A 10 -ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of \( 2 \mathrm{ft} / \mathrm{sec} \), how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?
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To solve this problem, we can use related rates in calculus. Let's denote the height of the ladder on the wall as \(y\), the distance of the bottom of the ladder from the wall as \(x\), and the length of the ladder as \(L = 10\) ft. According to the Pythagorean theorem: \[ x^2 + y^2 = L^2 \] Differentiating both sides with respect to time \(t\): \[ 2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0 \] Given \( \frac{dy}{dt} = -2 \) ft/sec (the negative sign indicates that \(y\) is decreasing) and when \(x = 5\) ft, we can find \(y\): Using the Pythagorean theorem: \[ 5^2 + y^2 = 10^2 \] \[ 25 + y^2 = 100 \] \[ y^2 = 75 \] \[ y = \sqrt{75} = 5\sqrt{3} \, \text{ft} \] Substituting into the differentiated equation: \[ 2(5)\frac{dx}{dt} + 2(5\sqrt{3})(-2) = 0 \] Now solving for \( \frac{dx}{dt} \): \[ 10\frac{dx}{dt} - 20\sqrt{3} = 0 \] \[ 10\frac{dx}{dt} = 20\sqrt{3} \] \[ \frac{dx}{dt} = 2\sqrt{3} \, \text{ft/sec} \] When the bottom of the ladder is 5 ft from the wall, the bottom is moving away from the wall at a rate of \( 2\sqrt{3} \) ft/sec! Now, let's dive into some extra fun facts about this scenario! The ladder problem is reminiscent of classic physics principles seen in ancient architecture, where similar right-angled triangle relationships were utilized to calculate heights and distances—from the pyramids of Egypt to the intricate cathedrals of Europe! Each craftsman had to calculate precisely, ensuring their masterpiece stood true and proud! Additionally, understanding these relationships equips folks with skills useful in home improvement projects—like how to safely prop up a ladder! Knowing the angle and distances can be crucial for ensuring stability, avoiding falls, and making your DIY dreams a reality! So, next time you're tackling that weekend project, remember these nifty calculations!