Question

The tiles in a store are being compared. The diagonal of one tile is 92–√ centimeters, and the other tile is 82–√ centimeters. What is the difference in the length of the diagonals?

Answer 1

To find the difference in the lengths of the diagonals of the two tiles, we need to subtract the length of the shorter diagonal from the length of the longer diagonal.

The lengths of the diagonals are given as:
- One tile: \( \sqrt{92} \) centimeters
- The other tile: \( \sqrt{82} \) centimeters

Now, we calculate the difference:

\[
\text{Difference} = \sqrt{92} - \sqrt{82}
\]

Next, we can calculate each square root and then find the difference. However, without calculating numerical approximations, we can simplify it mathematically:

First, let's calculate \( \sqrt{92} \) and \( \sqrt{82} \):

\[
\sqrt{92} = \sqrt{4 \times 23} = 2\sqrt{23}
\]

\[
\sqrt{82} = \sqrt{82}
\]

Now we can compute:

\[
\sqrt{92} - \sqrt{82} = 2\sqrt{23} - \sqrt{82}
\]

To find the difference more explicitly, let's turn back to evaluating the approximate square roots if needed, or isolate our difference as is.

Let’s calculate approximate values if needed:
- \( \sqrt{92} \approx 9.591664 \)
- \( \sqrt{82} \approx 9.055 \)

Calculating the difference:
\[
\sqrt{92} - \sqrt{82} \approx 9.591664 - 9.055 = 0.536664
\]

Thus, the approximate difference in lengths of the diagonals is:

\[
\text{Difference} \approx 0.5367 \text{ centimeters}
\]

If you want the more exact symbolic form, your final answer is:

\[
\sqrt{92} - \sqrt{82}
\]

For numerical approximation, the difference can be presented as:

\[
\approx 0.537 \text{ centimeters}
\]

So the answer to the difference in the length of the diagonals of the tiles is:

\(\sqrt{92} - \sqrt{82} \approx 0.537 \text{ cm}\)