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What is the measurement of the longest line segment in a right rectangular prism that is 26 inches long, 2 inches wide, and 2 in

EastWind [94]

Answer:

$6\sqrt{19} \approx 26.153$ inches.

Step-by-step explanation:

The longest line segment in a right rectangular prism is the diagonal that connects two opposite vertices. On the first diagram attached, the green line segment connecting A and G is one such diagonals. The goal is to find the length of segment $\mathsf{AG}$ .

In this diagram (not to scale,) $\mathsf{AB} = 26$ (length of prism,) $\mathsf{AC} = 2$ (width of prism,) $\mathsf{AE} = 2$ (height of prism.)

Pythagorean Theorem can help find the length of $\mathsf{AG}$ , one of the longest line segments in this prism. However, note that this theorem is intended for right triangles in 2D, not the diagonal in a 3D prism. The workaround is to simply apply this theorem on two different right triangles.

Start by finding the length of line segment $\mathsf{AD}$ . That's the black dotted line in the diagram. In right triangle $\triangle\mathsf{ABD}$ (second diagram,)

Segment $\mathsf{AD}$ is the hypotenuse.
One of the legs of $\triangle\mathsf{ABD}$ is $\mathsf{AB}$ . The length of $\mathsf{AB}$ is $26$ , same as the length of this prism.
Segment $\mathsf{BD}$ is the other leg of this triangle. The length of $\mathsf{BD}$ is $2$ , same as the width of this prism.

Apply the Pythagorean Theorem to right triangle $\triangle\mathsf{ABD}$ to find the length of $\mathsf{AB}$ , the hypotenuse of this triangle:

$\mathsf{AD} = \sqrt{\mathsf{AB}^2 + \mathsf{BD}^2} = \sqrt{26^2 + 2^2}$ .

Consider right triangle $\triangle \mathsf{ADG}$ (third diagram.) In this triangle,

Segment $\mathsf{AG}$ is the hypotenuse, while
$\mathsf{AD}$ and $\mathsf{DG}$ are the two legs.

$\mathsf{AD} = \sqrt{26^2 + 2^2}$ . The length of segment $\mathsf{DG}$ is the same as the height of the rectangular prism, $2$ (inches.) Apply the Pythagorean Theorem to right triangle $\triangle \mathsf{ADG}$ to find the length of the hypotenuse $\mathsf{AG}$ :

$\begin{aligned}\mathsf{AG} &= \sqrt{\mathsf{AD}^2 + \mathsf{GD}^2} \\ &= \sqrt{\left(\sqrt{26^2 + 2^2}\right)^2 + 2^2}\\ &= \sqrt{\left(26^2 + 2^2\right) + 2^2} \\&= 6\sqrt{19} \\&\approx 26.153\end{aligned}$ .

Hence, the length of the longest line segment in this prism is $6\sqrt{19} \approx 26.153$ inches.

5 0

3 years ago

Party favors are on sale for $2.50 each. You have $380 to spend on the decorations and gifts, and you have already spent $272 on

Alenkinab [10]

Answer:

we could buy 43 party favors

Step-by-step explanation:

initially we have an amount of $380

if we already spend an amount of $272 this we have to subtract it from the total

$380 - $272 = $108

if each party favor comes out $ 2.50 and we have $ 108 we have to divide what we have by what each one comes out to know how many we can buy

$108 / $2.50 = 43.2

this means we could buy 43 party favors

3 0

3 years ago

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If a baking tray is 700mm by 600mm, how many of 25cm in diameter can fit into the baking tray?

Trava [24]

1cm=10mm
700mm= 70cm
600mm=60cm

70cm:25cm=2,8 - on length only 2 fit into

60cm:25=2,4 - on width only 2 fit into

Two 25cm in diameter fit into the baking tray.

7 0

3 years ago

A Gallup poll of 1236 adults showed that 12% of the respondents believe that it is bad luck to walk under a ladder. Consider th

SOVA2 [1]

Answer:

c

Step-by-step explanation:

8 0

3 years ago

The function t relates the age of a plant fossil, in years, to the percentage, c, of carbon-14 remaining in the fossil relative

pshichka [43]

Answer: For plato family

A fossil with 47% of its carbon-14 remaining is approximately __2998_ years old.

A fossil that is 7,000 years old will have approximately _17%__ of its carbon-14 remaining

Step-by-step explanation:

Just took the test

3 0

3 years ago

Read 2 more answers