Question

select the correct answer.
on a map where each unit represents one kilometer, two marinas are located at ( p(4,2) ) and ( q(8,12) ). if a boat travels in a straight line from one marina to the other, how far does the boat travel?

a. 14 kilometers
b. ( 2sqrt{29} ) kilometers
c. 6 kilometers
d. ( 2sqrt{5} ) kilometers

Explanation:

Step1: Recall distance formula

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Step2: Substitute the given points

Let $(x_1,y_1)=(4,2)$ and $(x_2,y_2)=(8,12)$. Substitute into the formula:
$d=\sqrt{(8-4)^2+(12-2)^2}$

Step3: Calculate differences inside the root

$d=\sqrt{(4)^2+(10)^2}$

Step4: Compute squared terms

$d=\sqrt{16+100}$

Step5: Sum and simplify the root

$d=\sqrt{116}=\sqrt{4\times29}=2\sqrt{29}$

Answer:

B. $2\sqrt{29}$ kilometers