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stealth61 [152]
3 years ago
14

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.

(Let x, y, and z be the dimensions of the cardboard box.)
Mathematics
1 answer:
Montano1993 [528]3 years ago
7 0

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)

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what is the rate of change of the linear function that has a graph that passes through the points (-5, -2) and (-2, 6)
sp2606 [1]

The rate of change of the linear function whose graph passes through points (-5, -2) and (-2, 6) is: 2.7.

<h3>What is the Rate of Change of a Linear Function?</h3>

The rate of change of a linear function is given as change in y over change in x.

Given the points:

(-5, -2) and (-2, 6)

Change in y = 6 -(-2) = 8 units

Change in x = -2 -(-5) = 3 units

Rate of change = 8/3 = 2.7.

Learn more about rate of change on:

brainly.com/question/8728504

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2 years ago
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Vinil7 [7]

Answer:

Value 1: -2

Value 2: 10

Step-by-step explanation:

To make y negative, -2 ≤ x ≤ ∞

In other words, x must be equal to or greater than -2 for the value of y to be negative.

Value 1: -2

y = -2(-2) - 5

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y = -20 - 5

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7 0
3 years ago
PLEASE HELP ME WITH THIS QUESTION
GREYUIT [131]

Answer:

x = 2√58

Step-by-step explanation:

Pythagoras' Theorem:  a² + b² = c²

(where a and b are the legs and c is the hypotenuse of a right triangle)

Find the height of the triangle by using the base and hypotenuse of the smaller interior triangle and Pythagoras' Theorem:

   a² + b² = c²

⇒ a² + 6² = (6√2)²

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⇒ a² = 36

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Now we have the base and height of the larger interior triangle, we can calculate x:

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7 0
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andre [41]

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So, we have to set y=0 in our equation and look for the positive solution. We have

-\dfrac{1}{14}x^2+6x+3=0 \iff x=42\pm\sqrt{1806}

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6 0
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GalinKa [24]
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