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Alexxandr [17]
2 years ago
12

A rectangular prism is sliced by a horizontal plane. What is the shape of the cross-section?

Mathematics
1 answer:
Leni [432]2 years ago
5 0
The shape would be a rectangle
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A cardboard box without a lid is to have a volume of 8,788 cm3. Find the dimensions that minimize the amount of cardboard used.
Montano1993 [528]

Answer:

  x = y = 26 cm; z = 13 cm

Step-by-step explanation:

The generic solution for the minimum material in an open-top box is that the box is square and half as high as it is wide. It is half a cube of twice the volume.

The dimensions of the square base are ...

  ∛(2·8788 cm³) = 26 cm

Then the height is half that, or 13 cm.

  x = y = 26 cm; z = 13 cm

_____

If you need to see the development, you can use the method of Lagrange multipliers to find the minimum area for the given volume;

  area = xy +2(xz +yz)

  volume = xyz = 8788

We require each of the partial derivatives of L with respect to x, y, z, and λ to be zero.

  L = xy +2(xz +yz) +λ(xyz -8788)

  partial with respect to x: 0 = y+2z +λyz

  partial with respect to y: 0 = x +2z +λxz

  partial with respect to z: 0 = 2x+2y +λxy

  partial with respect to λ: 0 = xyz -8788

From the first two equations, we have ...

  λ = (y +2z)/(yz) = 1/z +2/y

  λ = (x +2z)/(xz) = 1/z +2/x

Equating these expressions for λ, we find ...

  1/z +2/y = 1/z +2/x   ⇒   x = y

The third equation then tells us ...

  λ = (2x +2y)/(xy) = 2/y +2/x

Comparing this to either of the first two expressions for λ, we see ...

  1/z +2y = 2/x +2/y   ⇒   z = x/2

This is the result we started the answer with:

  x = y = 2z = ∛(2·8788 cm³)

7 0
3 years ago
Simplify (4n^4 - 3n^2 + n)+ (2n^4 -7n^2 + 9n). Express your answer without factoring.
Pie

Answer:

6n^4-10n^2+10n

Step-by-step explanation:

4 n^4+2n^4-3n^2-7n^2+9n+n

6n^4-10n^2+10n

4 0
3 years ago
Read 2 more answers
Gavin goes to the market and buys one rectangle shaped board. The length of the board is 16 cm and width is 10 cm. if he wants t
Ira Lisetskai [31]
Check the picture below.

doesn't that make it just a 20 x 14?  well, surely you know what that area is.

8 0
2 years ago
Why do elephants have pointy tails
Naddika [18.5K]

Answer:

Balance -

Step-by-step explanation:

Long tail is most likely to a fast and wanna light animal or to jump and climb long tail needs.Buy in case of elephants short tail

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2 years ago
Lake in East Texas was stocked with 3,000 trout in the year 2000. The number of fish in the lake is modeled by f(x) = 3000 + 100
krok68 [10]

Answer:

Step-by-step explanation:

A

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3 years ago
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