We are given that the volume of the swimming pool is:

$V = 2x^3 + 30x^2 + 100x \, \text{cubic centimeters}$

The dimensions of the pool are:

• Width $W = 2x \, \text{cm}$
• Length $L = (x + 10) \, \text{cm}$

We are asked to find the height $h$ of the swimming pool.

The volume $V$ of a rectangular prism (like a swimming pool) is given by the formula:

$V = L \times W \times h$

Substituting the given values:

$2x^3 + 30x^2 + 100x = (x + 10) \times 2x \times h$

Simplify the right-hand side:

$(x + 10) \times 2x = 2x^2 + 20x$

Now, substitute this back into the equation:

$2x^3 + 30x^2 + 100x = (2x^2 + 20x) \times h$

To find $h$, divide both sides of the equation by $2x^2 + 20x$:

$h = \frac{2x^3 + 30x^2 + 100x}{2x^2 + 20x}$

Factor both the numerator and denominator:

Numerator:

$2x^3 + 30x^2 + 100x = 2x(x^2 + 15x + 50)$

Denominator:

$2x^2 + 20x = 2x(x + 10)$

Now, simplify the equation for $h$:

$h = \frac{2x(x^2 + 15x + 50)}{2x(x + 10)}$

Cancel out the common terms:

$h = \frac{x^2 + 15x + 50}{x + 10}$

Finally, perform the division:

$h = x + 5$

Thus, the height of the swimming pool is:

$h = x + 5 \, \text{cm}$

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Would you like further details on this solution or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the height change if the volume of the pool doubled?
2. What is the height when $x = 3$ cm?
3. How does the relationship between volume, length, and width affect the height?
4. What would happen to the height if the pool's width was tripled?
5. How do you factor polynomials like $x^2 + 15x + 50$?

Tip: When solving for a variable, try factoring out common terms to simplify the equation before performing division.