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.