Solved on Feb 02, 2024

Find the length of the longest diagonal in a right rectangular prism with dimensions 12.6 cm x 3.2 cm x 6 cm. The longest diagonal is $\sqrt{12.6^2 + 3.2^2 + 6^2}$ cm (rounded to the nearest tenth).

STEP 1

Assumptions
1. The right rectangular prism has dimensions of length $l = 12.6$ cm, width $w = 3.2$ cm, and height $h = 6$ cm.
2. The longest line segment in a right rectangular prism is the space diagonal, which connects two opposite vertices of the prism.
3. The space diagonal can be found using the Pythagorean theorem in three dimensions.

STEP 2

The formula for the space diagonal $d$ of a right rectangular prism is given by the 3D Pythagorean theorem:
$d = \sqrt{l^2 + w^2 + h^2}$

STEP 3

Now, plug in the given values for the length $l$ , width $w$ , and height $h$ to calculate the space diagonal.
$d = \sqrt{12.6^2 + 3.2^2 + 6^2}$

STEP 4

First, calculate the squares of the individual dimensions.
$12.6^2 = 158.76$ $3.2^2 = 10.24$ $6^2 = 36$

STEP 5

Now, add the squares of the dimensions together.
$Sum\, of\, squares = 158.76 + 10.24 + 36$

STEP 6

Calculate the sum of the squares.
$Sum\, of\, squares = 158.76 + 10.24 + 36 = 205$

STEP 7

Take the square root of the sum of the squares to find the length of the space diagonal.
$d = \sqrt{205}$

STEP 8

Calculate the square root.
$d \approx 14.3$

STEP 9

Round the result to the nearest tenth as needed.
$d \approx 14.3\, \mathrm{cm}$
The measurement of the longest line segment in the right rectangular prism is approximately $14.3 \mathrm{cm}$ .

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Find the length of the longest diagonal in a right rectangular prism with dimensions 12.6 cm x 3.2 cm x 6 cm. The longest diagonal is $\sqrt{12.6^2 + 3.2^2 + 6^2}$ cm (rounded to the nearest tenth).

STEP 1

STEP 2

The formula for the space diagonal $d$ of a right rectangular prism is given by the 3D Pythagorean theorem:
$d = \sqrt{l^2 + w^2 + h^2}$

STEP 3

Now, plug in the given values for the length $l$ , width $w$ , and height $h$ to calculate the space diagonal.
$d = \sqrt{12.6^2 + 3.2^2 + 6^2}$

STEP 4

First, calculate the squares of the individual dimensions.
$12.6^2 = 158.76$ $3.2^2 = 10.24$ $6^2 = 36$

STEP 5

Now, add the squares of the dimensions together.
$Sum\, of\, squares = 158.76 + 10.24 + 36$

STEP 6

Calculate the sum of the squares.
$Sum\, of\, squares = 158.76 + 10.24 + 36 = 205$

STEP 7

Take the square root of the sum of the squares to find the length of the space diagonal.
$d = \sqrt{205}$

STEP 8

Calculate the square root.
$d \approx 14.3$

STEP 9

Round the result to the nearest tenth as needed.
$d \approx 14.3\, \mathrm{cm}$
The measurement of the longest line segment in the right rectangular prism is approximately $14.3 \mathrm{cm}$ .

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Find the length of the longest diagonal in a right rectangular prism with dimensions 12.6 cm x 3.2 cm x 6 cm. The longest diagonal is 12.62+3.22+62\sqrt{12.6^2 + 3.2^2 + 6^2}12.62+3.22+62​ cm (rounded to the nearest tenth).

STEP 1

STEP 2

The formula for the space diagonal ddd of a right rectangular prism is given by the 3D Pythagorean theorem:d=l2+w2+h2d = \sqrt{l^2 + w^2 + h^2}d=l2+w2+h2​

STEP 3

Now, plug in the given values for the length lll, width www, and height hhh to calculate the space diagonal.d=12.62+3.22+62d = \sqrt{12.6^2 + 3.2^2 + 6^2}d=12.62+3.22+62​

STEP 4

First, calculate the squares of the individual dimensions.12.62=158.7612.6^2 = 158.7612.62=158.76 3.22=10.243.2^2 = 10.243.22=10.24 62=366^2 = 3662=36

STEP 5

Now, add the squares of the dimensions together.Sum of squares=158.76+10.24+36Sum\, of\, squares = 158.76 + 10.24 + 36Sumofsquares=158.76+10.24+36

STEP 6

Calculate the sum of the squares.Sum of squares=158.76+10.24+36=205Sum\, of\, squares = 158.76 + 10.24 + 36 = 205Sumofsquares=158.76+10.24+36=205

STEP 7

Take the square root of the sum of the squares to find the length of the space diagonal.d=205d = \sqrt{205}d=205​

STEP 8

Calculate the square root.d≈14.3d \approx 14.3d≈14.3

STEP 9

Round the result to the nearest tenth as needed.d≈14.3 cmd \approx 14.3\, \mathrm{cm}d≈14.3cmThe measurement of the longest line segment in the right rectangular prism is approximately 14.3cm14.3 \mathrm{cm}14.3cm.

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Find the length of the longest diagonal in a right rectangular prism with dimensions 12.6 cm x 3.2 cm x 6 cm. The longest diagonal is $\sqrt{12.6^2 + 3.2^2 + 6^2}$ cm (rounded to the nearest tenth).

The formula for the space diagonal $d$ of a right rectangular prism is given by the 3D Pythagorean theorem:
$d = \sqrt{l^2 + w^2 + h^2}$

Now, plug in the given values for the length $l$ , width $w$ , and height $h$ to calculate the space diagonal.
$d = \sqrt{12.6^2 + 3.2^2 + 6^2}$

First, calculate the squares of the individual dimensions.
$12.6^2 = 158.76$ $3.2^2 = 10.24$ $6^2 = 36$

Now, add the squares of the dimensions together.
$Sum\, of\, squares = 158.76 + 10.24 + 36$

Calculate the sum of the squares.
$Sum\, of\, squares = 158.76 + 10.24 + 36 = 205$

Take the square root of the sum of the squares to find the length of the space diagonal.
$d = \sqrt{205}$

Calculate the square root.
$d \approx 14.3$

Round the result to the nearest tenth as needed.
$d \approx 14.3\, \mathrm{cm}$
The measurement of the longest line segment in the right rectangular prism is approximately $14.3 \mathrm{cm}$ .