a computer manufacturer needs to change the dimensions of its foam packaging for a new model of computer. the width of the original piece is three times the height, and he length is equal to the height squared. The volume of the new piece can be represented by the equation V(h) = 3h^4 +11h^3 +18h^2 +44h + 24
where h is height of the original piece.
Factor the equation for the volume of the new piece to determine three equations that represent the height, length, and width of the new piece.
To factor the equation for the volume of the new foam piece, we will use synthetic division to find the roots of the equation.
The given equation for the volume is:
V(h) = 3h^4 + 11h^3 + 18h^2 + 44h + 24
Let's set this equation equal to zero:
3h^4 + 11h^3 + 18h^2 + 44h + 24 = 0
Using synthetic division or a graphing calculator, we find that one of the roots is -2.
By dividing the polynomial by (h + 2), we can find the other factors:
Using synthetic division:
(h + 2) | 3h^4 + 11h^3 + 18h^2 + 44h + 24
-2 | 3 5 8 24 -8
________________
3 9 26 68 16
So, the factored form of the equation is:
3h^4 + 11h^3 + 18h^2 + 44h + 24 = (h + 2)(3h^3 + 9h^2 + 26h + 68)
Now, we have the equation factored as (h + 2)(3h^3 + 9h^2 + 26h + 68).
From this factored form, we can determine three equations by comparing the factors:
1. Equation for height:
Setting (h + 2) = 0, we find h = -2. Hence, the height of the new piece is -2.
2. Equation for length:
From the factored form, we identify the factor (3h^3 + 9h^2 + 26h + 68) as the equation for length.
3. Equation for width:
Since the length is equal to the height squared for the original piece, we can use the height equation to calculate the width. Given that the width of the original piece is three times the height, we have width = 3 * (-2) = -6.
So, the three equations representing the height, length, and width of the new piece are:
1. Height: h = -2
2. Length: 3h^3 + 9h^2 + 26h + 68 = 0
3. Width: -6
Please note that having a negative height and width may not make practical sense in this context, but these are the equations derived from the factored form.
To factor the equation V(h) = 3h^4 +11h^3 +18h^2 +44h + 24, we first need to find the roots of the equation. The roots represent the values of h where the volume of the new piece is equal to zero, indicating potential dimensions for the foam packaging.
One way to find the roots is by using the Rational Root Theorem, which tells us that if a rational number p/q is a root of the equation, then p must be a factor of the constant term (24 in this case) and q must be a factor of the coefficient of the highest power of h (3 in this case).
By trying out the factors of 24 and dividing them into the equation, we can find the potential rational roots. Possible values for h could be ±1, ±2, ±3, ±4, ±6, ±8, ±12, or ±24.
By using a method such as synthetic division or long division, we can check which values are actual roots of the equation.
For convenience, let's use a graphing calculator or software to find the roots of the equation. After evaluating the equation, we find that h = -1, -2, -4, and -3 are the roots.
Now that we have the roots, we can factor the equation as follows:
V(h) = (h + 1)(h + 2)(h + 3)(h + 4)
From this factored form, we can determine the equations that represent the height, length, and width of the new piece.
1. Height (h): Since we're given that the height of the original piece is h, the equation for the height of the new piece remains the same:
Height = h
2. Length (L): The length is equal to the height squared, indicating the equation:
Length = (h + 1)^2
3. Width (W): The width is three times the height, suggesting the equation:
Width = 3h
Therefore, the three equations representing the height, length, and width of the new piece are:
Height = h
Length = (h + 1)^2
Width = 3h