We can solve both parts of the question using right triangle geometry. Let's break it down.

Part a) How far away is the foot of the ladder from the base of the wall?

This is a simple application of the Pythagorean theorem. In this case, the ladder, the height it reaches on the wall, and the distance from the wall form a right triangle.

The Pythagorean theorem states:

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

Where:

• $c$ is the length of the ladder (the hypotenuse),
• $a$ is the height of the ladder on the wall (vertical distance),
• $b$ is the horizontal distance (the distance we need to find).

Given:

• $c = 13.5 \, \text{m}$
• $a = 6.1 \, \text{m}$

We need to find $b$ (the distance from the foot of the ladder to the wall).

Rearranging the Pythagorean theorem:

$b = \sqrt{c^2 - a^2}$

Substituting the values:

$b = \sqrt{13.5^2 - 6.1^2}$ $b = \sqrt{182.25 - 37.21}$ $b = \sqrt{145.04}$ $b \approx 12.04 \, \text{m}$

So, the foot of the ladder is approximately 12.04 meters away from the base of the wall.

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Part b) What angle does the ladder make with the ground?

To find the angle, we can use the trigonometric function cosine. Cosine is defined as:

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Where:

• The adjacent side is the distance from the foot of the ladder to the wall ($b$),
• The hypotenuse is the length of the ladder ($c$).

Rearranging the formula to find $\theta$:

$\theta = \cos^{-1} \left( \frac{b}{c} \right)$

Substitute the values:

$\theta = \cos^{-1} \left( \frac{12.04}{13.5} \right)$ $\theta = \cos^{-1} (0.892)$ $\theta \approx 27.13^\circ$

So, the ladder makes an angle of approximately 27.13 degrees with the ground.

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Let me know if you'd like further details or clarification!

Follow-up questions:

1. How does the Pythagorean theorem apply to other real-world problems?
2. Can this problem be solved using the sine or tangent function instead of cosine?
3. How would the answer change if the length of the ladder were different?
4. How would you solve for the height the ladder reaches if you knew the angle and the length of the ladder?
5. How would the result change if the angle of the ladder were increased?

Tip:

When solving problems involving right triangles, it's often useful to start by labeling the sides and angles clearly before applying Pythagoras or trigonometric ratios.