Question

A small radio transmitter broadcasts in a 64 mile radius. If you drive along a straight line from a city 71 miles north of the transmitter to a second city 87 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter

Answer 1

Delay time traveling 50.677 miles

Explanation:

Assuming that the transmitter is at the origin (0, 0), then we have to:

City to the north (0.71)

East City (87.0)

The slope would then be:

m = - (71/87)

m = -0.816

To calculate B, the intercept we have to:

B = y - m * x =

B = 71 - (-0.816) * (0)

B = 71

Therefore the equation of the line between cities is

y = - 0.816 * x + 71

We also have the following equation, which would be that of the radio of the transmitter.

x ^ 2 + y ^ 2 = 64 ^ 2

Replacing the value of y, we are left with:

x ^ 2 + (- 0.816 * x + 71) ^ 2 = 64 ^ 2

We graph this on desmos.com

And we have that the distance would come being the intercepts, which would be

9.435 and 60.112

time will be what it takes to travel then this distance:

60.112 - 9.435 = 50.677

Answer 2

Step-by-step explanation

Suppose the transmitter is at the origin of a co-ordinate system. Then the range of the signal is the interior of a circle with the equation

x^2 + y^2 = 64^2.

The route the driver would take is from

City to the north at (0,71) as (x1,y1)

City to the east at (87,0) as (x2,y2), which is a line with the equation

y = mx+c where m is the slope

Slop m = (y2--y1)/(x2-x1) = (0-71)/(87-0) = -71/87

Intercept C = y - mx = 71 - (-71/87)(0) = 71

So the equation of line is y = (-71/87)x+71

Now we need to find the points of intersection of the circle and the line,

So substitute y = -(71/87)*x + 71 in the equation of circle

which is x^2 + y^2 = 64^2:

x^2 + (-(71/87)*x + 71)^2 = 64^2

x^2 + (71/87)^2 * x^2 - (2*(71/87)*71x) + 71^2 = 64^2

(1 + (71/87)^2)*x^2 - (10082/87)*x + (71^2 - 64^2) = 0

(12610/7569)*x^2 - (10082/87)*x + 945 = 0.

Use the quadratic formula: to find the roots of the equation

We get,

x = 9.437 to x = 59.957 miles.

So the signal will be picked up from x = 24.098 to x = 45.794 miles.