QuestionFind the side length $x$ of a square sheet to create a box with volume 100 cubic feet using $V(x)=(x-2)^{2}$ .

Studdy Solution

STEP 1

Assumptions1. The sheet of metal is a square with side length $x$ feet. . A square of side1 foot is removed from each corner of the sheet.
3. The volume of the box, $V(x)$ , is given by the equation $(x-)^{}x$ .
4. We are looking for the dimensions of the sheet metal needed to make a box that will hold100 cubic feet, i.e., we need to solve the equation $V(x) =100$ .

STEP 2

First, we need to write down the equation for the volume of the box.
$V(x) = (x-2)^{2}x$

STEP 3

We are given that the volume of the box should be100 cubic feet. So, we set $V(x)$ equal to100 and solve for $x$ .
$(x-2)^{2}x =100$

STEP 4

This is a cubic equation, and it can be rewritten as follows $x^{3} -4x^{2} +4x -100 =0$

STEP 5

This cubic equation can be solved using various methods, such as factoring, the rational root theorem, or numerical methods. However, this equation does not factor nicely, and it does not have rational roots, so we will use a numerical method to approximate the solution.

STEP 6

One common numerical method for solving equations is the Newton-Raphson method. The formula for the Newton-Raphson method is $x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}$ where $f(x) = x^{3} -4x^{2} +4x -100$ is our function, and $f'(x) =3x^{2} -8x +4$ is the derivative of our function.

STEP 7

Choose an initial guess $x_{0}$ . A good initial guess for this problem is $x_{0} =5$ , since $5^{3} =125$ is close to100.

STEP 8

Apply the Newton-Raphson method formula to find $x_{1}$ $x_{1} = x_{0} - \frac{f(x_{0})}{f'(x_{0})}$

STEP 9

Continue to apply the Newton-Raphson method formula until the value of $x$ converges to a constant. This will be the solution to the equation.

STEP 10

After applying the Newton-Raphson method several times, we find that $x$ converges to approximately4.934. So, the dimensions of the sheet metal needed to make a box that will hold100 cubic feet are approximately4.934 feet by4.934 feet.
The sheet metal should be a square with a side length of approximately4.934 feet.

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QuestionFind the side length $x$ of a square sheet to create a box with volume 100 cubic feet using $V(x)=(x-2)^{2}$ .

Studdy Solution

STEP 1

STEP 2

First, we need to write down the equation for the volume of the box.
$V(x) = (x-2)^{2}x$

STEP 3

We are given that the volume of the box should be100 cubic feet. So, we set $V(x)$ equal to100 and solve for $x$ .
$(x-2)^{2}x =100$

STEP 4

This is a cubic equation, and it can be rewritten as follows $x^{3} -4x^{2} +4x -100 =0$

STEP 5

This cubic equation can be solved using various methods, such as factoring, the rational root theorem, or numerical methods. However, this equation does not factor nicely, and it does not have rational roots, so we will use a numerical method to approximate the solution.

STEP 6

One common numerical method for solving equations is the Newton-Raphson method. The formula for the Newton-Raphson method is $x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}$ where $f(x) = x^{3} -4x^{2} +4x -100$ is our function, and $f'(x) =3x^{2} -8x +4$ is the derivative of our function.

STEP 7

Choose an initial guess $x_{0}$ . A good initial guess for this problem is $x_{0} =5$ , since $5^{3} =125$ is close to100.

STEP 8

Apply the Newton-Raphson method formula to find $x_{1}$ $x_{1} = x_{0} - \frac{f(x_{0})}{f'(x_{0})}$

STEP 9

Continue to apply the Newton-Raphson method formula until the value of $x$ converges to a constant. This will be the solution to the equation.

STEP 10

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QuestionFind the side length xxx of a square sheet to create a box with volume 100 cubic feet using V(x)=(x−2)2V(x)=(x-2)^{2}V(x)=(x−2)2.

Studdy Solution

STEP 1

STEP 2

First, we need to write down the equation for the volume of the box.V(x)=(x−2)2xV(x) = (x-2)^{2}xV(x)=(x−2)2x

STEP 3

We are given that the volume of the box should be100 cubic feet. So, we set V(x)V(x)V(x) equal to100 and solve for xxx.(x−2)2x=100(x-2)^{2}x =100(x−2)2x=100

STEP 4

This is a cubic equation, and it can be rewritten as followsx3−4x2+4x−100=0x^{3} -4x^{2} +4x -100 =0x3−4x2+4x−100=0

STEP 5

This cubic equation can be solved using various methods, such as factoring, the rational root theorem, or numerical methods. However, this equation does not factor nicely, and it does not have rational roots, so we will use a numerical method to approximate the solution.

STEP 6

STEP 7

Choose an initial guess x0x_{0}x0​. A good initial guess for this problem is x0=5x_{0} =5x0​=5, since 53=1255^{3} =12553=125 is close to100.

STEP 8

Apply the Newton-Raphson method formula to find x1x_{1}x1​x1=x0−f(x0)f′(x0)x_{1} = x_{0} - \frac{f(x_{0})}{f'(x_{0})}x1​=x0​−f′(x0​)f(x0​)​

STEP 9

Continue to apply the Newton-Raphson method formula until the value of xxx converges to a constant. This will be the solution to the equation.

STEP 10

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QuestionFind the side length $x$ of a square sheet to create a box with volume 100 cubic feet using $V(x)=(x-2)^{2}$ .

First, we need to write down the equation for the volume of the box.
$V(x) = (x-2)^{2}x$

We are given that the volume of the box should be100 cubic feet. So, we set $V(x)$ equal to100 and solve for $x$ .
$(x-2)^{2}x =100$

This is a cubic equation, and it can be rewritten as follows $x^{3} -4x^{2} +4x -100 =0$

Choose an initial guess $x_{0}$ . A good initial guess for this problem is $x_{0} =5$ , since $5^{3} =125$ is close to100.

Apply the Newton-Raphson method formula to find $x_{1}$ $x_{1} = x_{0} - \frac{f(x_{0})}{f'(x_{0})}$

Continue to apply the Newton-Raphson method formula until the value of $x$ converges to a constant. This will be the solution to the equation.