A pool, that is 12 feet wide and 18 feet long, is to have a sidewalk of uniform length constructed around its perimeter.
a. write the equation for the area A, of the pool with the sidewalk
b. determine the domain of the area of the pool with the sidewalk
c. what would the width of the sidewalk need to be in order for the area to be 364 sq. ft.
I am confused about what they want and not sure how to begin. If you could guide me, I would appreciate it.
OOPS, I meant Pre-Calc not ap Calc sorry
Draw the pool as a rectangle.
Now draw a rectangle around that, such that each side is equally distant from the sides of the pool. This represents the sidewalk around the pool.
The area of the pool is 12 * 18 = 216 sq.ft.
If the sidewalk were 3 feet wide everywhere, the width of the whole area would be (12 + 3 + 3) and the length (18 + 2 + 3). But we don't know the width, so call it x.
Now the width is:
12 + x + x = 12 + 2x
Fill in the length yourself similarly.
Now, the equation for the area of the whole thing is:
A = (2x + 12)(length you worked out)
Multiply that out, and you'll get a quadratic equation.
I'm a little unsure about what the question means by b. myself. I'd be inclined to say R+, but maybe I'm missing the point.
Given the formula you worked out in a., just set 364 = the quadratic equation, and solve for x.
I hope this gives you enough to go on.
Sure, I can guide you through the problem. Let's break it down step by step:
a. To write the equation for the area, we need to consider the pool and the sidewalk separately.
The area of the pool is the product of its width and length:
A_pool = width_pool * length_pool
The area of the sidewalk is the product of its length and width. Since the sidewalk has a uniform width, we can express its width as "x":
A_sidewalk = width_sidewalk * length_sidewalk
To find the equation for the area of the pool with the sidewalk, we need to add the areas of the pool and the sidewalk:
A = A_pool + A_sidewalk
Notice that the length of the pool with the sidewalk is the same as the length of the pool alone, and the width of the pool with the sidewalk is the sum of the pool's width and twice the width of the sidewalk:
length_pool_with_sidewalk = length_pool
width_pool_with_sidewalk = width_pool + 2 * width_sidewalk
Therefore, the equation for the area A of the pool with the sidewalk becomes:
A = (width_pool + 2 * width_sidewalk) * length_pool
b. To determine the domain of the area of the pool with the sidewalk, we must consider any restrictions on the width of the sidewalk.
In this case, since the sidewalk is constructed around the perimeter of the pool, it cannot have a negative width or be wider than the pool itself. So we need to consider the possible values for the width of the sidewalk.
The width of the sidewalk cannot be negative: width_sidewalk ≥ 0
The width of the sidewalk cannot be greater than half the length of the pool (to ensure the sidewalk remains within the perimeter): width_sidewalk ≤ length_pool / 2
Therefore, the domain of the area of the pool with the sidewalk is 0 ≤ width_sidewalk ≤ length_pool / 2.
c. To determine the width of the sidewalk needed for the area to be 364 sq. ft., we can substitute this value into the equation for the area and solve for the width_sidewalk.
Using the equation for the area:
A = (width_pool + 2 * width_sidewalk) * length_pool
Substituting A = 364 and length_pool = 18:
364 = (width_pool + 2 * width_sidewalk) * 18
Simplifying the equation:
364 = 18 * width_pool + 36 * width_sidewalk
Now, you will need additional information about the pool's original width (width_pool) to solve for the width of the sidewalk (width_sidewalk). If you have that information, you can substitute it into the equation and solve for width_sidewalk.