The top of a 13 foot ladder is sliding down a vertical wall at a constant rate of 4 feet per minute. When the top of the ladder is 5 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall

In just about any introductory Calculus book this type of question appears as a lead-in to the topic of "rates of change"

Make a diagram, label the height y and the the base length x
then x^2 + y^2 = 13^2
differentiate with respect to t (time)
2x dx/dt + 2y dy/dt = 0

when y = 5 , dy/dx = -4 (negative to show it is getting smaller)
and x^2 + 5^2 = 13^2
x = 12

2x dx/dt = - 2y dy/dt
dx/dt = - (y/x) dy/dt = -5/12 (-4) = 5/3 ft/min

(notice dx/dt is positive, showing the distance to be increasing)

Well, isn't that a slippery situation? Let's see if I can help you with that!

To find the rate of change of the distance between the bottom of the ladder and the wall, we can use the Pythagorean theorem. The ladder, the wall, and the ground form a right triangle.

When the top of the ladder is 5 feet from the ground, we know that the height of the triangle is also 5 feet. Now, we need to find the length of the base of the triangle, which represents the distance between the bottom of the ladder and the wall.

Since the top of the ladder is sliding down at a constant rate of 4 feet per minute, we can say that the height of the triangle is changing at a rate of -4 feet per minute. This is because the top of the ladder is moving downwards.

Using the Pythagorean theorem (a^2 + b^2 = c^2), we can express the relationship between the height (h), the base (b), and the hypotenuse (c) of the triangle:

5^2 + b^2 = 13^2

25 + b^2 = 169

Now, let's differentiate both sides of the equation with respect to time:

0 + 2b(db/dt) = 0

Simplifying, we get:

2b(db/dt) = 0

Since we're looking for the rate of change of the distance between the bottom of the ladder and the wall (db/dt), we can solve for it:

db/dt = 0 / (2b) = 0

So, the rate of change of the distance between the bottom of the ladder and the wall is 0 feet per minute. This means that the distance remains constant while the ladder is sliding down.

To find the rate of change of the distance between the bottom of the ladder and the wall, we can use the Pythagorean Theorem.

Let's assume that the distance between the bottom of the ladder and the wall is represented by x.

According to the Pythagorean Theorem, the relationship between the distances in a right-angled triangle is given by the equation:

x^2 + 5^2 = 13^2

Simplifying the equation, we have:

x^2 + 25 = 169

To find the rate of change of x, we need to differentiate both sides of the equation with respect to time (t) since the height of the ladder is changing with time:

D(x^2 + 25)/dt = D(169)/dt

Differentiating each term using the chain rule:

2x * dx/dt = 0

Since the ladder is sliding down the wall at a constant rate of 4 feet per minute, which means dx/dt = -4 (negative because x is decreasing), we can substitute this value into the equation:

2x * (-4) = 0

-8x = 0

Solving for x, we find:

x = 0

Since x = 0, it means that the ladder is touching the ground, and the distance between the bottom of the ladder and the wall is 0 feet.

Therefore, the rate of change of the distance between the bottom of the ladder and the wall is 0 feet per minute.

To find the rate of change of the distance between the bottom of the ladder and the wall, let's consider the situation diagrammatically.

Let's assume that the distance between the bottom of the ladder and the wall is represented by the variable "x" at a certain time. Also, at that time, let's assume the distance between the top of the ladder and the wall is represented by the variable "y".

We are given that the ladder is sliding down the wall at a constant rate of 4 feet per minute. This means that the rate of change of "y", denoted as dy/dt (where t represents time), is -4 feet per minute. The negative sign indicates that the distance "y" is decreasing.

By using the Pythagorean theorem, we can relate "x" and "y" with the length of the ladder, which is 13 feet:

x^2 + y^2 = 13^2

Since we are looking for the rate of change of "x", we need to differentiate both sides of the equation with respect to time t.

Taking the derivative of both sides, we have:

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

Since dx/dt represents the rate of change of "x", which we want to find, and we know dy/dt = -4 feet per minute, we can rearrange the equation to solve for dx/dt:

2x(dx/dt) = -2y(dy/dt)
dx/dt = (-2y(dy/dt)) / (2x)

Substituting the given values, when the top of the ladder is 5 feet from the ground (y = 5), we can plug these values into the equation:

dx/dt = (-2(5)(-4)) / (2x)
dx/dt = 40 / (2x)

We need to determine the value of "x" at this point. Using the Pythagorean theorem, we can find "x" when y = 5:

x^2 + 5^2 = 13^2
x^2 = 13^2 - 5^2
x^2 = 144
x = 12

Substituting x = 12 into the equation for dx/dt, we have:

dx/dt = 40 / (2*12)
dx/dt = 40 / 24
dx/dt = 5/3 feet per minute

Therefore, when the top of the ladder is 5 feet from the ground, the rate of change of the distance between the bottom of the ladder and the wall is 5/3 feet per minute.