Let's determine the coordinates of the treasure using the provided partition ratios and coordinates.
1. First, let's set up the information we have:
- The coordinates of the rock are [tex]\((x_1, y_1) = (0, 0)\)[/tex].
- The coordinates of the tree are [tex]\((x_2, y_2) = (10, 15)\)[/tex].
- The ratio of the distances is [tex]\(m:n = 5:9\)[/tex].
2. The formula to find the coordinates that partition the segment in the given ratio is:
[tex]\[
\left( \frac{m}{m+n}(x_2 - x_1) + x_1, \frac{m}{m+n}(y_2 - y_1) + y_1 \right)
\][/tex]
3. Plug the values [tex]\(m = 5\)[/tex] and [tex]\(n = 9\)[/tex] into the formula:
Coordinates:
[tex]\[
x = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1
\][/tex]
[tex]\[
y = \left( \frac{m}{m+n} \right) (y_2 - y_1) + y_1
\][/tex]
4. Calculate the x-coordinate of the treasure:
[tex]\[
x = \left( \frac{5}{5+9} \right) (10 - 0) + 0 = \left( \frac{5}{14} \right) \cdot 10
\][/tex]
[tex]\[
x = \left( \frac{50}{14} \right) = 3.5714285714285716
\][/tex]
Rounding to the nearest tenth, we get:
[tex]\[
x \approx 3.6
\][/tex]
5. Calculate the y-coordinate of the treasure:
[tex]\[
y = \left( \frac{5}{5+9} \right) (15 - 0) + 0 = \left( \frac{5}{14} \right) \cdot 15
\][/tex]
[tex]\[
y = \left( \frac{75}{14} \right) = 5.357142857142857
\][/tex]
Rounding to the nearest tenth, we get:
[tex]\[
y \approx 5.4
\][/tex]
Thus, the coordinates of the treasure are:
[tex]\[
(x, y) = (3.6, 5.4)
\][/tex]
Therefore, based on the given options, none of them exactly match the coordinates calculated. The rounded coordinates of the treasure are:
[tex]\[
(3.6, 5.4)
\][/tex]
