A 10-foot ladder is leaning straight up against a wall when a person begins pulling the base of the ladder away from the wall at the rate of 1 foot per second. Calculate the true distance between the top of the ladder and the ground when the base of the ladder is 9 feet from the wall.

The true statement about the true distance between the top of the ladder and the ground when the base is 9 feets from the wall is - 9/√19

Let :

Ladder forms a right angle triangle as it leans against the wall :

Hence, we have :

Taking the implicit derivative of both sides:

2xdx/dt + 2ydy/dt = 0

xdx/dt + ydy/dt = 0

ydy/dt = - xdx/dt

dy/dt = (-xdx/dt) / y

When the ladder is 9 ft from the wall;

81 + y² = 100

y² = 100 - 81 = 19

y = √19

Substituting into the equation :

dy/dt = -9(1)/√19

Hence, dy/dt = - 9/√19

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Explanation:

Let the height of the ladder be L

Also:

When the ladder leans against the wall, it forms a triangle and the length of the ladder forms the hypotenuse.

So, we have:

--- Pythagoras Theorem

When the base is 9ft from the wall, this means that:

Substitute 9 for x and 10 for L in

Make the subject

Make y the subject

Hence, the true distance at that point is approximately 4.36ft

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angle is from normal turned to line of ray