To solve this problem, we can apply the Pythagorean Theorem because the ladder, the ground, and the wall form a right triangle.

Given data:

1. The ladder's length (hypotenuse) = 35 feet
2. Initial distance up the wall = 21 feet

Let's calculate the initial horizontal distance (distance from the wall to the base of the ladder).

Step 1: Find the Initial Horizontal Distance

Using the Pythagorean theorem: $x^2 + 21^2 = 35^2$ where $x$ is the initial horizontal distance from the wall.

$x^2 + 441 = 1225$ $x^2 = 1225 - 441 = 784$ $x = \sqrt{784} = 28 \text{ feet}$

So, initially, the ladder’s base was 28 feet from the wall.

Step 2: New Horizontal Distance

Now, Elijah moves the base 4 feet farther from the house: $\text{New horizontal distance} = 28 + 4 = 32 \text{ feet}$

Step 3: Calculate the New Height Up the Wall

Using the Pythagorean theorem again with the new horizontal distance: $32^2 + y^2 = 35^2$ where $y$ is the new vertical distance up the wall.

$1024 + y^2 = 1225$ $y^2 = 1225 - 1024 = 201$ $y = \sqrt{201} \approx 14.2 \text{ feet}$

Answer

The ladder will now reach approximately 14.2 feet up the side of the house.

Would you like further clarification on any step?

Here are some follow-up questions you might explore:

1. How does moving the base of the ladder affect the height it reaches on the wall?
2. What would the height be if the ladder were moved 6 feet farther instead?
3. Can this problem be solved using trigonometric ratios instead of the Pythagorean theorem?
4. How would the height change if a different ladder length were used?
5. What are practical implications of this calculation in terms of ladder safety?

Tip: When using the Pythagorean theorem, ensure the hypotenuse is always the longest side (opposite the right angle).