Question

Math

Posted 5 months ago

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Squares of width $x$ are removed from a 20 - $\mathrm{cm}$ by 27 - $\mathrm{cm}$ piece of cardboard, and the resulting edges are folded up to form a box with no top. Determine all values of $x$ so that the volume of the resulting box is at most $450 \mathrm{~cm}^{3}$.
The values of $x$ are
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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

We start by setting up the equation for the volume of the box. The volume $V$ of the box is given by the product of its length, width, and height. The length and width of the box are reduced by $2x$ each due to the squares cut out from each corner, and the height of the box is $x$. Therefore, the volume $V$ is given by:
$V = (20 - 2x)(27 - 2x)x$

step 2

We need to find the values of $x$ such that the volume $V$ is at most $450 \, \text{cm}^3$. This gives us the inequality:
$(20 - 2x)(27 - 2x)x \leq 450$

step 3

We solve the inequality:
$(20 - 2x)(27 - 2x)x \leq 450$
Expanding the left-hand side:
$(20 - 2x)(27 - 2x)x = (540 - 40x - 54x + 4x^2)x = 4x^3 - 94x^2 + 540x$
So the inequality becomes:
$4x^3 - 94x^2 + 540x \leq 450$

step 4

We simplify the inequality:
$4x^3 - 94x^2 + 540x - 450 \leq 0$

step 5

We solve the cubic inequality. First, we find the roots of the equation:
$4x^3 - 94x^2 + 540x - 450 = 0$
Using numerical methods or a graphing calculator, we find the roots to be approximately:
$x \approx 1, \quad x \approx 7.5, \quad x \approx 15$

step 6

We determine the intervals where the inequality holds. Testing intervals around the roots, we find that:
$1 \leq x \leq 7.5$

Answer

The values of $x$ are $1 \leq x \leq 7.5$

Key Concept

Solving a cubic inequality

Explanation

To determine the values of $x$ that satisfy the volume constraint, we set up the inequality, expand and simplify it, find the roots, and test the intervals around the roots.

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