Pregunta
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A housing committee wants a rectangular swimming pool and surrounding deck with a total area of . The
deck is 3 m wide at each end of the pool and 2 m wide on each side. Find the dimensions of the pool that has the
maximum area.

Ask by Pearson Morrison. in Canada
Mar 29,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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The pool should be 9 meters in length and 6 meters in width to maximize the area.

Solución

We start by letting the length and width of the pool be and , respectively. The deck surrounds the pool with a width of 3 m at each end (adding 6 m to the length) and 2 m on each side (adding 4 m to the width). This means the overall dimensions (pool plus deck) are:
The total area (pool plus deck) is given as 150 m , so we have the constraint:
Our goal is to maximize the pool area
subject to the area constraint above.

Step 1. Express in Terms of

From the constraint:
Subtract 4 from both sides to get:

Step 2. Write the Area of the Pool as a Function of

The pool area becomes:
Simplify:

Step 3. Find the Critical Points by Differentiating

Differentiate with respect to . First, write the derivative of the first term using the quotient rule:
The derivative of the second term is .
Thus, the derivative of is:
Set the derivative equal to zero for maximization:
Solve for :
Taking the positive square root (since lengths must be positive):

Step 4. Compute Using the Constraint

Substitute into the expression for :

Final Answer

The dimensions of the pool that give the maximum area are:

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Bonus Knowledge

To solve this problem, let’s define the dimensions of the pool. Let be the length of the pool and be the width of the pool. The total area of the pool plus the deck is given as . The deck surrounds the pool, adding 3 meters to the length (2 meters on either side) and 3 meters to the width (3 meters on either end).
The total area can be expressed as:
This equation accounts for the deck’s width.
Expanding, we get:
Simplifying, we have:
To maximize the pool’s area , we can express in terms of using the equation above:
Substituting this into the area formula results in a single-variable function for area, which can be maximized using calculus.
To find the dimensions of the pool, you’ll need to derive this function, set the derivative to zero, and solve for and .
This can get a bit tricky, but trust that it’s like rearranging a puzzle – one piece at a time to see how they fit together!
To ensure you don’t make common mistakes when maximizing area, remember that clearly define your variables, double-check your substitutions, and ensure you respect the total area constraint of . Small algebraic errors can lead to big discrepancies in your final dimensions, so take a step back and verify each stage of your calculations!

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