Question

what variable represents the horizontal difference between two points in the distance formula?
a. z
b. δy
c. δx
d. h

which component of the distance formula represents the vertical difference between two points?
a. δa
b. δs
c. δz
d. δy

in logistics, the distance formula is used to optimize:
a. delivery routes
b. warehouse temperatures
c. vehicle load capacities
d. package sizes

what is the primary role of the distance formula in computer graphics?
a. sound engineering
b. text formatting
c. color rendering
d. object positioning

calculate the distance between the points (1, 2) and (4, 6)
a. 2 units
b. 3 units
c. 4 units
d. 5 units

the distance between (10, 15) and (2, 9) is:
a. 10 units
b. 9 units
c. 11 units
d. 12 units

which pair of coordinates would have the longest distance between them based on the distance formula?

Explanation:

Step1: Recall distance - formula concepts

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(\Delta x=x_2 - x_1\) represents the horizontal difference and \(\Delta y=y_2 - y_1\) represents the vertical difference.

Step2: Answer the first question

The variable that represents the horizontal difference between two points in the distance formula is \(\Delta x\). So the answer to the first question is c. \(\Delta x\).

Step3: Answer the second question

The component of the distance formula that represents the vertical difference between two points is \(\Delta y\). So the answer to the second question is d. \(\Delta y\).

Step4: Consider logistics application

In logistics, the distance formula is used to optimize delivery - routes. So the answer to the third question is a. Delivery routes.

Step5: Think about computer - graphics application

In computer graphics, the primary role of the distance formula is in object positioning. So the answer to the fourth question is d. Object positioning.

Step6: Calculate distance for points \((1,2)\) and \((4,6)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(x_1 = 1,y_1 = 2,x_2 = 4,y_2 = 6\). Then \(d=\sqrt{(4 - 1)^2+(6 - 2)^2}=\sqrt{3^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5\). So the answer to the fifth question is d. 5 units.

Step7: Calculate distance for points \((10,15)\) and \((2,9)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(x_1 = 10,y_1 = 15,x_2 = 2,y_2 = 9\). Then \(d=\sqrt{(2 - 10)^2+(9 - 15)^2}=\sqrt{(-8)^2+(-6)^2}=\sqrt{64 + 36}=\sqrt{100}=10\). So the answer to the sixth question is a. 10 units.

Answer:

1. c. \(\Delta x\)
2. d. \(\Delta y\)
3. a. Delivery routes
4. d. Object positioning
5. d. 5 units
6. a. 10 units