Anatoli was given this problem: Two cars are driving towards an intersection from perpendicular directions. The first car's velocity is 2 meters per second and the second car's velocity is 9 meters per second. At a certain instant t_(0), the first car is a distance x(t_(0)) of 8 meters from the intersection and the second car is a distance y(t_(0)) of 6 meters from the intersection. What is the rate of change of the distance d(t) between the cars at that instant? Which equation should Anatoli use to solve the problem? Choose 1 answer: (A) tan[d(t)]=(y(t))/(x(t)) (B) d(t)=(x(t)*y(t))/(2) (C) d(t)+x(t)+y(t)=180 (D) [d(t)]^(2)=[x(t)]^(2)+[y(t)]^(2)

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Anatoli was given this problem: Two cars are driving towards an intersection from perpendicular directions. The first car's velocity is 2 meters per second and the second car's velocity is 9 meters per second. At a certain instant  t_(0), the first car is a distance  x(t_(0)) of 8 meters from the intersection and the second car is a distance  y(t_(0)) of 6 meters from the intersection. What is the rate of change of the distance  d(t) between the cars at that instant? Which equation should Anatoli use to solve the problem? Choose 1 answer: (A)  tan[d(t)]=(y(t))/(x(t)) (B)  d(t)=(x(t)*y(t))/(2) (C)  d(t)+x(t)+y(t)=180 (D)  [d(t)]^(2)=[x(t)]^(2)+[y(t)]^(2)

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Apply Pythagorean Theorem: To find the rate of change of the distance between the two cars, we need to use the Pythagorean theorem because the cars are moving along perpendicular paths. The Pythagorean theorem relates the sides of a right triangle, which in this case are the distances x(t) and y(t) from the intersection, to the hypotenuse, which is the distance d(t) between the cars.Derive Equation: The correct equation that represents the Pythagorean theorem is d(t)^2 = x(t)^2 + y(t)^2. This equation will allow us to find the distance d(t) between the cars at any time t.Differentiate with Respect to Time: We can differentiate both sides of the equation with respect to time t to find the rate of change of d(t). This will give us the derivative of d(t)^2 with respect to t, which is 2d(t) * (dd(t)/dt), and the derivative of x(t)^2 + y(t)^2 with respect to t, which is 2x(t) * (dx(t)/dt) + 2y(t) * (dy(t)/dt).Substitute Velocities: We know the velocities of the cars, which are the rates of change of x(t) and y(t) with respect to time t. The first car's velocity is 2 meters per second, so dx(t)/dt = -2 (negative because it is approaching the intersection), and the second car's velocity is 9 meters per second, so dy(t)/dt = -9 (also negative because it is approaching the intersection).Simplify the Equation: Substituting the known values into the differentiated equation, we get 2d(t) * (dd(t)/dt) = 2 * 8 * (-2) + 2 * 6 * (-9). This simplifies to 2d(t) * (dd(t)/dt) = -16 - 54, which further simplifies to 2d(t) * (dd(t)/dt) = -70.Find d(t_0): To find dd(t)/dt, which is the rate of change of the distance between the cars, we need to divide both sides of the equation by 2d(t). However, we first need to find the value of d(t) at the instant t_0 using the Pythagorean theorem: d(t_0)^2 = 8^2 + 6^2, which simplifies to d(t_0)^2 = 64 + 36, and further simplifies to d(t_0)^2 = 100.Calculate dd(t)/dt: Taking the square root of both sides of the equation d(t_0)^2 = 100 gives us d(t_0) = 10 meters.Divide to Find Rate: Now we can find dd(t)/dt by dividing -70 by 2 * 10, which gives us dd(t)/dt = -70 / 20, and this simplifies to dd(t)/dt = -3.5 meters per second. This is the rate of change of the distance between the cars at the instant t_0.Final Equation: The correct equation that Anatoli should use to solve the problem is therefore (D) d(t)^2 = x(t)^2 + y(t)^2, because it is the Pythagorean theorem applied to the situation described.

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