Chinedu was given this problem:A person stands at a distance of 18 meters east of an intersection and watches a car driving away from the intersection to the north at 3 meters per second. The function y(t) gives the car's distance, in meters, north of the intersection at time t. At a certain instant t0, the angle θ(t0) between the line of sight from the person to the car and the line of sight from the person to the intersection is 0.7 radians. What is the rate of change of the angle θ(t) at that instant?Which equation should Chinedu use to solve the problem?Choose 1 answer:(A) tan[θ(t)]=18y(t)(B) θ(t)=218⋅y(t)(C) θ(t)+18+y(t)=180(D) [θ(t)]2=182+[y(t)]2
Q. Chinedu was given this problem:A person stands at a distance of 18 meters east of an intersection and watches a car driving away from the intersection to the north at 3 meters per second. The function y(t) gives the car's distance, in meters, north of the intersection at time t. At a certain instant t0, the angle θ(t0) between the line of sight from the person to the car and the line of sight from the person to the intersection is 0.7 radians. What is the rate of change of the angle θ(t) at that instant?Which equation should Chinedu use to solve the problem?Choose 1 answer:(A) tan[θ(t)]=18y(t)(B) θ(t)=218⋅y(t)(C) θ(t)+18+y(t)=180(D) [θ(t)]2=182+[y(t)]2
Identify Relationship: Chinedu needs to find the relationship between the angle θ(t), the distance of the car from the intersection y(t), and the fixed distance of the person east of the intersection, which is 18 meters. To find the rate of change of the angle, we need to use a trigonometric function that relates these quantities. The tangent function relates an angle in a right triangle to the lengths of the opposite side and the adjacent side. Therefore, the correct equation should involve the tangent of the angle θ(t).
Apply Tangent Function: The tangent of the angle theta(t), which is tan[θ(t)], is equal to the opposite side over the adjacent side in a right triangle. In this scenario, the opposite side is the distance y(t) that the car has traveled north from the intersection, and the adjacent side is the fixed distance of 18 meters east of the intersection where the person is standing. Therefore, the correct equation is tan[θ(t)]=18y(t).
Verify Correct Equation: Now we need to check the given options to see which one correctly represents the tangent function as described. Option (A) tan[θ(t)]=18y(t) matches our description and is the correct equation to use for solving the problem. Options (B), (C), and (D) do not represent the tangent function correctly in the context of this problem.
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