To find out how high the top of the ladder is touching the house, we can use the Pythagorean theorem. In this scenario, we can think of the ladder, the height it reaches on the house, and the distance from the base of the house to the bottom of the ladder as forming a right triangle.
The Pythagorean theorem states that: \[ a^2 + b^2 = c^2 \] where:
- \(c\) is the length of the hypotenuse (the ladder),
- \(a\) is one leg of the triangle (the height from the ground to the top of the ladder),
- \(b\) is the other leg of the triangle (the distance from the base of the house to the bottom of the ladder).
In this case:
- \(c = 20\) feet (the length of the ladder),
- \(b = 5\) feet (the distance from the base of the house).
We want to find \(a\), the height at which the ladder touches the house. Plugging our known values into the equation gives us: \[ a^2 + 5^2 = 20^2 \]
Calculating the squares: \[ a^2 + 25 = 400 \]
Now, we can solve for \(a^2\) by subtracting 25 from both sides: \[ a^2 = 400 - 25 \] \[ a^2 = 375 \]
Next, we take the square root of both sides to find \(a\): \[ a = \sqrt{375} \]
We simplify \(\sqrt{375}\): \[ \sqrt{375} = \sqrt{25 \times 15} = 5\sqrt{15} \]
Now, calculating \(5\sqrt{15}\): \[ \sqrt{15} \approx 3.872 \] Thus, \[ a \approx 5 \times 3.872 \approx 19.36 \]
Rounding \(19.36\) to the nearest tenth gives: \[ \boxed{19.4} \]
So, the height above the ground at which the top of the ladder is touching the house is approximately 19.4 feet.
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