91Ó°ÊÓ

In this exercise, we're considering a monkey, a chain, and a vertical pole forming a right triangle. A right triangle is a special type of triangle with one angle measuring exactly 90 degrees. This allows us to apply the Pythagorean theorem, a key mathematical principle for calculating distances.

A right triangle consists of three sides:

• The hypotenuse, which is the longest side and opposite the right angle.
• Two shorter sides, or legs, which meet at the right angle.

In our scenario, the chain is the hypotenuse, the pole is vertical forming one leg, and the distance on the ground from the stake to the pole forms the other leg. Understanding this setup is crucial for applying the theorem to find out how high the monkey can climb.

The concept of chain length is essential in solving this problem. The chain represents the hypotenuse of the triangle and is given as 3.40 meters.

This length is pivotal because, in the triangle, it dictates how far the monkey can reach. The chain is fixed at one end to the stake, and the other end is free for the monkey. Thus, determining the permissible height, known as the other leg, becomes a matter of figuring out which combination of height and distance meets the entire chain's length. This entire calculation revolves around understanding how to balance these sides within the constraints that are given. This setup refers to the classic geometric rule where the combined squares of the two legs equal the square of the hypotenuse.

Distance calculation in this problem involves using the Pythagorean theorem. The core formula for this theorem is:

• \[ a^2 + b^2 = c^2 \]

Where:

• \(a\) is one leg, the fixed distance from the stake to the pole, which is 3.00 meters.
• \(b\) is the unknown leg, the vertical height on the pole that the monkey can climb.
• \(c\) is the length of the chain, which is 3.40 meters.

By substituting these numbers into the equation, we calculate the potential height \(h\) (equivalent to \(b\)) the monkey can climb.

This involves solving for \(h\) through basic algebra steps:

• Substituting known values into the equation results in \(3.00^2 + h^2 = 3.40^2\).
• Solving for \(h\) involves finding the difference between the square of the hypotenuse and the square of the known leg.
• Lastly, taking the square root provides the vertical height: \(h = \sqrt{2.56} = 1.6\) meters.

This calculation shows the importance of accurate measurements to decide the monkey's climbing range.

Height determination is the ultimate goal here. It involves applying the previous concepts of right triangle geometry and chain length. The aim is to conclude how high the monkey can ascend the pole. Beginning with the formula:

• \[ h^2 = c^2 - a^2 \]

We previously computed:

• \(c^2 = 3.40^2 = 11.56\)
• \(a^2 = 3.00^2 = 9.00\)
• Therefore, \(h^2 = 11.56 - 9.00 = 2.56\)

Taking the square root yields the height \(h = \sqrt{2.56} = 1.6\) meters.

Through this straightforward calculation, based on the Pythagorean theorem, we show that understanding and applying mathematical principles can lead to practical solutions, like determining reachable heights in real-life situations.