A Rectangular Prism Has Dimensions Of \[$(x-3)\$\], \[$(x-1)\$\], And \[$(x-8)\$\], And A Volume Of \($33x$\) Cubic Units. What Are The Dimensions Of The Prism?
Introduction
In mathematics, a rectangular prism is a three-dimensional solid object with six rectangular faces. The dimensions of a rectangular prism are the lengths of its sides, and the volume is the amount of space inside the prism. In this problem, we are given the dimensions of a rectangular prism as {(x-3)$}$, {(x-1)$}$, and {(x-8)$}$, and its volume as ${}$ cubic units. We need to find the dimensions of the prism.
The Formula for the Volume of a Rectangular Prism
The volume of a rectangular prism is given by the formula:
V = lwh
where V is the volume, l is the length, w is the width, and h is the height. In this problem, the volume is given as ${}$ cubic units, and the dimensions are given as {(x-3)$}$, {(x-1)$}$, and {(x-8)$}$. We can substitute these values into the formula:
33x = (x-3)(x-1)(x-8)
Expanding the Equation
To solve for x, we need to expand the equation:
33x = (x^2 - 4x + 3)(x - 8)
Using the distributive property, we can expand the right-hand side of the equation:
33x = x^3 - 12x^2 + 25x - 24
Simplifying the Equation
We can simplify the equation by combining like terms:
33x = x^3 - 12x^2 + 25x - 24
Subtracting 33x from both sides gives us:
0 = x^3 - 12x^2 + 25x - 57
Factoring the Equation
We can factor the equation by grouping terms:
0 = (x^3 - 12x^2) + (25x - 57)
Factoring out x^2 from the first group and 1 from the second group gives us:
0 = x^2(x - 12) + 1(25x - 57)
Using the Rational Root Theorem
The rational root theorem states that if a rational number p/q is a root of the polynomial equation a_nx^n + a_(n-1)x^(n-1) + ... + a_0 = 0, then p must be a factor of a_0 and q must be a factor of a_n. In this case, the factors of 57 are ±1, ±3, ±19, and ±57, and the factors of 1 are ±1.
Trying Possible Roots
We can try the possible roots to see if any of them are actually roots of the equation. We can start by trying the factors of 57:
x = 1: 0 = (1)^2(1 - 12) + 1(25(1) - 57) = -11 x = -1: 0 = (-1)^2(-1 - 12) + 1(25(-1) - 57) = -82 x = 3: 0 = (3)^2(3 - 12) + 1(25(3) - 57) = 0 x = -3: 0 = (-3)^2(-3 - 12) + 1(25(-3) - 57) = 0 x = 19: 0 = (19)^2(19 - 12) + 1(25(19) - 57) = 0 x = -19: 0 = (-19)^2(-19 - 12) + 1(25(-19) - 57) = 0 x = 57: 0 = (57)^2(57 - 12) + 1(25(57) - 57) = 0 x = -57: 0 = (-57)^2(-57 - 12) + 1(25(-57) - 57) = 0
Finding the Value of x
We can see that x = 3 is a root of the equation. We can factor the equation as:
0 = (x - 3)(x^2 - 9x + 19)
Solving for x
We can solve for x by setting each factor equal to 0:
x - 3 = 0 --> x = 3 x^2 - 9x + 19 = 0
Using the Quadratic Formula
We can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -9, and c = 19. Plugging these values into the formula gives us:
x = (9 ± √((-9)^2 - 4(1)(19))) / 2(1) x = (9 ± √(81 - 76)) / 2 x = (9 ± √5) / 2
Finding the Dimensions of the Prism
Now that we have the value of x, we can find the dimensions of the prism. We are given that the dimensions are {(x-3)$}$, {(x-1)$}$, and {(x-8)$}$. Plugging in x = 3, we get:
l = 3 - 3 = 0 w = 3 - 1 = 2 h = 3 - 8 = -5
However, the dimensions of a prism cannot be negative. Therefore, we must try the other value of x, which is x = (9 + √5) / 2.
Finding the Dimensions of the Prism
Plugging in x = (9 + √5) / 2, we get:
l = (9 + √5) / 2 - 3 w = (9 + √5) / 2 - 1 h = (9 + √5) / 2 - 8
Simplifying these expressions gives us:
l = (√5 - 3) / 2 w = (√5 - 1) / 2 h = (√5 - 8) / 2
Conclusion
In this problem, we were given the dimensions of a rectangular prism as {(x-3)$}$, {(x-1)$}$, and {(x-8)$}$, and its volume as ${}$ cubic units. We used the formula for the volume of a rectangular prism to set up an equation, and then solved for x. We found that x = 3 is a root of the equation, and then used the quadratic formula to find the other value of x. Finally, we used the value of x to find the dimensions of the prism.