A pilot has crashed in the Sahara Desert. She still has

Chapter 8, Problem 8.3.17

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A pilot has crashed in the Sahara Desert. She still has her maps and knows her position, but her radio is destroyed. Her only hope for rescue is to hike out to a highway that passes near her position. She needs to determine the closest point on the highway and how far away it is. a. The highway is a straight line passing through a point 15 miles due north of her and another point 20 miles due east. Draw a sketch of the situation on graph paper, placing the pilot at the origin and labeling the two points on the highway. b. Construct an equation that represents the highway (using x for miles east and y for miles north). c. Now use the Pythagorean Theorem to describe the square of the distance, d, of the pilot to any point (x, y) on the highway. d. Substitute the expression for y from part (b) into the equation from part (c) in order to write d2 as a quadratic in x. e. If we minimize d2, we minimize the distance d. So let D 5 d2 and write D as a quadratic function in x. Now find the minimum value for D. f. What are the coordinates of the closest point on the highway, and what is the distance, d, to that point?

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