This problem involves using the Pythagorean Theorem to determine the height $d$ of the ladder against the house. The Pythagorean Theorem states:

$a^2 + b^2 = c^2$

Where:

• $a$ is one leg of the triangle (3 feet, the base of the ladder),
• $b$ is the other leg of the triangle ($d$, the vertical height against the house),
• $c$ is the hypotenuse (10 feet, the length of the ladder).

Step 1: Plug in the known values

$3^2 + d^2 = 10^2$

$9 + d^2 = 100$

Step 2: Solve for $d^2$

$d^2 = 100 - 9$

$d^2 = 91$

Step 3: Solve for $d$

$d = \sqrt{91}$

Final Answer:

$d \approx 9.53 \, \text{feet}$

Let me know if you want further details or explanations. Here are 5 related questions for further exploration:

1. What would be the height $d$ if the base of the ladder was 4 feet away from the house?
2. How does the Pythagorean Theorem apply to real-world problems like this one?
3. What safety recommendations might apply when placing a ladder at different angles?
4. How can you calculate the angle of the ladder using trigonometry?
5. What is the area of the triangle formed by the ladder, house, and ground?

Tip: Always make sure your ladder is placed at a safe angle, which is often recommended as 75 degrees to the ground.