Question

Rate of Change A 25 -foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate \(r\) of \(r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec}\) where \(x\) is the distance between the ladder base and the house. (a) Find \(r\) when \(x\) is 7 feet. (b) Find \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x \rightarrow 25^{-}\).

Short answer

The rate \(r\) when \(x\) is 7 feet is 0.567 ft/sec, when \(x\) is 15 feet, the rate is 0.6 ft/sec, and the limit of \(r\) as \(x \rightarrow 25^{-}\) is infinite.

Step-by-step solution

Calculate rate for \(x = 7 ft\)

Substitute \(x = 7 ft\) into the equation \(r=\frac{2 x}{\sqrt{625-x^{2}}}\) to find \(r\). The result is: \(r_{7}\) = \(\frac{2 * 7}{\sqrt{625-7^{2}}}\) = \(0.567 ft/sec\).

Calculate rate for \(x = 15 ft\)

Similarly to step 1, substitute \(x = 15 ft\) into the equation \(r=\frac{2 x}{\sqrt{625-x^{2}}}\) to find \(r\). The result is: \(r_{15}\) = \(\frac{2 * 15}{\sqrt{625-15^{2}}}\) = \(0.6 ft/sec\).

Calculate limit as \(x \rightarrow 25^{-}\)

You are required to find the limit as \(x\) approaches 25 from the left \(x \rightarrow 25^{-}\). In this case, you can use the limit definition. Substitute \(x = 25 \) into the equation \(r=\frac{2 x}{\sqrt{625-x^{2}}}\). Since substitution gives a division by zero scenario, this indicates that the limit is infinite.