Let's denote the length of the ladder as \( L \) ft. According to the problem, we have two conditions based on the description:
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The distance from the bottom of the ladder to the building is 10 ft shorter than the length of the ladder: \[ d = L - 10 \] where \( d \) is the distance from the bottom of the ladder to the building.
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The height the ladder reaches up the building is 5 ft less than the length of the ladder: \[ h = L - 5 \] where \( h \) is the height the ladder reaches.
Since we have a right triangle formed by the ladder, the distance from the bottom of the ladder to the building, and the height up the building, we can apply the Pythagorean theorem: \[ L^2 = d^2 + h^2 \]
Substituting the expressions for \( d \) and \( h \): \[ L^2 = (L - 10)^2 + (L - 5)^2 \]
Now, let's expand the expressions on the right side:
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Expanding \( (L - 10)^2 \): \[ (L - 10)^2 = L^2 - 20L + 100 \]
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Expanding \( (L - 5)^2 \): \[ (L - 5)^2 = L^2 - 10L + 25 \]
Now, we can combine these results: \[ L^2 = (L^2 - 20L + 100) + (L^2 - 10L + 25) \] \[ L^2 = 2L^2 - 30L + 125 \]
Next, let's rearrange the equation: \[ L^2 - 2L^2 + 30L - 125 = 0 \] \[ -L^2 + 30L - 125 = 0 \] Multiplying through by -1 gives: \[ L^2 - 30L + 125 = 0 \]
Now, let's solve this quadratic equation using the quadratic formula: \[ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = -30, c = 125 \): \[ L = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 1 \cdot 125}}{2 \cdot 1} \] \[ L = \frac{30 \pm \sqrt{900 - 500}}{2} \] \[ L = \frac{30 \pm \sqrt{400}}{2} \] \[ L = \frac{30 \pm 20}{2} \]
We have two potential solutions:
- \( L = \frac{50}{2} = 25 \)
- \( L = \frac{10}{2} = 5 \) (not valid since the distance and height can't be negative)
Thus, the length of the ladder is \( L = 25 \) ft.
Now, we can find the height \( h \): \[ h = L - 5 = 25 - 5 = 20 \text{ ft} \]
Thus, the height up the side of the building is \( \boxed{20} \) feet.