Question

every day a food truck sets up at a location that is conveniently half - way between the entrance and the exit. plot and label the location of the food truck. after getting lunch from the food truck, you decide to head to the milk jug. give the coordinates of the half - way point. explain how you know. are the bumper cars closer to the entrance, to the darts, or the same distance from both? give a convincing argument. a new ride, the intimidator, will be built half - way between the milk jug and ferris wheel. a. what is the x - coordinate of this new ride? b. what is the y - coordinate of this new ride? c. plot and label the intimidator on the map. the exit is exactly half - way between the ferris wheel and where you parked your car. give the coordinates of your parking spot.

Explanation:

Step1: Find food - truck location

The entrance is at (-3,0) and the exit is at (7,0). The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here, $x_1=-3,y_1 = 0,x_2=7,y_2 = 0$. So the mid - point (food truck location) is $(\frac{-3 + 7}{2},\frac{0+0}{2})=(2,0)$.

Step2: Find mid - point between food truck and Milk Jug

The food truck is at (2,0) and the Milk Jug is at (6,-5). Using the mid - point formula, $x=\frac{2 + 6}{2}=4$ and $y=\frac{0+( - 5)}{2}=-\frac{5}{2}$. So the mid - point is $(4,-\frac{5}{2})$.

Step3: Compare distances of Bumper Cars

The Bumper Cars are at (-2,-3), the entrance is at (-3,0) and the Darts are at (0,-6).
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
Distance between Bumper Cars and Entrance: $d_1=\sqrt{(-2+3)^2+(-3 - 0)^2}=\sqrt{1 + 9}=\sqrt{10}$.
Distance between Bumper Cars and Darts: $d_2=\sqrt{(-2-0)^2+(-3 + 6)^2}=\sqrt{4 + 9}=\sqrt{13}$. Since $\sqrt{10}<\sqrt{13}$, the Bumper Cars are closer to the Entrance.

Step4: Find coordinates of Intimidator

The Milk Jug is at (6,-5) and the Ferris Wheel is at (1,6).
a. The x - coordinate of the mid - point (Intimidator) is $\frac{6 + 1}{2}=\frac{7}{2}$.
b. The y - coordinate of the mid - point (Intimidator) is $\frac{-5+6}{2}=\frac{1}{2}$.
c. Plot the point $(\frac{7}{2},\frac{1}{2})$ and label it as Intimidator.

Step5: Find parking spot coordinates

The exit is at (7,0) and the Ferris Wheel is at (1,6). Let the parking spot be $(x,y)$. Using the mid - point formula, $7=\frac{1 + x}{2}$ and $0=\frac{6 + y}{2}$.
From $7=\frac{1 + x}{2}$, we get $14=1 + x$, so $x = 13$. From $0=\frac{6 + y}{2}$, we get $y=-6$. So the parking spot is at (13,-6).

Answer:

1. Food truck location: (2,0)
2. Mid - point between food truck and Milk Jug: $(4,-\frac{5}{2})$
3. Bumper Cars are closer to the Entrance.
4. a. x - coordinate of Intimidator: $\frac{7}{2}$

b. y - coordinate of Intimidator: $\frac{1}{2}$
c. Plot the point $(\frac{7}{2},\frac{1}{2})$ and label it as Intimidator.

1. Parking spot coordinates: (13,-6)