Angela Mary and Tiff are all standing near the intersection

Angela, Mary, and Tiff are all standing near the intersection of University and 42nd streets. Mary and Tiff do not move, but Angela runs toward Tiff at 12 ft/sec along a straight line, as pictured. Assume the roads are 50 feet wide and Tiff is 50 feet north of the nearest comer Tiff Mary 42 nd St. Angela University Way Where is Angela lot ated wten she Es dosest to Mary? l et Angela\'s initial ccatian be the origin. R und your answers to two decimal plates, When does she reach this spot? (Round your answer to one decimal place.)

Solution

Angela\'s intial location is (0,0) and as the roads are 50\' wide then Mary\'s location is (0,50) and Tiff\' is at (50,110).
The line joining Angela to Tiff has the equation

y = 2.2x

[Hint - use slope intercept form and Tiff\'s coordinates ; 110 = 2.2 * 50]
Angela is closest to Mary when a perpendicular from the line y = 2.2x passes though the point where Mary is standing.
The product of the slopes of perpendicular lines is -1.
The slope of the perpendicular line is thus( -1/2.2)
In slope-intercept form the equation of the perpendicular line is y = mx + c.
Substituting -1/2.2 for m and using Mary\'s coordinates this gives, 50 = -0/2.2 + c.
Thus c = 50 and the line equation becomes y = -x/2.2 + 50.
At the point where the two lines intercept we can put, 2.2x = -x/2.2 + 50
Then, 4.84x = -x + 110 : 5.84x = 110 : x = 18.8356
Using (say) the initial line equation, the corresponding value of y is : y = 2.2x = 2.2*18.8356 = 41.4384
Use Pythagoras to determine the distance (L) travelled by Angela to get to this point.
L² = 18.8356² + 41.4384² = 354.78045 + 1717.1374
L = 45.518
As Angela is running at 12 ft/s then she covers this distance in 45.518/12 = 3.79 seconds.