Question

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 transmitter 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?

Answer 1

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.

Answer 2

To solve the given problem first we need to find the equation for line then we have to find the intersection point on circle.

You will pick up a signal from the transmitter in 29% of the drive.

Given:

The radius is 53 mile.

The origin point of transmitter is $(0,0)$.

Calculate the slope of line with point $(0,70)$ and $(74,0)$.

$m=\dfrac{\rm change \:in\ Y}{\rm change \:in\ X}$

Substitute the value.

$m=\dfrac{-70}{74}=\dfrac{-35}{37}$

Write the general equation of line.

$y=mx+c$

Now we got the equation of line,

$y=\dfrac{-35}{37}x+70$

Write the transmitter reach the area enclosed by the next circle.

$x^2+y^2=53^2$

Substitute the value of $y$.

$x^2 + \left[\dfrac{-35}{37}x + 70\right]^2 = 53^2$

Further solving we get the point $(24.1, 47.2)$ and $(45.8, 26.7)$.

Please refer the attached figure.

Apply the Pythagoras theorem to find the total distance.

$d_1=\sqrt{70^2+74^2}=101.9\:\rm miles$

Apply the Pythagoras theorem to find distance when the signal is picked up.

$d_2 =\sqrt{45.8 - 24.1)^2 + (47.2 - 26.7)^2} \\d_2=29.9\:\rm miles$

Calculate the percentage of distance covered.

$\dfrac{d_1}{d_2}\times 100=\dfrac{101.9}{29.9}\times 100=29\%$

Thus, You will pick up a signal from the transmitter in 29% of the drive.

Learn more about what equation of line is here:

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