Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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]
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]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.