Math Word problem with factoring...HELP!!! This questions has me stumped!? I have been working on it for days...

A 10ft-by-15ft pool was installed in your backyard. You have $3000 left that you would like to spend on surrounding the pool with a patio, equally wide on all sides. Write an expression in terms of x for the length and width of the patio.

1.Find the area of the swimming pool

2.Write an algebraic expression for the total area of the region containing both the pool and the patio

3.Use subtraction to find an algebraic expression for the area of just the patio.

4.Find the perimeter of the swimming pool alone.

5.For each patio material option, write an algebraic expression for the total cost of installing the patio based o its area and the given price information.
a) poured cement $5.00 per square foot
b) Bricks $7.50 per squre foot pluse a $30 flat fee for delivering
c)Outdoor carpeting $4.50 per squre foot plus $10.86 per foot of the pool's perimeter to install edging

6. If you plan to speed the entire $3000 on the patio, how wide would the patio in option A be?

7.If you plan to speed the entire $3000 on the patio, how wide would the patio in option B be?

8.If you plan to speed the entire $3000 on the patio, how wide would the patio in option C be?

9.Which option wou choose?What are some of the pros and cons for each options?

length= 15+x

width=10+x

total area of just patio= (15+x)(10+x)-10*15

perimeter spool= 2*10+2*15

6. $45

7. $55
8.65

1. To find the area of the swimming pool, multiply the length and width: Area = 10ft * 15ft = 150ft².

2. The total area of the region containing both the pool and the patio can be expressed as the sum of the pool area and the patio area. Let's call the width of the patio "x", so the length of the pool combined with the patio will be 10ft + 2x (because the patio is equally wide on all sides) and the width will be 15ft + 2x. Therefore, the total area equation is: Total Area = (10ft + 2x) * (15ft + 2x).

3. To find the area of just the patio, we need to subtract the pool area from the total area. Area of just the patio = Total Area - Pool Area = (10ft + 2x) * (15ft + 2x) - 150ft².

4. The perimeter of the swimming pool can be calculated by adding up all the sides. Perimeter of the pool = 2 * (length + width) = 2 * (10ft + 15ft) = 2 * 25ft = 50ft.

5. Let's write the algebraic expressions for the total cost of each patio material option:
a) Poured cement: Total Cost = Area * $5.00 per square foot.
b) Bricks: Total Cost = (Area + 2x * Perimeter + $30) * $7.50 per square foot.
c) Outdoor carpeting: Total Cost = (Area + 10.86ft * Perimeter) * $4.50 per square foot.

6. Option A: Since you plan to spend the entire $3000 on the patio, you need to set the total cost equal to $3000 and solve for x in the algebraic expression for option A.
Total Cost for option A = Area * $5.00 per square foot = 150ft² * $5.00 = $750. Setting this equal to $3000:
$750 = 3000, divide by $5.00
150ft² = x / $150, multiply by $150
x ≈ 150ft².

7. Option B: Set the total cost equal to $3000 and solve for x in the algebraic expression for option B.
Total Cost for option B = (Area + 2x * Perimeter + $30) * $7.50 per square foot = (150ft² + 2x * 50ft + $30) * $7.50 per square foot = ($7500 + 100x + $30) * $7.50 per square foot.
Set this equal to $3000:
($7500 + 100x + $30) * $7.50 = $3000,
$7500 + 100x + $30 = $3000 / $7.50,
$7500 + 100x + $30 = $400,
100x = $400 - $7500 - $30,
x = ($400 - $7500 - $30) / 100.

8. Option C: Set the total cost equal to $3000 and solve for x in the algebraic expression for option C.
Total Cost for option C = (Area + 10.86ft * Perimeter) * $4.50 per square foot = (150ft² + 10.86ft * 50ft) * $4.50 per square foot = ($6750 + 543ft) * $4.50 per square foot.
Set this equal to $3000:
($6750 + 543ft) * $4.50 = $3000,
$6750 + 543ft = $3000 / $4.50,
$6750 + 543ft = $666.67,
543ft = $666.67 - $6750,
x = ($666.67 - $6750) / 543.

