# A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is #pi/4# rad?

By signing up, you agree to our Terms of Service and Privacy Policy

We can use related rates to solve this problem. Let ( x ) represent the distance between the bottom of the ladder and the wall, and ( \theta ) represent the angle between the ladder and the wall. We are given ( \frac{{dx}}{{dt}} = 2 ) ft/s, and we want to find ( \frac{{d\theta}}{{dt}} ) when ( \theta = \frac{{\pi}}{{4}} ) rad.

Using the Pythagorean theorem, ( x^2 + 10^2 = (10\cos\theta)^2 ). Differentiating both sides with respect to time ( t ), we get ( 2x\frac{{dx}}{{dt}} = 20\cos\theta\frac{{d\theta}}{{dt}} ).

Substitute the given values: ( 2(10)\cdot2 = 20\cos\frac{{\pi}}{{4}}\frac{{d\theta}}{{dt}} ). Solve for ( \frac{{d\theta}}{{dt}} ):

[ \frac{{d\theta}}{{dt}} = \frac{{40}}{{20\cdot\frac{{\sqrt{2}}}{{2}}} ]

[ \frac{{d\theta}}{{dt}} = \frac{{40}}{{10\sqrt{2}}} ]

[ \frac{{d\theta}}{{dt}} = \frac{{4}}{{\sqrt{2}}} ]

[ \frac{{d\theta}}{{dt}} = 2\sqrt{2} ]

So, ( \frac{{d\theta}}{{dt}} = 2\sqrt{2} ) rad/s when ( \theta = \frac{{\pi}}{{4}} ) rad.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How o do you determine the instantaneous rate of change of #V(r) = 4/3πr^3# for #r= 5#?
- What is the equation of the line normal to #f(x)=xe^x# at #x=7#?
- How do you find the derivative of #f(x) = (x^2-1) / (2x-3)# using the limit definition?
- How do you use the definition to find the derivative of #3x^2-5x+2#?
- What is the equation of the line normal to # f(x)=-(x+6)(x+3)+4x^2-8x+2# at # x=0#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7