The situation described can be solved using the Pythagorean theorem, as the banner, the distance between the hook and the pole, and the height of the pole form a right triangle.

In this right triangle:

• The banner is the hypotenuse $c = 17 \, \text{ft}$,
• The horizontal distance between the pole and the hook is one leg $b = 15 \, \text{ft}$,
• The height of the pole is the other leg $a$, which we want to calculate.

According to the Pythagorean theorem:

$c^2 = a^2 + b^2$

Substitute the values for $c$ and $b$:

$17^2 = a^2 + 15^2$

$289 = a^2 + 225$

Now, solve for $a^2$:

$a^2 = 289 - 225 = 64$

Finally, take the square root of both sides to find $a$:

$a = \sqrt{64} = 8 \, \text{ft}$

Therefore, the height of the pole required to hang the banner is 8 feet.

However, you mentioned that the pole is 22.7 feet tall, which means Tony is using a taller pole than needed for this banner length.

Do you want more details or have any questions about this?

Here are 5 related questions:

1. What would happen if the banner length changes to 20 feet but the distance between hook and pole remains the same?
2. How much extra height does Tony have with a 22.7-foot pole compared to the 8-foot requirement?
3. Can the Pythagorean theorem be used if the banner isn’t hanging perfectly straight? Why or why not?
4. What is the maximum distance the hook can be from the pole with the same 17-foot banner?
5. If Tony needed the banner at 12 feet high, how long should the banner be?

Tip: Always ensure that you are working with the correct right triangle dimensions to apply the Pythagorean theorem correctly.