A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at the rate of 0.5ft/s. What is the configuration of the ladder at the instant that the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?

At the instant that the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder, the ladder will be forming an angle of 45 degrees with the wall.

To determine the configuration of the ladder at the given instant, we need to find the relationship between the vertical speed of the top of the ladder and the horizontal speed of the foot of the ladder. Let's divide the ladder into two parts – the vertical distance from the ground to the foot of the ladder (x) and the distance from the foot of the ladder to the top of the ladder (y).

Let's assume that the ladder makes an angle θ with the ground. Since the ladder is leaning against a vertical wall, it forms a right triangle with the wall. Therefore, we can use trigonometry to find a relationship between x and y.

By using the Pythagorean theorem, we know that:

x^2 + y^2 = 12^2

To find this relationship in terms of x, we can differentiate both sides of the equation with respect to time (t):

2x * dx/dt + 2y * dy/dt = 0

Since we're interested in the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder, we can set dy/dt = 0.5 ft/s (given in the problem).

Considering the relationship between x and y obtained earlier from the Pythagorean theorem, we can differentiate x^2 + y^2 = 12^2 with respect to time:

2x * dx/dt + 2y * dy/dt = 0

Substituting the known values, we have:

2x * dx/dt + 2y * 0.5 = 0

Simplifying:

2x * dx/dt = -y

Now, we need to find the value of x such that the vertical speed of the top (dy/dt) equals the horizontal speed of the foot (dx/dt). Therefore, we can substitute dy/dt = dx/dt = 0.5 ft/s into the equation:

2x * 0.5 = -y

This gives us:

x = -y

Thus, the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder is such that x is equal to -y.

Additional information, such as specific measurements or angles, would be required to determine the exact lengths of x and y.

To find the configuration of the ladder at the instant that the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder, we can use the concept of related rates.

Let's denote the distance between the foot of the ladder and the wall as x, and the height of the ladder as y. We are given that the ladder is 12 ft long, so we know that x^2 + y^2 = 12^2.

We are also given that Jack is pulling the foot of the ladder away from the wall at a rate of 0.5 ft/s. This means that dx/dt = 0.5 ft/s.

To find the vertical speed of the top of the ladder, we can differentiate the equation x^2 + y^2 = 12^2 with respect to time (t):

2x(dx/dt) + 2y(dy/dt) = 0

Since the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder, we have dy/dt = dx/dt = 0.5 ft/s.

Substituting dx/dt = 0.5 ft/s into the equation, we get:

2x(0.5) + 2y(0.5) = 0
x + y = 0

Since the ladder is against a vertical wall, the foot of the ladder is not moving horizontally, which means dx/dt = 0. Therefore, x = 0.

Substituting x = 0 into the equation x^2 + y^2 = 12^2, we get:

0^2 + y^2 = 12^2
y^2 = 12^2
y = 12 ft

Therefore, the configuration of the ladder at the instant that the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder is with the foot of the ladder against the wall and the ladder standing vertically with a height of 12 ft.