How fast is the top of a 13-foot ladder sliding down a wall at the instant the base is 12 feet from the wall, assuming the base is moving away from the wall at 4 feet per second?

How fast is the top of a 13-foot ladder sliding down a wall at the instant the base is 12 feet from the wall, assuming the base is moving away from the wall at 4 feet per second?

• A. 9.2 feet per second
• B. 8.3 feet per second
• C. -9.6 feet per second
• D. -10 feet per second

Answer: (C)

To determine how fast the top of the ladder is sliding down the wall, we can apply the Pythagorean theorem and related rates of change. Given that the ladder forms a right triangle with the wall and the ground, we can represent the lengths using variables. Let \( y \) be the height of the top of the ladder on the wall, and \( x \) be the distance from the wall to the base of the ladder on the ground. According to the Pythagorean theorem, we have: \[ x^2 + y^2 = L^2 \] Where \( L \) is the length of the ladder. In this case, \( L = 13 \) feet. We know that \( x = 12 \) feet and that the base is moving away from the wall at a rate of \( \frac{dx}{dt} = 4 \) feet per second. First, we can find \( y \) by plugging in our value for \( x \): \[ 12^2 + y^2 = 13^2 \\ 144 + y^2 = 169 \\ y^2 = 169 - 144 \\ y^2 = 25

To determine how fast the top of the ladder is sliding down the wall, we can apply the Pythagorean theorem and related rates of change.

Given that the ladder forms a right triangle with the wall and the ground, we can represent the lengths using variables. Let ( y ) be the height of the top of the ladder on the wall, and ( x ) be the distance from the wall to the base of the ladder on the ground. According to the Pythagorean theorem, we have:

[

x^2 + y^2 = L^2

]

Where ( L ) is the length of the ladder. In this case, ( L = 13 ) feet. We know that ( x = 12 ) feet and that the base is moving away from the wall at a rate of ( \frac{dx}{dt} = 4 ) feet per second.

First, we can find ( y ) by plugging in our value for ( x ):

[

12^2 + y^2 = 13^2 \

144 + y^2 = 169 \

y^2 = 169 - 144 \

y^2 = 25