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rosijanka [135]
3 years ago
6

A designer is making a rectangular prism box with maximum volume, with the sum of its length, width, and height 8 in. The length

must be 2
times the width. What should each dimension be? Round to the nearest tenth of an inch if necessary.
Mathematics
1 answer:
Sever21 [200]3 years ago
6 0

Answer:

width = 1.8 in

length =   3.6 in

height =8-3w =8-3(1.8)= 2.6 in

Step-by-step explanation:

A designer is making a rectangular prism box with maximum volume, with the sum of its length, width, and height 8 in

Let l , w and h are the length , width and height

l +w+h=8\\l=2w

Plug it in the first equation

2w+w+h=8\\3w+h=8\\h=8-3w

volume the box is length times width times height

volume = (2w)(w)(8-3w)\\16w^2-6w^3

To get maximum volume we take derivatived

v'=32w-18w^2

set the derivative =0 and solve for w

0=32w-18w^2\\0=-2w(9w-16)\\w=0 , w= \frac{16}{9}

width = 1.8 in

length = 2w= 3.6 in

height =8-3w =8-3(1.8)= 2.6 in

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The answer: The 3 (three) consecutive odd integers are: -3, -1, 1.
_____________________________________________
Explanation:
_______________
To represent 3 (three consecutive odd integers):
____________________________________
Let "x" be the first odd integer.
____________________________________
Let "(x+2)" be next consecutive odd integer.
____________________________________
Let "(x+4") be the third odd integer.
____________________________________
The sum of these three consecutive odd integers:
__________________________________________
x + (x + 2) + (x + 4) = x + x + 2 + x + 4 = 3x + 6 ;
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Six ("6") times the sum of these 3 (three) consecutive odd integers =
_________________
6*(3x+6) = 6(3x + 6) = -18 (as given in the problem).
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Given: 6(3x + 6) = -18 ; We can divide EACH SIDE of the equation by "6", to cancel the "6" on the left-hand side into a "1"; 
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{6(3x + 6) } / 6 = -18 / 6 ;  to get:
________________________________________
3x + 6 = -3 ; Now, we can subtract "6" from EACH SIDE of the equation:
__________________________________
3x + 6 - 6 = -3 - 6 ; to get:
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3x = -9 ; Now, we can divide EACH SIDE of the equation by "3"; to isolate "x" on one side of the question; and solve for "x" ;
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_____________________
Remember, from above:
__________________________
Let "x" be the first odd integer.  We know that "x = -3".
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____________________________________
Let "(x+2)" be next consecutive odd integer.  So (x+2) = (-3+2) = -1.
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____________________________________
Let "(x+4") be the third odd integer. So (x+4) = (-3+4) = 1.
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So, our 3 (three) consecutive odd integers are: -3, -1, 1.
___________________________
To check our work: Is 6 times the sum of our 3 consecutive odd integers, equal to "(-18)" ?

The sum of our 3 consecutive odd integers = -3 + (-1) + 1 = -3 - 1 + 1 = -3.

6 * -3 = ? -18? Yes!
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