Answer:
- depth: 5 cm
- width: 15 cm
- length: 13 cm
Explanation:
You want to know the dimensions of a cuboid cereal box with volume 975 cm³ that is 3 times as wide as deep, and 2 cm shorter than wide.
Volume
Let x represent the depth of the box. Then the width is 3 times that, or 3x. The length is 2 cm less than the width, so is (3x -2). The volume is the product of these dimensions.
V = DWL
975 = x(3x)(3x -2)
Solution
We like to graph an equation like this in the form ...
f(x) = 0
so the solution is the x-intercept of the graph of f(x). Subtracting 975 puts the equation in that form:
x(3x)(3x -2) -975 = 0
The attachment shows the graph of x(3x)(3x -2), and that it has its only real solution at x=5. This means the dimensions are ...
- depth: 5 cm
- width: 15 cm
- length: 13 cm
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Alternate solution
Sometimes finding the solution to a cubic isn't easy. As a starting point, we can recognize that 3x-2 is about the same as 3x, so the value of x will be approximately the solution to ...
975 = x(3x)(3x) = 9x³
x = ∛(975/9) ≈ 4.8
The nearest integer to this value is x=5, so that can be a good value to check: 5(3·5)(3·5 -2) = 5·15·13 =975. (x = 5 works!)
Iterated solution
We can rewrite the equation as ...
975 = 3x²(3x -2)
325 = x²(3x -2)
325/(3x -2) = x²
x = √(325/(3x -2))
A value of x will be a solution to this equation if it makes the right side equal to the left side. This equation works as an "iterator," meaning we can use a value of x on the right side, and the value of that function will give a value of x on the left that is closer to the solution. The table in the second attachment shows this convergence to x=5.
It isn't always easy to find a suitable iteration function for a cubic or higher degree polynomial equation. The usual tools are Descartes' Rule of Signs, and the Rational Root Theorem. The best iteration functions are found using calculus.