Question

John’s company sells two styles of propane tanks. The ratio of the lengths, in feet, of the smaller tank to the larger one is 5/√34+2 . Express the ratio of the smaller tank to the larger one with a rationalized denominator in simplified form.

Answer 1

To rationalize the denominator of the given ratio \( \frac{5}{\sqrt{34} + 2} \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{34} - 2 \).

Here are the steps:

1. Start with the expression:
\[
\frac{5}{\sqrt{34} + 2}
\]

2. Multiply the numerator and denominator by \( \sqrt{34} - 2 \):
\[
\frac{5(\sqrt{34} - 2)}{(\sqrt{34} + 2)(\sqrt{34} - 2)}
\]

3. Calculate the denominator using the difference of squares:
\[
(\sqrt{34})^2 - (2)^2 = 34 - 4 = 30
\]

4. Now, simplify the numerator:
\[
5(\sqrt{34} - 2) = 5\sqrt{34} - 10
\]

5. Combine everything:
\[
\frac{5\sqrt{34} - 10}{30}
\]

6. Simplify the fraction:
\[
\frac{5\sqrt{34}}{30} - \frac{10}{30} = \frac{\sqrt{34}}{6} - \frac{1}{3}
\]

Thus, the final expression for the ratio of the smaller tank to the larger one, with a rationalized denominator in simplified form, is:
\[
\frac{5\sqrt{34} - 10}{30}
\]
Or in a compact form:
\[
\frac{\sqrt{34}}{6} - \frac{1}{3}
\]
However, when presenting the complete rationalized form of the initial ratio \( \frac{5}{\sqrt{34} + 2} \), it's more straightforward to leave it in the form:
\[
\frac{5(\sqrt{34} - 2)}{30}
\]
which can also be represented as:
\[
\frac{\sqrt{34} - 2}{6}
\]

Hence, the expression you would like is \( \frac{5\sqrt{34} - 10}{30} \) which maintains the ratio's original characteristics effectively while expressing it with rational numbers.