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An engineer is designing a storage compartment in a spacecraft .The compartment must be 2 meters longer than it is wide, and its depth must be 1 meter less than its width. The volume of the compartment must be 8 cubic meters.Write an equation to model the volume of the compartment. Determine the dimensions where it produces the maximum volume. Sketch the graph

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Answer:

Let x be the width of the compartment,

Then according to the question,

The length of the compartment = x + 2

And the depth of the compartment = x -1

Thus, the volume of the compartment,  [tex]V = (x+2)x(x-1) = x^3 + x^2 - 2x[/tex]

But, the volume of the compartment must be 8 cubic meters.

⇒ [tex]x^3 + x^2 - 2x = 8[/tex]

⇒ [tex]x^3 + x^2 - 2x - 8=0[/tex]

⇒ [tex](x-2)(x^2+3x+4)=0[/tex]

If [tex]x-2=0\implies x = 2[/tex] and if  [tex]x^2+3x+4=0\implies x = \text{complex number}[/tex]

But, width can not be the complex number.

Therefore, width of the compartment = 2 meter.

Length of the compartment = 2 + 2 = 4 meter.

And, Depth of the compartment = 2 - 1 = 1 meter.

Since, the function that shows the volume of the compartment is,

[tex]V(x) = x^3 + x^2 - 2x[/tex]

When we lot the graph of that function we found,

V(x) is maximum for infinite.

But width can not infinite,

Therefore, the maximum value of V(x) will be 8.




Ver imagen parmesanchilliwack

Answer:  The dimensions are 1 m × 2 m × 4 m.


Step-by-step explanation: Given that an Engineer is designing a storage compartment in a spacecraft. The length of the spacecraft is 2 m more than its width and depth is 1 m less than its width.

We need to find the dimensions of the compartment where it produces the maximum volume.

Let, 'b' m be the width of the compartment. Then, its length will be (b + 2) and depth will be (b-1). Since the volume of the compartment is 8 cubic metres, so

[tex]b\times (b+2)\times (b-1)=8\\\\\Rightarrow b(b^2+b-2)=8\\\\\Rightarrow b^3+b^2-2b-8=0\\\\\Rightarrow b^2(b-2)+3b(b-2)+4(b-2)=0\\\\\Rightarrow (b-2)(b^2+3b+4)=0.[/tex]

So, b = 2  or b² +3b + 4 = 0. Since the second quadratic equation will not give real roots and length of any thing cannot be imaginary, it must be real, so we will consider b = 2.

Hence, width = 2 m, length = 2+2 = 4 m  and depth = 2-1 = 1 m.

Thus, the equation which model the volume of the compartment is

[tex]b(b-2)(b-1)=0,[/tex]

and the dimensions are 1 m × 2 m × 4 m. Also, the sketch is attached herewith.


Ver imagen ColinJacobus

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