(1) A particle on the -ais is moving to the right at 2 units per second. At a ce

Question

(1) A particle on the -ais is moving to the right at 2 units per second. At a certain instant on the r-axis is moving to the right at 2 units per second. At a certain instant it is at the point (5,0). How rapidly it the distance between the particle and the point (0,9) on the y-axis changing at that instant? (2) The length of a rectangle is increasing at the rate of 5 meters per minute while the width is decreasing at the rate of 3 meters per minute. At a certain instant, the length is 20 meters and the width is 10 meters. At that instant determine (a) the rate of change of the area of the rectangle, (b) the rate of change of the perimeter of the rectangle 3) A particle on the r-axis is moving to the right at 2 units per second. A second particle is moving down the y-axis at the rate of 3 units per second. At a certain instant the first particle is at the point (5,0), and the second is a the point (0,7). How rapidly it the angle the r-axis and the line joining the two particles changing at that instant? Are etween the particles moving towards or away from each other at that instant? 4) The beacon on a lighthouse makes one revolution every 20 seconds. The beacon is 300 feet from the nearest point, P,on a straight shoreline. Find the rate at which the ray of light

Explanation / Answer

answer-1 So the particle is always on the x-axis, so use the distance formula to write down an expression (function really) for the distance, then find the derivative of that expression with respect to time:
Using Pythagorean theorem:

d² = (x - x)² + y² ----------------------------1
(since it's always on the x-axis, y is always zero --> (y - 0)² = y²)
Use implicit differentiation in equation 1
--> differentiate with respect to t (remember x and y are constants, therefore dx/dt = dy/dt = 0)

2d * dd/dt = 2(x - x)dx/dt + 0
--> 2's cancel

dd/dt = (x - x) / d * dx/dt
So find x, x, d and dx/dt
Given x, x, and dx/dt (x = 0 because the "reference point" is (0, 9) --> x = 0, y = 9)
x = 5, dx/dt = 2
Calculate d at that point:
d² = x² + y²
--> d² = 5² + 9² = 25 + 81 = 106
--> d = 106
dd/dt = 5/106 * 2 = 10 / 106 --------------answer

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