An ant crawls straight from $(-4,6)$ to the origin, and then continues straight on to $(4,3)$. How far does it travel?
To find the distance the ant travels, we can break down the journey into two steps - from $(-4, 6)$ to the origin $(0, 0)$, and then from the origin to $(4, 3)$.
Step 1: To find the distance from $(-4, 6)$ to $(0, 0)$, we can use the distance formula, which is given by:
Distance $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Plugging in the values, we get:
Distance $= \sqrt{(0 - (-4))^2 + (0 - 6)^2}$
Distance $= \sqrt{(4)^2 + (-6)^2}$
Distance $= \sqrt{16 + 36}$
Distance $= \sqrt{52}$
Step 2: To find the distance from $(0, 0)$ to $(4, 3)$, we use the distance formula again:
Distance $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Plugging in the values, we get:
Distance $= \sqrt{(4 - 0)^2 + (3 - 0)^2}$
Distance $= \sqrt{(4)^2 + (3)^2}$
Distance $= \sqrt{16 + 9}$
Distance $= \sqrt{25}$
Distance $= 5$
Therefore, the ant travels a total distance of $\sqrt{52} + 5$.
To find the distance traveled by the ant, we can break down the distance into two parts: the distance from $(-4, 6)$ to the origin, and the distance from the origin to $(4, 3)$.
To calculate the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, we can use the distance formula:
distance = $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Let's start with the first part, the distance from $(-4, 6)$ to the origin (0, 0).
Distance from $(-4, 6)$ to the origin = $\sqrt{(0 - (-4))^2 + (0 - 6)^2} \\
= \sqrt{4^2 + 6^2} \\
= \sqrt{16 + 36} \\
= \sqrt{52}$
Now let's calculate the second part, the distance from the origin to $(4, 3)$.
Distance from the origin to $(4, 3)$ = $\sqrt{(4 - 0)^2 + (3 - 0)^2} \\
= \sqrt{4^2 + 3^2} \\
= \sqrt{16 + 9} \\
= \sqrt{25} \\
= 5$
Finally, to find the total distance traveled by the ant, we add the distance from $(-4, 6)$ to the origin and the distance from the origin to $(4, 3)$:
Total distance traveled = $\sqrt{52} + 5 \\
\approx 7.2111 + 5 \\
\approx 12.2111$
Therefore, the ant travels approximately 12.2111 units in total.
first leg
4^2 + 6^2 = 16 + 36 = 54
sqrt 52 = 7.21
second leg
3,4,5 triangle so 5
7.21 + 5 = 12.2
That is how far it TRAVELLED (sum on the ant odometer)
However the straight line displacement from (-4,6) to (4,3) is
sqrt (8^2 + 3^2) = sqrt(64+9) = sqrt (73) = 8.54