Question

Whenever he visits Danville, Tony has to drive 5 miles due north from home. Whenever he visits Bluepoint, he has to drive 8 miles due east from home. How far apart are Danville and Bluepoint, measured in a straight line? If necessary, round to the nearest tenth.

Answer 1

To find the straight-line distance between Danville and Bluepoint, we can represent the locations using a coordinate system.

Let's assume:
- Tony's home is at the origin (0, 0).
- Danville is located at (0, 5) since it is 5 miles due north.
- Bluepoint is located at (8, 0) as it is 8 miles due east.

Now we need to calculate the straight-line distance between Danville (0, 5) and Bluepoint (8, 0) using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting in the coordinates of Danville and Bluepoint:
- \(x_1 = 0\), \(y_1 = 5\)
- \(x_2 = 8\), \(y_2 = 0\)

Now plug in the values:

\[
d = \sqrt{(8 - 0)^2 + (0 - 5)^2}
\]
\[
d = \sqrt{8^2 + (-5)^2}
\]
\[
d = \sqrt{64 + 25}
\]
\[
d = \sqrt{89}
\]

Calculating the square root:

\[
d \approx 9.433981
\]

Rounding to the nearest tenth:

\[
d \approx 9.4
\]

Therefore, the straight-line distance between Danville and Bluepoint is approximately \( \boxed{9.4} \) miles.