Stephan has a set of plans to build a wooden box. he wants to reduce the volume of the box to 105 cubic inches. he would like to reduce the length of each dimension in the plan by the same amount. the plans call got the box to he 10 inches by 8 inches by 6 inches. write and solve a polynomial equation to find out how much stephan should take from each dimension

x³ - 24x² + 188x - 375 = 0;  x = 3

Step-by-step explanation:

Data:

l = 10 in; w = 8 in; h = 6 in

Calculations:

(a) Polynomial

Let x = amount to be removed from each dimension

Then, the volume of the new box is

V = lwh

  = (10 - x)(8 - x)(6 - x)

  = 105

Here's the polynomial by long multiplication:

10       -     x

 8      -     x

80     -   8x

          - 10x  +  x²

80      - 18x  +  x²

 6     -      x                    

480  - 108x + 6x²

       -  80x + 18x²  - x³

480 - 188x + 24x² - x³

So,

  480 - 188x + 24x² - x³ = 105     Subtract 105 from each side  

 x³ + 24 x² - 188x + 375 = 0        Multiply each side by -1

x³  - 24x² + 188x  - 375 = 0        This is the polynomial equation

(b) Solution to the equation

We can use the rational root theorem to help find a root.

The general formula for a third-degree polynomial is

f(x) = ax³ + bx² + cx + d

Your polynomial is  

f(x) = x³  - 24x² + 188x  - 375 = 0

a = 1; d = -375  

Factors of a = ±1

Factors of d = ±1, ±3. ±5, ±15, ±25, ±75, ±125, ±375

Possible roots are x = ±1, ±3, ±5, ±15, ±25, ±75, ±125, ±375

Now, it's a matter trial and error in seeing which value of x is a zero of the polynomial.

f(3) =  3³ - 24×3² + 188×3 - 375

f(3) = 27 - 24×9   +   564   -375

f(3) = 27 -  216     +   564 - 375

f(3) = 0

So, x = 3 is one root of the polynomial, and there are no other real roots.

A graph of the polynomial shows one zero at x = 3.

Stefan should take 3 in from each dimension of the box.

Then, the dimensions of the box are

 l = 10 - 3 = 7 in

w =  8 - 3 = 5 in

h =  6 - 3 = 3 in

Check:

V = lwh = 7 × 5 × 3 = 105 in³

throw the computer

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