Find solutions to your problems with the expert advice available on IDNLearn.com. Join our interactive community and access reliable, detailed answers from experienced professionals across a variety of topics.
Sagot :
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.
[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.
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.