This problem involves the Pythagorean theorem, as the ladder forms a right triangle with the house and the ground.

Step 1: Set up the original scenario

Let:

• The height the ladder reaches up the side of the house = $27$ feet.
• The length of the ladder = $45$ feet.
• The distance of the ladder's base from the house = $x$.

From the Pythagorean theorem: $x^2 + 27^2 = 45^2$

Step 2: Solve for $x$

$x^2 + 729 = 2025$ $x^2 = 2025 - 729$ $x^2 = 1296 \implies x = \sqrt{1296} = 36 \, \text{feet}.$

So initially, the base of the ladder is $36$ feet from the house.

Step 3: Adjust the base distance

When Latanya pulls the base 3 feet farther from the house, the new distance of the base becomes: $x' = 36 + 3 = 39 \, \text{feet}.$

Step 4: Solve for the new height

Let the new height the ladder reaches be $h$. Using the Pythagorean theorem again: $39^2 + h^2 = 45^2$ $1521 + h^2 = 2025$ $h^2 = 2025 - 1521$ $h^2 = 504 \implies h = \sqrt{504} \approx 22.5 \, \text{feet}.$

Final Answer:

The ladder will reach approximately 22.5 feet up the side of the house.

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Follow-up:

Would you like me to explain further or verify these steps?

Related Questions:

1. How would the height change if the base were pulled 5 feet farther instead of 3 feet?
2. What is the angle of the ladder with the ground in the initial position?
3. How does the height compare when using a shorter ladder, such as 30 feet?
4. Can you calculate the area of the triangle formed by the ladder, the wall, and the ground?
5. If the ladder's length changed to 50 feet, how would the height change?

Tip:

The Pythagorean theorem is a powerful tool for right triangles—always identify which sides correspond to $a$, $b$, and $c$.