the length of a rare jeweled rectangular prism has a length equal to its width and a height that is 5 cm greater than its width.volume of the prism is 72 cc
a) write a polynomial equation that can be used to solve for the width (x) of the prism
b) solve the equation. what are the actual dimensions of the prism?
length --- x
width ---- x
height ---- x+5
x(x)(x+5) = 72
x^3 + 5x^2 - 72 = 0
I tried ±2, and ±3
x = 3 works
so by synthetic long division
x^3 + 5x^2 - 72 = 0
(x-3)(x^2+ 4x + 24) = 0
the quadratic has no real solution, so
x = 3
the prism is 3 by 3 by 8
To solve for the width (x) of the prism, let's first understand the given information.
Given:
- The length of the rectangular prism = its width = x cm.
- The height of the prism = width + 5 cm.
- The volume of the prism = 72 cc.
Now, let's proceed to solve the problem step by step:
a) Write a polynomial equation to solve for the width (x) of the prism:
Volume of a rectangular prism is given by:
Volume = Length * Width * Height
In this case, the length = width = x cm, and the height = width + 5 cm.
So, the equation becomes:
72 = x * x * (x + 5)
b) Solve the equation to find the actual dimensions of the prism:
Let's simplify the equation and solve for x:
72 = x^2 * (x + 5)
72 = x^3 + 5x^2
Rearranging the equation:
x^3 + 5x^2 - 72 = 0
To solve this equation, you can use different methods like factoring, the rational root theorem, or numerical methods. Since it may not be easily factorable, let's solve it using numerical methods or a graphing calculator:
By using a graphing calculator, you can graph the equation y = x^3 + 5x^2 - 72 and find the x-intercept(s) or zero(s). The x-intercept(s) would represent the values of x that satisfy the equation.
Alternatively, you can also use online tools or software that can solve the equation numerically, such as Wolfram Alpha or numerical equation solvers.
Upon solving, the possible solutions for x are:
x ≈ -8.180, x ≈ 3.397, and x ≈ 4.783
Since the width of the prism cannot be negative, the only valid solution is:
x ≈ 4.783 cm (rounded to three decimal places)
Therefore, the actual dimensions of the prism are:
Width = Length = x ≈ 4.783 cm
Height = Width + 5 ≈ 4.783 + 5 ≈ 9.783 cm