Sam has a patio that is in the shape of a rectangle with an area of $$ 600 \mathrm{ft}^{2} $$ will be a larger rectangle. The current patio has a length of 30 ft and a wi new width 3 times the current width. (a) Fill in the blanks to describe the size of Sam's new patio. New length (b) How will the expansion change the area of the patio? The area of the new patio will be $$ \square $$ times the area of the curre (c) Sam's friend Lisa also wants to expand her rectangular patio. Li current patio. She goes to Sam for advice. Fill in the blank so th Make the new length 7 times the current length, and make the (tt

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Let's break down the problem step by step.

### Known Conditions:
1. Current patio area: $$ 600 \, \text{ft}^2 $$
2. Current length of the patio: $$ L = 30 \, \text{ft} $$
3. Current width of the patio: $$ W $$ can be calculated using the area formula $$ A = L \times W $$.

### Step 1: Calculate the Current Width
Using the area formula:
$$
A = L \times W \implies 600 = 30 \times W
$$
To find $$ W $$:
$$
W = \frac{600}{30} = 20 \, \text{ft}
$$

### Step 2: Determine the New Dimensions of Sam's Patio
(a) The new width is 3 times the current width:
$$
\text{New Width} = 3 \times W = 3 \times 20 = 60 \, \text{ft}
$$
The new length remains the same as the current length:
$$
\text{New Length} = L = 30 \, \text{ft}
$$

So, the new dimensions of Sam's patio are:
- New Length: $$ 30 \, \text{ft} $$
- New Width: $$ 60 \, \text{ft} $$

### Step 3: Calculate the Area of the New Patio
The area of the new patio can be calculated as:
$$
\text{New Area} = \text{New Length} \times \text{New Width} = 30 \times 60
$$

Let's calculate this area:
$$
\text{New Area} = 1800 \, \text{ft}^2
$$

(b) To find how the expansion changes the area of the patio:
$$
\text{Area Change Factor} = \frac{\text{New Area}}{\text{Current Area}} = \frac{1800}{600} = 3
$$

### Step 4: Advice for Lisa's Patio Expansion
(c) Lisa wants to make her new length 7 times the current length and make the new width the same as the current width.

Let’s denote Lisa's current length as $$ L_L $$ and current width as $$ W_L $$. The new dimensions will be:
- New Length: $$ 7 \times L_L $$
- New Width: $$ W_L $$

The area of Lisa's new patio will be:
$$
\text{New Area}_L = (7 \times L_L) \times W_L = 7 \times (L_L \times W_L) = 7 \times \text{Current Area}_L
$$

Thus, the area of Lisa's new patio will be $$ 7 $$ times the area of her current patio.

### Summary of Answers:
(a) New Length: $$ 30 \, \text{ft} $$, New Width: $$ 60 \, \text{ft} $$  
(b) The area of the new patio will be $$ 3 $$ times the area of the current patio.  
(c) The area of Lisa's new patio will be $$ 7 $$ times the area of her current patio.
(a) New Length: 30 ft, New Width: 60 ft (b) The area of the new patio will be 3 times the current area. (c) The area of Lisa's new patio will be 7 times her current patio area.

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