Answer:
11.82 mph
Step-by-step explanation:
The angle between the direction of travel and the direction to the city of Hartville is ...
arctan(180/49) ≈ 74.77°
The speed from the direction of Hartville is the the actual speed multiplied by the cosine of this angle:
speed from Hartville = (45 mph)×cos(74.77°) ≈ 11.82 mph
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<em>Comment on the solution</em>
There are other approaches you can use to solve this. One is to compute the change in distance over a small period of time, such as 0.02 hours.
0.01 hours before the time of interest, the distance to the city is ...
d1 = √(180² +(49-.01(45))²) ≈ 186.4326 . . . miles
0.01 hours after the time of interest, the distance to the city is ...
d2 = √(180² +(49+.01(45))²) ≈ 186.6690 . . . miles
Then the rate of change of distance is ...
(d2 -d1)/(t2 -t1) = (186.6690 -186.4326)/0.02 = 11.82 . . . mi/h
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Another is to write the distance equation and differentiate it. (You will find the solution looks very much like the trig solution above.)
d = √(180² +(45t)²)
dd/dt = 45²t/√(180² +(45t)²) . . . . . to be evaluated at t=49/45
rate of change = 45(49/√(180²+49²)) ≈ 11.82 . . . mi/h
