A small radio transmitter broadcasts in a 53 mile radius. If you drive along a straight line from a city 70 miles north of the t
ransmitter to a second city 74 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?
1 answer:
Answer:
29% of the drive
Step-by-step explanation:
The plot of the problem is:
Transmitter at origin (0,0)
City to the north at (0,70)
City to the east at (74,0)
The path from city to city is a line with slope:
m = (-70)/74 = -35/37
and y-intercept at y = 70, so the equation is y = (-35/37)*x + 70.
The transmitter reach the area enclosed by the next circle:
x^2 + y^2 = 53^2
See the picture attached
The intersection is gotten from the picture or solving:
x^2 + [(-35/37)*x + 70]^2 = 53^2
the points approximately are: (24.1, 47.2) and (45.8, 26.7)
From Pythagorean theorem the total distance of the trip is:
d1 = √(70^2 + 74^2) ≈ 101.9 miles
And the distance when the signal is picked up is:
d2 =√ [(45.8 - 24.1)^2 + (47.2 - 26.7)^2] ≈ 29.9 miles
You will pick up a signal from the transmitter in (d2/d1)*100 = 29% of the drive.
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