Expert: Scott A Wilson - 4/21/2010

Question
Hello,
I have 2 questions I just can't figure out.  I hope you don't mind that I'm asking more than one and that they're word problems.

Q1. If 18*{square root of}18 = r*{square root of} t, where r and t are positive integers and r is greater than t, what is the value of rt?

Q2. Let the function h be defined by h(x)= 14+ ((x^2)/4).  If h(2m)=9m, what is one possible value of m?  

Thanks.

Answer
Note today, since I only have 3 other people with questions.

Q1. It is known that 18√18 = 18*√(9*2) = 18*3√2 = 54√2.
If 54√2 = r√t, and r is greater than t, then t must have a 2 in it.
That is the 1st choice: r=54, t=2.

If we factored out a 2 from 54, and put a 4 in the squareroot, we would get 27√8.
That is the 2nd choice: r=27, t=8.

If we factored a 3 out of 54 and put a 9 under the squareroot, we would get 18√18,
and here r would be equal to t, which can't happen.

I have found 2 answers.
1) r=54, t=2
2) r=27, t=8

The value of rt is two different choices.  They are (1) 108 (2) 216
I'm not sure it is of any relevance, but for my 2nd answer, 216, it is known that 216 = 6³.

Q2.  This means that 9m = 14 + (4m²/4) = 14 + m².
Taking the left and the right, we have 9m = 14 + m².
Subtracting of 9m from both sides gives 0 = 14 - 9m + m².

This is the same as m² - 9m + 14 = 0.
That factors into (m-7)(m-2).
Form here, m could be 7 or m could be 2.

Lets try both of these values.
The 1st, m=7, gives 2m=14, so h(14) = 14 + 196/4 = 14 + 49 = 63.  This is 9m, so that works.
The 2nd, m=2, gives 2m=4, so h(4) = 14 + 16/4 = 14 + 4 = 18.  This is 9m, so that works.

Either answer would be true.  That is, m=2 or m=7.