A person on a bicycle is 100 meters west of a person in a car. the person on the bike starts to ride at 8 meters per second directly east and at the same time the person in the car starts to drive directly north at 18 meters per second. Write the distance between the bicycle and the car as a function of time and then find the time when the distance between the bicycle and the car is a minimum. verify that this time does give a minimum distance and find this minimum distance ( hint: after x seconds the cyclist has ridden 8x meters and the car has driven 18x meters.)
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To find the distance between the bicycle and the car as a function of time, we can use the Pythagorean theorem. Let's assume the horizontal distance between them at any time t is x.
1. At time t, the person on the bicycle has traveled 8t meters to the east, and the person in the car has traveled 18t meters to the north.
2. By the Pythagorean theorem, the distance between them is the hypotenuse of a right-angled triangle formed by their positions. So, the distance between them can be calculated as follows:
distance = √((100 + 8t)^2 + (18t)^2)
Now, let's find the time when the distance between the bicycle and the car is at a minimum.
To find the minimum, we can take the derivative of the function with respect to t, set it equal to zero, and solve for t.
1. Differentiate the distance function with respect to t:
d(distance)/dt = (1/2) * (100 + 8t)^(-1/2) * 2 * 8 * (100 + 8t) * 8 + (1/2) * (18t)^(-1/2) * 2 * 18 * 18
= (64 / √(100 + 8t)) + (324 / √(18t))
2. Set the derivative equal to zero:
(64 / √(100 + 8t)) + (324 / √(18t)) = 0
Now we can solve this equation for t.
1. Multiply both sides of the equation by √(100 + 8t) * √(18t) to eliminate the radicals:
64 * √(18t) + 324 * √(100 + 8t) = 0
2. Square both sides of the equation to eliminate the square roots:
(64 * √(18t) + 324 * √(100 + 8t))^2 = 0
3. Expand and simplify the equation:
18t * 64^2 + 2 * 64 * 324 * √(18t) * √(100 + 8t) + (100 + 8t) * 324^2 = 0
4. Simplify further:
18t * 64^2 + 648 * 64 * √(18t * (100 + 8t)) + (100 + 8t) * 324^2 = 0
Now we can solve this equation to find the value of t when the distance between the bicycle and the car is at a minimum.
Once we find the value of t, we can substitute it back into the distance function to calculate the minimum distance.