9. The choice of patio material option depends on personal preference and budget. Here are some pros and cons for each option:
a) Poured cement
Pros:
- Relatively low cost compared to other options.
- Durable and long-lasting.
- Versatile for different design options.

Cons:
- Can crack or shift over time.
- Can be slippery when wet.
- Might require regular maintenance.

b) Bricks
Pros:
- Aesthetically appealing and visually pleasing.
- Durability and longevity.
- Good for creating a traditional or classic look.

Cons:
- Higher cost compared to poured cement or outdoor carpeting.
- Requires skilled labor and precise installation.
- May need regular maintenance to prevent weed growth between the bricks.

c) Outdoor carpeting
Pros:
- Offers a softer and more comfortable surface.
- Less expensive than bricks.
- Can be easily DIY installed.

Cons:
- Less durable than poured cement or bricks.
- Can fade or wear down over time.
- Limited design options compared to other materials.

Consider your priorities, budget, and maintenance preferences when making a decision.

1. To find the area of the swimming pool, you can use the formula for the area of a rectangle, which is length multiplied by width. In this case, the length of the pool is 15ft and the width is 10ft. Therefore, the area of the swimming pool is 15ft * 10ft = 150 square feet.

2. To find the total area of the region containing both the pool and the patio, you need to add the area of the pool to the area of the patio. Let's call the width of the patio x. The length and width of the region containing both the pool and the patio would be 10ft + 2x (since there are two sides of the patio along the length) and 15ft + 2x (since there are two sides of the patio along the width). Therefore, the area is (10ft + 2x)(15ft + 2x).

3. To find the area of just the patio, you need to subtract the area of the pool from the total area of the region. The area of just the patio would be (10ft + 2x)(15ft + 2x) - 150 square feet.

4. The perimeter of the swimming pool alone is found by adding up all the sides of the pool. The perimeter of a rectangle is 2 times the length plus 2 times the width. In this case, the perimeter of the swimming pool alone would be 2 * 10ft + 2 * 15ft = 20ft + 30ft = 50 feet.

5. To write the algebraic expressions for the cost of installing each option, we need to use the area of the patio and the given price information.
a) The cost of poured cement patio would be the area of the patio multiplied by the price per square foot, which is $5.00. So the expression would be 5 * area of the patio.
b) The cost of bricks patio would be the area of the patio multiplied by the price per square foot, which is $7.50, plus a flat fee of $30 for delivery. So the expression would be 7.50 * area of the patio + 30.
c) The cost of outdoor carpeting patio would be the area of the patio multiplied by the price per square foot, which is $4.50, plus the length of the pool's perimeter multiplied by $10.86 for installing edging. So the expression would be 4.50 * area of the patio + 10.86 * perimeter of the pool.

6. To find the width of the patio in option A if you plan to spend the entire $3000, you need to set the expression for the cost of option A equal to $3000 and solve for x (the width of the patio).

7. To find the width of the patio in option B if you plan to spend the entire $3000, you need to set the expression for the cost of option B equal to $3000 and solve for x (the width of the patio).

8. To find the width of the patio in option C if you plan to spend the entire $3000, you need to set the expression for the cost of option C equal to $3000 and solve for x (the width of the patio).

9. To choose the best option, you can compare the cost, functionality, and aesthetic appeal of each option. Some pros and cons to consider could be:
- Option A (poured cement): Pros could include durability and versatility. Cons could include a higher cost per square foot compared to other options.
- Option B (bricks): Pros could include a classic and timeless look. Cons could include a higher cost per square foot and an extra flat fee for delivery.
- Option C (outdoor carpeting): Pros could include a lower cost per square foot and the added benefit of edging. Cons could include durability concerns and higher installation costs due to the pool's perimeter.

Ultimately, the choice would depend on personal preference, budget constraints, and desired longevity of the patio.