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3 years ago

11

An ant needs to travel along a 20cm × 20cm cube to get from point A to point B. What is the shortest path he can take, and how l

ong will it be (in cm)?

1 answer:

malfutka [58]3 years ago

6 0

Answer:

1. Along a diagonal, and then along an edge.

2. 48.28 cm.

Step-by-step explanation:

Assume points A and B are at diagonally opposite corners, as in the diagram below.

1. Shortest path

The shortest path is to go along the diagonal AC and continue along the edge CB.

2. Distance

The distance travelled is

d = AC + CB

According to Pythagoras,

AC² = AD² + CD² = 20² + 20² = 400 + 400 = 800

AC = √800 = 20√2

d = AC + CB = 20√ 2+ 20 ≈ 2 × 1.414 + 20 = 48.28 cm

The ant travels 48.28 cm.

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The corners of a meadow are shown on a coordinate grid. Ethan wants to fence the meadow. What length of fencing is required?

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Step-by-step explanation:

The lenght of fencing required is the total distance between point A to B, B to C, C to D, and D to A. That is the distance between all 4 corners of the meadow.

The coordinates of the corners of the meadow is shown on a coordinate plane in the attachment. (See attachment below).

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Distance between point A(-6, 2) and point B(2, 6):

$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Let,

$A(-6, 2)) = (x_1, y_1)$

$B(2, 6) = (x_2, y_2)$

$AB = \sqrt{(2 - (-6))^2 + (6 - 2)^2}$

$AB = \sqrt{(8)^2 + (4)^2}$

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$AB = 8.9$ (nearest tenth)

Distance between B(2, 6) and C(7, 1):

$BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Let,

$B(2, 6) = (x_1, y_1)$

$C(7, 1) = (x_2, y_2)$

$BC = \sqrt{(7 - 2)^2 + (1 - 6)^2}$

$BC = \sqrt{(5)^2 + (-5)^2}$

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$BC = 7.1$ (nearest tenth)

Distance between C(7, 1) and D(3, -5):

$CD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Let,

$C(7, 1) = (x_1, y_1)$

$D(3, -5) = (x_2, y_2)$

$CD = \sqrt{(3 - 7)^2 + (-5 - 1)^2}$

$CD = \sqrt{(-4)^2 + (-6)^2}$

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Distance between D(3, -5) and A(-6, 2):

$DA = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Let,

$D(3, -5) = (x_1, y_1)$

$A(-6, 2) = (x_2, y_2)$

$DA = \sqrt{(-6 - 3)^2 + (2 - (-5))^2}$

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Length of fencing required = 8.9 + 7.1 + 7.2 + 11.4 = 34.6 units

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