1) A small rectangular pool has dimensions 12 m by 6 m by 2 m . A landscaper wishes to increase the length, width and height by the same amount so that the surface area is at most $$ 440 \mathrm{~m}^{2} $$. What are the maximum dimensions of the larger pool?

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To determine the maximum dimensions of the larger pool, let's follow these steps:

### 1. Original Pool Dimensions:
- **Length (L)** = 12 m
- **Width (W)** = 6 m
- **Height (H)** = 2 m

### 2. Increased Dimensions:
Let $$ x $$ be the amount by which each dimension is increased. Then, the new dimensions are:
- **New Length** = $$ 12 + x $$ m
- **New Width** = $$ 6 + x $$ m
- **New Height** = $$ 2 + x $$ m

### 3. Surface Area Constraint:
The surface area $$ S $$ of a rectangular prism is given by:
$$
S = 2[(L \times W) + (L \times H) + (W \times H)]
$$
For the larger pool:
$$
2[(12 + x)(6 + x) + (12 + x)(2 + x) + (6 + x)(2 + x)] \leq 440
$$
Divide both sides by 2:
$$
(12 + x)(6 + x) + (12 + x)(2 + x) + (6 + x)(2 + x) \leq 220
$$

### 4. Expanding and Simplifying:
$$
(12 + x)(6 + x) + (12 + x)(2 + x) + (6 + x)(2 + x) = 3x^2 + 40x + 108 \leq 220
$$
Subtract 220 from both sides:
$$
3x^2 + 40x - 112 \leq 0
$$
Solve the quadratic equation $$ 3x^2 + 40x - 112 = 0 $$ using the quadratic formula:
$$
x = \frac{-40 \pm \sqrt{40^2 - 4 \times 3 \times (-112)}}{2 \times 3} = \frac{-40 \pm \sqrt{1600 + 1344}}{6} = \frac{-40 \pm \sqrt{2944}}{6} = \frac{-40 \pm 8\sqrt{46}}{6} = \frac{-20 \pm 4\sqrt{46}}{3}
$$
Since $$ x $$ must be positive:
$$
x = \frac{-20 + 4\sqrt{46}}{3} \approx 2.38 \, \text{m}
$$

### 5. Maximum Dimensions:
Adding $$ x $$ to each original dimension:
$$
\begin{align*}
\text{New Length} &= 12 + 2.38 \approx 14.38 \, \text{m} \
\text{New Width} &= 6 + 2.38 \approx 8.38 \, \text{m} \
\text{New Height} &= 2 + 2.38 \approx 4.38 \, \text{m}
\end{align*}
$$

### **Final Answer:**
The largest possible pool measures approximately 14.38 m × 8.38 m × 4.38 m.
The largest possible pool dimensions are approximately 14.38 m in length, 8.38 m in width, and 4.38 m in height.

A small rectangular pool has dimensions 12 m by 6 m by 2 m . A
landscaper wishes to increase the length, width and height by the same
amount so that the surface area is at most . What are the
maximum dimensions of the larger pool?

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Solución

1. Original Pool Dimensions:

2. Increased Dimensions:

3. Surface Area Constraint:

4. Expanding and Simplifying:

5. Maximum Dimensions:

Final Answer:

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A small rectangular pool has dimensions 12 m by 6 m by 2 m . A landscaper wishes to increase the length, width and height by the same amount so that the surface area is at most . What are the maximum dimensions of the larger pool?