# Precalculus Circular Motion Question?

Elend stands at the western most point of a circular forest of radius 65 miles. Vin is 39  miles north and 2 miles west of Elend. Vin drives due east for 100 miles, then turns and drives due south for 100 miles. She drives at a constant speed of 65 miles per hour. How much time does Vin spend inside the forest? Give your answer in minutes.

https://sites.math.washington.edu/~m120/Win18/ostr... Relevance

All distances in units of miles.

A clear, diagram (with xy axes and roughly to scale) will help you a lot.

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Centre of circle is O(0,0). Equation of circle (edge of forest) is x² + y² = 65².

E is (-65, 0).

Vin’s initial position: A = (-65-2, 0+39) = (-67, 39).

Vin’s position after travelling from A, 100 east is B = (-67+100, 39)  = (33, 39).

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All points on AB have y=39 so AB intersects circle when x² +39² = 65²

x = ±√(65² – 39²) = ±52

AB first enters circle on left at P(-52, 39), Distance AP = 67 - 52 = 15

AB’S projection leaves circle on right at Q(52, 39) and we know B is (33, 39).

This means B is inside circle and the part of AB inside circle has length PB = AB -AP = 100 – 15 = 85.

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Vin’s position after travelling from B 100 south is C =(33, 39-100) = (33, -61)

All points on BC have x=33 so BC intersects circle when 33² +y² = 65²

y = ±√(65² – 33²) = ±56 (positive solution clearly not relevant).

So BC leaves circle at R(33, -56). Since B is (33, 39) this means the part of BC inside circle is length 39 +56 = 95

Total distance travelled inside circle = 85 + 95 = 180

Time taken = 180miles/(65 miles per hour)

= (180*60/65) mins = 166mins (rounded)

Check my working/arithmetic though

• let the circle be x² + y² = 65² so E(-65 , 0) and V(-67 , 39)....Vin meets the circle at ( -52 , 39)...a distance of 13miles , and turns at (33,39), a distance of 85 miles in the 'forest '. Turns { without loosing speed } and travels to (33 , - 61) , a distance of ≈100 miles.Thus leaves the forest at ( 33 ,  - 56 ). Hence the distance inside the ' forest ' is ( 85 + 95). Divide by 65 to get the hours.