To solve this problem, we need to find the base distance from the wall to the ladder when the ladder has to climb to a height of 12 feet. The problem provides a hint stating that for every 4 feet upwards, the ladder needs to go 1 foot across horizontally. We'll use this ratio to find our answer step-by-step:
1. Understand the ratio:
- For every 4 feet the ladder goes upwards, it moves 1 foot horizontally away from the wall. This means the ratio of the height to the base is 4:1.
2. Calculate the proportion for 1-foot height:
- If the ladder moves horizontally 1 foot for every 4 feet it moves upwards, we need to determine the horizontal movement per foot climbed. This can be written as `1 foot (horizontal) per 4 feet (vertical)`.
- Therefore, for every 1 foot climbed vertically, the ladder moves `1/4` or 0.25 feet horizontally.
3. Find the base distance for 12 feet height:
- If the ladder climbs 12 feet vertically, we can use the ratio calculated in the previous step to find out the total horizontal distance.
- Multiply the vertical height by the horizontal movement per foot of height:
[tex]\[
\text{Base distance} = 12 \text{ feet (height)} \times \frac{1 \text{ foot (horizontal)}}{4 \text{ feet (vertical)}}
\][/tex]
- Simplifying this:
[tex]\[
\text{Base distance} = 12 \times 0.25 = 3 \text{ feet}
\][/tex]
Therefore, the base of the ladder should be placed 3 feet away from the point directly below the top of the ladder.
