On paper, the capital of a small city corresponds to the origin . A radio transmitter, located 1 mile east and
4 miles south of the capital, broadcasts in a mile radius. If you drive on a straight road from a city 4 miles
south of the capita
signal from the transmitter? Box your answer.
Problem 3: (5 noints)

Question

On paper, the capital of a small city corresponds to the origin

. A radio transmitter, located 1 mile east and
4 miles south of the capital, broadcasts in a

mile radius. If you drive on a straight road from a city 4 miles
south of the capita
signal from the transmitter? Box your answer.
Problem 3: (5 noints)

UpStudy AI · Accepted Answer

To solve this problem, we need to determine the distance from the point where you start driving (4 miles south of the capital) to the radio transmitter, and then check if this distance is less than or equal to the radius of the signal from the transmitter.

### Step 1: Identify the coordinates of the capital and the transmitter

- The capital is at the origin: $$ (0, 0) $$.
- The transmitter is located 1 mile east and 4 miles south of the capital. Therefore, its coordinates are:
  $$
  (1, -4)
  $$

### Step 2: Identify the starting point of the drive

- The starting point is 4 miles south of the capital, which corresponds to the coordinates:
  $$
  (0, -4)
  $$

### Step 3: Calculate the distance from the starting point to the transmitter

We can use the distance formula to find the distance $$ d $$ between the two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$:
$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$
Substituting the coordinates of the starting point $$ (0, -4) $$ and the transmitter $$ (1, -4) $$:
$$
d = \sqrt{(1 - 0)^2 + (-4 - (-4))^2} = \sqrt{(1)^2 + (0)^2} = \sqrt{1} = 1 \text{ mile}
$$

### Step 4: Compare the distance to the radius of the signal

The radius of the signal from the transmitter is $$ \sqrt{15} $$ miles. We need to check if the distance $$ d $$ is less than or equal to the radius:
$$
1 \leq \sqrt{15}
$$

Since $$ \sqrt{15} $$ is approximately $$ 3.87 $$, the condition holds true.

### Conclusion

You will receive the signal from the transmitter when you drive on a straight road from the city 4 miles south of the capital.

**Final Answer: Yes, you will receive the signal.**
You will receive the signal from the transmitter when you drive from the city 4 miles south of the capital.

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