=1 A 26 - f ladder is standing up against a wall. The distance between the base of the ladder and the wall is 14 ft less than the distance between the top of the ladder and the base of the wall. Find the distance between the base of the ladder and the wall. The distance between the base of the ladder and the wall is ft.
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To tackle this problem, we can use the Pythagorean theorem since we have a right triangle formed by the wall, the ground, and the ladder. Let’s denote the distance from the base of the ladder to the wall as \(x\) ft. According to the problem, the distance from the top of the ladder to the base of the wall is \(x + 14\) ft. Using the ladder length provided (26 ft), we can set up the equation: \[ x^2 + (x + 14)^2 = 26^2 \] Now, let’s solve for \(x\). First, expand the equation: \[ x^2 + (x^2 + 28x + 196) = 676 \] Combine like terms: \[ 2x^2 + 28x + 196 = 676 \] Subtract 676 from both sides: \[ 2x^2 + 28x - 480 = 0 \] We can simplify this equation by dividing everything by 2: \[ x^2 + 14x - 240 = 0 \] Now, we apply the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] Where \(a = 1\), \(b = 14\), and \(c = -240\): \[ b^2 - 4ac = 14^2 - 4(1)(-240) = 196 + 960 = 1156 \] Now, we substitute back: \[ x = \frac{{-14 \pm \sqrt{1156}}}{2} \] Calculating the square root: \[ \sqrt{1156} = 34 \] So: \[ x = \frac{{-14 \pm 34}}{2} \] We find two potential values for \(x\): 1. \(x = \frac{20}{2} = 10\) 2. \(x = \frac{-48}{2} = -24\) (not a valid physical distance) Thus, the distance between the base of the ladder and the wall is **10 ft**. --- The distance between the base of the ladder and the wall is 10 ft.
