a 14 foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 3 feet/second, how fast is the top of the ladder moving down when the foor of the ladder is 5 feet from the wall?
I got -1.083 when I ended by doing -15/sq rt 192
To solve this problem, we can use the Pythagorean Theorem. Let's denote the distance between the foot of the ladder and the wall as x, and the height of the ladder as y.
According to the problem, the bottom of the ladder is sliding along the pavement away from the building at a rate of 3 feet/second. This means dx/dt (the rate at which x is changing) is 3 ft/s.
We need to find dy/dt (the rate at which y is changing) when x = 5 ft.
Using the Pythagorean Theorem, we have:
x^2 + y^2 = 14^2
Differentiating both sides with respect to t (time), we get:
2x(dx/dt) + 2y(dy/dt) = 0
Substituting the given values and solving for dy/dt:
(2)(5)(3) + 2(√(14^2 - 5^2))(dy/dt) = 0
30 + 2(√(196 - 25))(dy/dt) = 0
30 + 2(√(171))(dy/dt) = 0
2(√(171))(dy/dt) = -30
√(171)(dy/dt) = -15
dy/dt = -15 / √(171)
Approximating this to three decimal places, dy/dt = -1.083 ft/s.
Therefore, the top of the ladder is moving down at a rate of approximately 1.083 feet per second.
To solve this problem, we can use the concept of related rates. Here's how you can approach it:
1. Draw a diagram to visualize the problem. Label the distance between the foot of the ladder and the wall as "x," and the height of the ladder as "y."
2. Understanding that the ladder forms a right triangle with the wall and the ground, you can use the Pythagorean theorem to relate x and y. The equation is: x^2 + y^2 = 14^2 (since the ladder is 14 feet long).
3. Differentiate both sides of the equation with respect to time (t) to find the rates of change. Remember, x is changing at a rate of 3 feet/second, and we need to find the rate of change of y with respect to t, dy/dt.
4. After differentiating, you will have: 2x(dx/dt) + 2y(dy/dt) = 0.
5. Now, substitute the given values: x = 5 (since the foot of the ladder is 5 feet from the wall), and you know that dx/dt = 3 feet/second.
6. Solve the equation for dy/dt, which represents the rate at which the top of the ladder is moving downward.
7. Plugging in the values, you should get: 2(5)(3) + 2y(dy/dt) = 0. Simplify this expression to get 30 + 2y(dy/dt) = 0.
8. Rearrange the equation and solve for dy/dt: dy/dt = -30 / (2y).
9. To find dy/dt when x = 5, substitute x = 5 into the original equation x^2 + y^2 = 14^2 to solve for y. You should get y = sqrt(14^2 - 5^2).
10. Finally, substitute this value of y into dy/dt = -30 / (2y) to calculate the rate at which the top of the ladder is moving downward when the foot of the ladder is 5 feet from the wall.
Note: It seems like there might be an error in your calculation since -1.083 is not a plausible answer in this context.
same as your other question.
check your work, that should have been √171 instead of √192
25 + y^2 = 14^2
y^2 = 196 - 25
etc