Johan has a 300-cm ladder that he leans up against a wall. The safety sticker on the side of the ladder shows that the bottom must be placed between 70 cm and 110 cm away from the wall. What are the minimum distance and maximum distance up the wall that the ladder can reach? Give your answers to the nearest tenth of a centimetre.

This problem involves using the Pythagorean theorem because the ladder, the distance from the wall, and the height up the wall form a right triangle.

Let's define:
- \( L = 300 \, \text{cm} \): the length of the ladder, which is the hypotenuse of the triangle.
- \( d \): the horizontal distance from the base of the ladder to the wall.
- \( h \): the vertical distance from the ground to the point where the ladder touches the wall.

The Pythagorean theorem states:
\[
L^2 = d^2 + h^2
\]
Rearranging to solve for \( h \):
\[
h = \sqrt{L^2 - d^2}
\]

### Step 1: Calculate the Minimum Height (when \( d = 110 \, \text{cm} \))
Using \( L = 300 \) cm and \( d = 110 \) cm:
\[
h = \sqrt{300^2 - 110^2} = \sqrt{90000 - 12100} = \sqrt{77900} \approx 279.1 \, \text{cm}
\]

### Step 2: Calculate the Maximum Height (when \( d = 70 \, \text{cm} \))
Using \( L = 300 \) cm and \( d = 70 \) cm:
\[
h = \sqrt{300^2 - 70^2} = \sqrt{90000 - 4900} = \sqrt{85100} \approx 291.7 \, \text{cm}
\]

### Final Answer
- **Minimum height up the wall:** \( 279.1 \, \text{cm} \)
- **Maximum height up the wall:** \( 291.7 \, \text{cm} \)

Would you like more details on the solution or have any questions?

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Here are 5 related questions for further exploration:
1. What if the ladder were longer? How would that affect the maximum and minimum heights?
2. How would the answer change if the ground were at an incline?
3. What happens if the distance to the wall exceeds the safe range?
4. Could we calculate the angle of the ladder in each case?
5. How would this change if the wall had a protrusion or obstacle?

**Tip:** Always check ladder safety guidelines, as angles impact stability.

Explore how to calculate the minimum and maximum height a 300-cm ladder can reach based on its distance from the wall, using the Pythagorean theorem. The safe distance from the wall ranges from 70 cm to 110 cm, and the corresponding heights up the wall are approximately 279.1 cm and 291.7 cm, respectively.

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