To determine the coordinates of the treasure, we need to partition the distance between the rock and the tree in a 5:9 ratio. The formula to find the x and y coordinates of the treasure is given by:
[tex]\[
\begin{array}{l}
x = \left(\frac{m}{m + n}\right) \left(x_2 - x_1\right) + x_1 \\
y = \left(\frac{m}{m + n}\right) \left(y_2 - y_1\right) + y_1
\end{array}
\][/tex]
In this problem, the values are given as:
- [tex]\( m = 5 \)[/tex]
- [tex]\( n = 9 \)[/tex]
- [tex]\( (x_1, x_2) = (2, 14) \)[/tex]
- [tex]\( (y_1, y_2) = (3, 17) \)[/tex]
### Step-by-step Solution:
1. First, calculate the ratio:
[tex]\[
\frac{m}{m+n} = \frac{5}{5+9} = \frac{5}{14}
\][/tex]
2. Use this ratio to calculate the x-coordinate of the treasure:
[tex]\[
x = \left( \frac{5}{14} \right) \left(14 - 2\right) + 2
\][/tex]
Simplify inside the parentheses:
[tex]\[
x = \left( \frac{5}{14} \right) \times 12 + 2
\][/tex]
Perform the multiplication:
[tex]\[
x = \left( \frac{60}{14} \right) + 2
\][/tex]
Simplify the fraction:
[tex]\[
x = \left( 4.285714285714286 \right) + 2
\][/tex]
Add:
[tex]\[
x = 6.285714285714286
\][/tex]
Round to the nearest tenth:
[tex]\[
x = 6.3
\][/tex]
3. Now, use the same ratio to calculate the y-coordinate of the treasure:
[tex]\[
y = \left( \frac{5}{14} \right) \left(17 - 3\right) + 3
\][/tex]
Simplify inside the parentheses:
[tex]\[
y = \left( \frac{5}{14} \right) \times 14 + 3
\][/tex]
Perform the multiplication:
[tex]\[
y = 5 + 3
\][/tex]
Add:
[tex]\[
y = 8
\`
Round to the nearest tenth:
\[
y = 8.0
\][/tex]
Thus, the coordinates of the treasure are [tex]\((6.3, 8.0)\)[/tex].
### Answer:
The coordinates of the treasure are [tex]\((6.3, 8.0)\)[/tex]. This is not an exact match with any of the provided options, but based on the detailed calculation, it is the correct answer.
