A treasure map says that a treasure is buried so that it partitions the distance between a rock and a tree in a 5:9 ratio. Marina traced the map onto a coordinate plane to find the exact location of the treasure.
[tex]x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1[/tex]
[tex]v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1[/tex]
What are the coordinates of the treasure? If necessary, round the coordinates to the nearest tenth.
A. (11.4, 14.2)
B. (7.6, 8.8)
C. (5.7, 7.5)
D. (10.2, 12.6)
Answer (2)
Substitute the given values into the formulas for the x and y coordinates.
Calculate the x-coordinate: x = ( 5 + 9 5 ) ( 11.4 − 7.6 ) + 7.6 ≈ 9.0 .
Calculate the y-coordinate: y = ( 5 + 9 5 ) ( 14.2 − 8.8 ) + 8.8 ≈ 10.7 .
State the coordinates of the treasure: ( 9.0 , 10.7 ) .
Explanation
Problem Analysis and Setup We are given the coordinates of a rock ( x 1 , y 1 ) = ( 7.6 , 8.8 ) and a tree ( x 2 , y 2 ) = ( 11.4 , 14.2 ) . The treasure is buried along the line segment connecting these two points, partitioning the distance in a ratio of m : n = 5 : 9 . We are given the formulas to find the coordinates of the treasure ( x , y ) :
x = ( m + n m ) ( x 2 − x 1 ) + x 1 y = ( m + n m ) ( y 2 − y 1 ) + y 1
Our goal is to substitute the given values into these formulas and calculate the coordinates of the treasure.
Calculating the x-coordinate First, let's calculate the x-coordinate of the treasure. We substitute m = 5 , n = 9 , x 1 = 7.6 , and x 2 = 11.4 into the formula:
x = ( 5 + 9 5 ) ( 11.4 − 7.6 ) + 7.6 x = ( 14 5 ) ( 3.8 ) + 7.6 x = 14 5 × 3.8 + 7.6 x = 14 19 + 7.6 x ≈ 1.357 + 7.6 x ≈ 8.957 Rounding to the nearest tenth, we get x ≈ 9.0 .
Calculating the y-coordinate Next, let's calculate the y-coordinate of the treasure. We substitute m = 5 , n = 9 , y 1 = 8.8 , and y 2 = 14.2 into the formula:
y = ( 5 + 9 5 ) ( 14.2 − 8.8 ) + 8.8 y = ( 14 5 ) ( 5.4 ) + 8.8 y = 14 5 × 5.4 + 8.8 y = 14 27 + 8.8 y ≈ 1.929 + 8.8 y ≈ 10.729 Rounding to the nearest tenth, we get y ≈ 10.7 .
Final Answer Therefore, the coordinates of the treasure are approximately ( 9.0 , 10.7 ) .
Examples
In urban planning, determining the location of a new facility (like a park or community center) can involve considering its accessibility to different neighborhoods. If we want the facility to be located proportionally between two residential areas, we can use a weighted average based on population sizes to find the ideal location. For instance, if neighborhood A has 5000 residents and neighborhood B has 9000 residents, we would want the facility to be closer to neighborhood B. The formulas used to find the treasure's location are analogous to those used to find the weighted average location of the facility, ensuring equitable access for both communities.
To find the treasure's coordinates, we calculated the weighted average based on the 5:9 ratio between the rock at (7.6, 8.8) and the tree at (11.4, 14.2). The calculations give the coordinates of the treasure as approximately (9.0, 10.7). Among the options provided, we do not have an exact match, but option B is the closest.
;
