Mathematics: A Sixteen Foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 2 feet/second, how fast is the top of the ladder moving down when the foot of the ladder is 5 feet from the wall? The Top of the ladder is moving down at a rate of ___feet/second when the foot of the ladder is 5 feet from the wall.

A Sixteen Foot ladder is leaning against a building.

hannahkharel2

21.01.2022

-0.69 feet per second

Step-by-step explanation:

We have dx/dt = 2 ft/sec, and we want to determine dy/dt.

x and y are related by the Pythagorean Theorem.

x²+y²=16²

Differentiate both sides of this equation with respect to t to get

$2x\frac{dx}{dt}+2y\frac{dy}{dt} =0$

$2y\frac{dy}{dt} =-2x\frac{dx}{dt}$

Divide both sides by 2y

$\frac{dy}{dt} =-\frac{x}{y}\frac{dx}{dt}$

When x = 5 ft, we have

x²+y²=16²

5²+y²=256

y²=256-25=231

therefore

y= $\sqrt{213}$

$\frac{dy}{dt} =-\frac{5}{\sqrt{213}}\frac{2 ft}{sec}$

=-0.69

The top of the ladder is sliding down (because of the negative sign in the result) at a rate of -0.69 feet per second.

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mrashrafkotkaat

11.02.2023

A parallel line would contain the same slope and have a different y-intercept.

Using the slope of the original line you would start with:

Y = -5x + b, where b is the new y-intercept.

Using the given point replace x and y and solve for b:

-9 = -5(-4) + b

Simplify:

-9 = 20 + b

Subtract 20 from both sides

-29 = b

Now replace b in the equation to get:

Y = -5x -29

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