A 30 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 a rate of r= 2x/(sqrt(900-x^2) ft/sec, where s ist the distance between the base of the ladder and the house. Find the rate r when x is 18 feet. Help!

I was confused about if I should just plug in the value or take the derivative of the function? But Thank you!

To find the rate r when x is 18 feet, we need to use the given equation r = 2x / √(900 - x^2) ft/sec.

Substitute x = 18 into the equation:
r = 2 * 18 / √(900 - 18^2) ft/sec

First, calculate 18^2 = 324:
r = 2 * 18 / √(900 - 324) ft/sec

Next, calculate 900 - 324 = 576:
r = 36 / √576 ft/sec

Next, calculate √576 = 24:
r = 36 / 24 ft/sec

Simplify 36/24 to get:
r = 3/2 ft/sec

Therefore, when x is 18 feet, the rate r is 3/2 ft/sec.

To find the rate r when x is 18 feet, we need to find the derivative of r with respect to time (dr/dt) and then substitute x = 18 into the expression.

First, let's analyze the given information. We have the rate at which the base of the ladder is moving away from the house, which is given as ds/dt = 2 ft/sec. We also have an equation relating the distance between the base of the ladder and the house (s) to the rate r at which the top of the ladder is moving down the wall.

Given: r = 2x / √(900 - x^2)

To find dr/dt, we'll use the chain rule. The chain rule states that if r = f(x) and x = g(t), then dr/dt = f'(x) * g'(t). In this case, x depends on t, which means that r also depends on t. So, we can differentiate both sides of the equation with respect to t.

Differentiating r = 2x / √(900 - x^2) with respect to t:

dr/dt = d/dt [2x / √(900 - x^2)]

Now, let's differentiate each part separately:

1. Differentiate 2x with respect to t:
d/dt [2x] = 2 * dx/dt

2. Differentiate √(900 - x^2) with respect to t:
d/dt [√(900 - x^2)] = (1/2) * (900 - x^2)^(-1/2) * (-2x) * dx/dt

Combining both differentiations:

dr/dt = 2 * dx/dt / √(900 - x^2) - x * dx/dt / (900 - x^2)^(1/2)

Now, substitute x = 18 into the equation to find the rate r when x is 18 feet:

r = 2 * 18 / √(900 - 18^2) - 18 * ds/dt / (900 - 18^2)^(1/2)

simplifying,

r = 2 * 18 / √(900 - 324) - 18 * 2 / (900 - 324)^(1/2)

Finally, calculate the value of r.

what's the problem? Just plug in your value of x:

r = 2(18)/24 = 3/2

This may be clearer if we consider the height h up the wall.

x^2 + h^2 = 900
2x dx/dt + 2h dh/dt = 0
when x=18, h=24, and since dx/dt = 2,

18(2) + 24 dh/dt = 0
dh/dt = -18(2)/24 = -3/2

Note that we have a negative value for dh/dt, which indicates the height is decreasing as the base is pulled away. The formula they gave gives a positive value, but it is stipulated that the top is moving down at that rate.