Mathematics, 15.04.2020 03:35 kiyomi58

. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 2323 ft by 1313 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. b. Suppose that in part (a) the original piece of cardboard is a square with sides of length s. Find the volume of the largest box that can be formed in this way.

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Answer from: andesmints5341

The volume of a box is the amount of space in it.

The maximum value of the box is $\mathbf{361.186in^3}$ The volume of the largest box is $\mathbf{22.559in^3}$

The dimension of the cardboard is given as:

$\mathbf{Length = 23}$

$\mathbf{Width = 13}$

Assume the cut-out is x.

So, the dimension of the box is:

$\mathbf{Length = 23 -2x}$

$\mathbf{Width = 13 - 2x}$

$\mathbf{Height = x}$

The volume of the box is:

$\mathbf{V = (23 -2x) (13 - 2x)x}$

Expand

$\mathbf{V = 299x -72x^2 +4x^3}$

Differentiate

$\mathbf{V' = 299 -144x +12x^2}$

Set to 0

$\mathbf{299 -144x +12x^2 = 0}$

Rewrite as:

$\mathbf{12x^2 - 144x + 299= 0}$

Using a calculator, we have:

$\mathbf{x = (2.67, 9.33)}$

9.33 is greater than the dimensions of the box.

So, the possible value of x is:

$\mathbf{x = 2.67}$

Recall that:

$\mathbf{V = 299x -72x^2 +4x^3}$

So, we have:

$\mathbf{V = 299(2.67) - 72(2.67)^2 + 4(2.67)^3}$

$\mathbf{V = 361.186}$

Hence, the maximum value of the box is $\mathbf{361.186in^3}$

If the dimension of the cardboard is s,

The dimension of the box would be:

$\mathbf{Length = s -2x}$

$\mathbf{Width = s - 2x}$

$\mathbf{Height = s}$

So, the volume is:

$\mathbf{V = (s - 2x)(s - 2x) s}$

Substitute $\mathbf{x = 2.67}$

$\mathbf{V = (s - 2 \times 2.67)(s - 2 \times 2.67) s}$

$\mathbf{V = (s - 5.34)(s - 5.34) s}$

Expand

$\mathbf{V = (s - 5.34)(s^2 - 5.34s)}$

$\mathbf{V = s^3 - 5.34s^2 - 5.34s^2 + 28.5156s}$

$\mathbf{V = s^3 -10.68s^2 + 28.5156s}$

Integrate

$\mathbf{V' = 3s^2 -21.36s + 28.5156}$

Set to 0

$\mathbf{3s^2 -21.36s + 28.5156 = 0}$

Using a calculator, we have:

$\mathbf{s = (1,78,5.34)}$

5.34 is greater than the side lengths. So, we have:

$\mathbf{s = 1.78}$

Recall that:

$\mathbf{V = s^3 -10.68s^2 + 28.5156s}$

So, we have:

$\mathbf{V = 1.78^3 - 10.68 \times 1.78^2 + 28.5156 \times 1.78}$

$\mathbf{V = 22.559}$

Hence, the volume of the largest box is $\mathbf{22.559in^3}$

Read more about volumes at:

link

Answer from: Calvinailove13

(a)Therefore the volume of the box is 361.19 cubic ft.

(b)Therefore the volume of the box is 361.19 cubic ft.

Step-by-step explanation:

Given that,

Squares with sides length x are cut out from of each corner of a rectangular piece of cardboard measuring 23 ft and 13 ft.

Now the length of the box is =(23-2x) ft

The width of the box is =(13-2x) ft

The height of the box is= x

The volume of the box is = Length×width×Height

=(23-2x)(13-2x)x cubic ft

= 299x-72x²+4x³ cubic ft

Let,

V=299x-72x²+4x³

Differentiating with respect to x

V'= 299-144x+12x²

Again differentiating with respect to x

V''= -144+24x

To find the maximum volume, we set V'=0

∴299-144x+12x²=0

Applying quadratic formula $x=\frac{-b\pm\sqrt {b^2-4ac}}{2a}$ , here a=12,b= -144 and c=299

$\therefore x=\frac{-(-144)\pm\sqrt{(-144)^2-4.12.299}}{2.12}$

$\Rightarrow x= 9.33, 2.67$

For x= 9.33 , the width of the box will negative which is impossible.

So, x= 2.67

$V''|_{x=2.67}=-144+24(2.67)$

So, at x=2.67, the volume of the box will be maximum.

Therefore the volume of the box is

=(299x-72x²+4x³) cubic ft

=361.19 cubic ft

(b)

The value of s will same with the value of x

so the volume of the box also remains same.

Therefore the volume of the box is 361.19 cubic ft

Answer from: Quest

step-by-step explanation:

if you want to solve percent error follow these steps.

1. subtract the accepted value from the experimental value.

2. take the absolute value of step 1.

3. divide that answer by the accepted value.

4. multiply that answer by 100 and add the % symbol to express the answer as a percentage.

if you want to solve percent of change just do this.

1. work out the difference (increase) between the two numbers you are comparing.

2. divide the increase by the original number and multiply the answer by 100. if your answer is a negative number then this is a percentage decrease.

hope this : )

Answer from: Quest

answer: r there any options?

step-by-step explanation:

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