41 cm by 25 cm

_____

The rational root theorem tells you that any rational root will be a factor of 20500. This number has 24 divisors. Fortunately there are only 5 that are about 20 or less, so the number of them that need to be tested is minimal. (The cubic has a value greater than 20500 for b > 20.)

You can guess that the solution will be near (slightly less than) ∛(20500/2), since the factors are all near "b" after dividing the second one by 2: (b)(b+1/2)(b+5). That root is about 21.7, so b=20 is a good first guess.

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