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Anjali was given this problem:
A person stands at a distance of 12 meters east of an intersection and watches a car driving away from the intersection to the north at 4 meters per second. At a certain instant 
t_(0), the car is at distance 
n(t_(0)) of 9 meters from the intersection. What is the rate of change of the distance 
d(t) between the car and the person at that instant?
Which equation should Anjali use to solve the problem?
Choose 1 answer:
(A) 
tan[d(t)]=(n(t))/(12)
(B) 
d(t)=(12*n(t))/(2)
(C) 
d(t)+12+n(t)=180
(D) 
[d(t)]^(2)=12^(2)+[n(t)]^(2)

Anjali was given this problem:\newlineA person stands at a distance of 1212 meters east of an intersection and watches a car driving away from the intersection to the north at 44 meters per second. At a certain instant t0 t_{0} , the car is at distance n(t0) n\left(t_{0}\right) of 99 meters from the intersection. What is the rate of change of the distance d(t) d(t) between the car and the person at that instant?\newlineWhich equation should Anjali use to solve the problem?\newlineChoose 11 answer:\newline(A) tan[d(t)]=n(t)12 \tan [d(t)]=\frac{n(t)}{12} \newline(B) d(t)=12n(t)2 d(t)=\frac{12 \cdot n(t)}{2} \newline(C) d(t)+12+n(t)=180 d(t)+12+n(t)=180 \newline(D) [d(t)]2=122+[n(t)]2 [d(t)]^{2}=12^{2}+[n(t)]^{2}

Full solution

Q. Anjali was given this problem:\newlineA person stands at a distance of 1212 meters east of an intersection and watches a car driving away from the intersection to the north at 44 meters per second. At a certain instant t0 t_{0} , the car is at distance n(t0) n\left(t_{0}\right) of 99 meters from the intersection. What is the rate of change of the distance d(t) d(t) between the car and the person at that instant?\newlineWhich equation should Anjali use to solve the problem?\newlineChoose 11 answer:\newline(A) tan[d(t)]=n(t)12 \tan [d(t)]=\frac{n(t)}{12} \newline(B) d(t)=12n(t)2 d(t)=\frac{12 \cdot n(t)}{2} \newline(C) d(t)+12+n(t)=180 d(t)+12+n(t)=180 \newline(D) [d(t)]2=122+[n(t)]2 [d(t)]^{2}=12^{2}+[n(t)]^{2}
  1. Identify Relationship: Anjali needs to find the rate of change of the distance between the car and the person at a certain instant. To do this, she needs to use a relationship that connects the distance of the car from the intersection, the distance of the person from the intersection, and the distance between the car and the person. The Pythagorean theorem is a suitable choice for this situation because we are dealing with a right triangle where the car's path and the person's position form the two perpendicular sides of the triangle.
  2. Apply Pythagorean Theorem: The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, the hypotenuse is the distance between the car and the person, d(t)d(t), one side is the constant distance of the person east of the intersection, which is 1212 meters, and the other side is the distance of the car from the intersection, n(t)n(t). Therefore, the correct equation to represent this relationship is d(t)2=122+n(t)2d(t)^2 = 12^2 + n(t)^2.
  3. Choose Correct Equation: Looking at the answer choices, the equation that matches the Pythagorean theorem is (D) [d(t)]2=122+[n(t)]2[d(t)]^2 = 12^2 + [n(t)]^2. This equation will allow Anjali to find the distance d(t)d(t) at any time tt, and by differentiating this equation with respect to time, she can find the rate of change of the distance d(t)d(t) with respect to time, which is what the problem is asking for.

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