Two automobiles are approaching the origin. The first one is traveling from the left on the x-axis at 30 mph. The second is traveling from the top on the y-axis at 45 mph. How fast is the distance between them changing when the first is at (-5, 0) and the second is at (0, 10)? (Both coordinates are in miles.)
Answer (2)
To find out how fast the distance between the two automobiles is changing, we'll use the Pythagorean theorem to express the distance between the two cars as a function of time and then differentiate with respect to time.
Let's define the variables:
Let x ( t ) be the position of the first car on the x-axis at time t . Since it starts at − 5 miles and moves toward the origin at 30 mph, the position x ( t ) = − 5 + 30 t .
Let y ( t ) be the position of the second car on the y-axis at time t . Since it starts at 10 miles and moves toward the origin at 45 mph, the position y ( t ) = 10 − 45 t .
The distance d ( t ) between the two cars at time t is given by the Pythagorean theorem:
d ( t ) = ( x ( t ) ) 2 + ( y ( t ) ) 2
The instantaneous rate of change of this distance with respect to time is given by d t dd , the derivative of the distance with respect to time. To find d t dd , we'll use the chain rule of differentiation.
First, find the derivative:
d t dd = 2 1 ( 2 x d t d x + 2 y d t d y ) ⋅ x 2 + y 2 1
Calculate d t d x = 30 mph (speed of the first car) and d t d y = − 45 mph (negative because the car moves towards the origin).
At the time of interest (when the first car is at ( − 5 , 0 ) and the second at ( 0 , 10 ) ), their positions are:
x = − 5
y = 10
Plug these values into the derivative formula:
d t dd = ( − 5 ) 2 + 1 0 2 1 (( − 5 ) ( 30 ) + ( 10 ) ( − 45 ))
Calculate:
d = 25 + 100 = 125 = 5 5
d t dd = 5 5 1 ( − 150 − 450 ) = 5 5 − 600
To simplify, multiply the numerator and denominator by 5 :
d t dd = 25 − 600 5 = − 24 5 mph
Thus, the speed at which the distance between the two cars is changing at the specified moment is − 24 5 mph, meaning the distance is decreasing at approximately 53.67 mph.
The distance between the two cars is changing at a rate of approximately -24√5 mph, indicating they are getting closer together at about 53.67 mph. This is calculated using the Pythagorean theorem and derivatives. The negative rate shows that the distance is decreasing.
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