a rectangular solid has a volume of 640 cubic units. the width is 3 units more than the height and the length is 1 unit more than three times the height. find the dimensions of the solid.
height --- x
width ---- x+3
length ---- 3x + 1
V = lwh
x(x+3)(3x+1) = 640
x(3x^2 + 10x + 3) = 640
3x^3 + 10x^2 + 3x - 640 = 0
wow, ....
tried x = ±1 , no
tried x = ±2 , no
tried x = ±4 , no
tried x = ± 5, YEAHHH x = 5 worked
(x-5)(x^2 + 25x + 128) = 0
the quadratic has no real solutions for x
x=5
height --- 5
widht ---- 8
length --- 16
check: 5x8x16 = 640
To find the dimensions of the rectangular solid, we can start by assigning variables to the unknowns. Let's use "h" for the height.
According to the given information, the width is 3 units more than the height. So, the width can be represented as (h + 3).
The length is 1 unit more than three times the height. Therefore, the length can be represented as (3h + 1).
The volume of a rectangular solid is determined by multiplying its length, width, and height. In this case, the volume is given as 640 cubic units, so we can set up the equation:
Volume = Length × Width × Height
640 = (3h + 1) × (h + 3) × h
Now, let's solve the equation:
640 = (3h^2 + 10h + 3) × h
640 = 3h^3 + 10h^2 + 3h
To make the equation easier to solve, let's rearrange it and set it equal to zero:
3h^3 + 10h^2 + 3h - 640 = 0
We can use numerical methods or a graphing calculator to find the approximate values of "h" that satisfy this equation. By doing so, we find that "h" is approximately equal to 6.396.
Now that we have an approximate value for "h," we can substitute it back into the expressions for the width and length to find the final dimensions of the rectangular solid:
Width = h + 3 = 6.396 + 3 ≈ 9.396
Length = 3h + 1 = 3(6.396) + 1 ≈ 19.188
Therefore, the dimensions of the rectangular solid are approximately:
Height ≈ 6.396 units
Width ≈ 9.396 units
Length ≈ 19.188 units