Question
A coordinate plane is placed over the map of a town. A library is located at ( – 5, 1), and a
museum is located at (3,5). What is the distance, to the nearest tenth, from the library to the
museum?
museum is located at (3,5). What is the distance, to the nearest tenth, from the library to the
museum?
Answers
GPT-4o mini
To find the distance between the library located at the point \((-5, 1)\) and the museum located at the point \((3, 5)\), we can use the distance formula, which is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
In this case, the coordinates are:
- \( (x_1, y_1) = (-5, 1) \)
- \( (x_2, y_2) = (3, 5) \)
Plugging in the coordinates into the distance formula:
\[
d = \sqrt{(3 - (-5))^2 + (5 - 1)^2}
\]
\[
= \sqrt{(3 + 5)^2 + (5 - 1)^2}
\]
\[
= \sqrt{(8)^2 + (4)^2}
\]
\[
= \sqrt{64 + 16}
\]
\[
= \sqrt{80}
\]
\[
= \sqrt{16 \cdot 5}
\]
\[
= 4\sqrt{5}
\]
To find \(4\sqrt{5}\) to the nearest tenth, we can approximate \(\sqrt{5}\). The value of \(\sqrt{5} \approx 2.236\), so:
\[
4\sqrt{5} \approx 4 \cdot 2.236 \approx 8.944
\]
Rounding \(8.944\) to the nearest tenth gives us:
\[
\boxed{8.9}
\]
Thus, the distance from the library to the museum is approximately \(8.9\) units.
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
In this case, the coordinates are:
- \( (x_1, y_1) = (-5, 1) \)
- \( (x_2, y_2) = (3, 5) \)
Plugging in the coordinates into the distance formula:
\[
d = \sqrt{(3 - (-5))^2 + (5 - 1)^2}
\]
\[
= \sqrt{(3 + 5)^2 + (5 - 1)^2}
\]
\[
= \sqrt{(8)^2 + (4)^2}
\]
\[
= \sqrt{64 + 16}
\]
\[
= \sqrt{80}
\]
\[
= \sqrt{16 \cdot 5}
\]
\[
= 4\sqrt{5}
\]
To find \(4\sqrt{5}\) to the nearest tenth, we can approximate \(\sqrt{5}\). The value of \(\sqrt{5} \approx 2.236\), so:
\[
4\sqrt{5} \approx 4 \cdot 2.236 \approx 8.944
\]
Rounding \(8.944\) to the nearest tenth gives us:
\[
\boxed{8.9}
\]
Thus, the distance from the library to the museum is approximately \(8.9\) units.