Thread: two related rate problem help please

1. two related rate problem help please

Hi could someone please help show me how to solve the following two questions, I've spent a couple of hours at these without success.

1)

A search light is 225m from a straight waal. As the beam moves along the wall, the angle between the beam and the perpendicular to the wall is increasing at a rate of 1.5 degree/sec. How fast is the length of the beam increasing when it is 315m long?

The diagram indicates the beam on the ground pointing up to the perpendicular wall and moving across the x axis.

2) A computer is programmed to inscribe a series of rectangles in the first quadrant under the curve of y= e^-x. What is the area of the larget rectangle that can be inscribed?

I know the second question isnt related rate but im still having difficulty solving. Thanks so much for your help its very much appreciated, im lucky to have a forum like this to reach out to.

Regards,

T

2. Originally Posted by tilterino
...

2) A computer is programmed to inscribe a series of rectangles in the first quadrant under the curve of y= e^-x. What is the area of the larget rectangle that can be inscribed?

...
1. I assume that 2 adjacent sides of the rectangle are placed on the coordinate axes.

2. The area of the rectangle is calculated by:

$A(x)=x \cdot e^{-x}$ (see attachment)

3. Calculate the first derivate of A and solve A(x) = 0 for x.

Be sure that you use the product rule and the chain rule when doing the derivation. I've got $A'(x)=e^{-x} (1-x)$

4. For confirmation only: $A_{max} = \dfrac1e$

3. thanks for your help with #2. Can someone please help me with #1 I really need to figure this out tonight and would really appreciate the help

thanks

4. Originally Posted by tilterino
Hi could someone please help show me how to solve the following two questions, I've spent a couple of hours at these without success.

1)

A search light is 225m from a straight waal. As the beam moves along the wall, the angle between the beam and the perpendicular to the wall is increasing at a rate of 1.5 degree/sec. How fast is the length of the beam increasing when it is 315m long?

The diagram indicates the beam on the ground pointing up to the perpendicular wall and moving across the x axis.
Let theta(t) be the angle of the beam with the perpendicular at time t, and let L(t) be the length of the beam at time t. Then you can see geometrically that cos(theta) = 225/L(t), or theta = arccos(225/L(t)). Theta is increasing at a constant rate of 1.5 degrees/sec, or pi/120 radians/sec. Therefore we have

d/dt theta = d/dt arccos(225/L(t)) = pi/120.

Using the chain rule, we can express d/dt arccos(225/L(t)) as $\frac{-1}{\sqrt{1 - (\frac{225}{L(t)})^2}} \cdot \frac{-255}{L(t)^2} \cdot L'(t)$. Set this equal to pi/120, solve for L'(t) in terms of L(t) and then plug in L(t) = 315.