Question

Please do not take up unless you are specialized in pre calculus and/or trig

Answer 1

The steps below will guide us to the answer

Step 1: Interprete the 49 miles radius

It was given that the radio broadcast in a 49 miles radius. This means that radio is broadcasting within a circle with radius of 49miles.

Step 2: Interpretation of the 55 miles north of the transmitter

Driving along a straight line from a city 55miles north of the transmitter, means a disatnce positive of the y-axis (i.e. point (0, 55))

Step 3: Interpretation of the 57 miles east of the transmitter

Driving 57 miles east of the transmitter, means a distance positive of the x-axis (i.e. (57, 0))

Step 4: Represent the information on x-y cartesian plane

This information can be represented using equation of a circle with radius 49 on a x-y cartesian plane as shown below:

Step 5: Locate the points where the line cut the circle

The above diagram showed that the signal would be picked up at the point where the line cust the cirle.

The line cut the cicle at points A(6.694, 48.541) and B(48.271, 8.423)

Step 6: Calculate the distance AB

The distance between these points is

`AB=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}`
`AB=\sqrt[]{(8.423-48.541)^2+(48.271-6.694)^2}`
`AB=\sqrt[]{1609.4539+1728.6469}`
`\begin{gathered} AB=\sqrt[]{3338.1008} \\ AB=57.7763\text{miles} \end{gathered}`

Therefore, the signal would be picked up at 57.78 miles or approximately 58miles