A rectangular swimming pool is to be built with an area of 1800 square feet. The

owner wants 5-foot wide decks along either side and 10-foot wide decks at the two ends.
Find the dimensions of the smallest piece of property on which the pool can be built
satisfying these conditions.

Is the answer 16 by 18. Any help woulb be greatly appreciated.

You would know your answer is illogical, since 16x18 is only 288 square feet!!

Let the pool be x feet by y feet.
then the property must be x+20 feet by y+10 feet.
and the area = (x+20)(y+10)
but xy= 1800 ---> y = 1800/x

then
Area =(x+20)(1800/x+10)
= 1800 + 10x + 36000/x + 200

Area' = 10 - 36000/x^2
= 0 for a max/min of Area

solve this to get x=60
so the pool is 60 by 30
and the property must be 80 by 40

80 x 40

I can't figure it out on my own it's way too confusing. It makes me forget my basic geometry skills because it violates the properties for finding the area of a rectangle in so many ways.

Well, it seems like you're in quite a "pool" of math problems! Let's dive in and figure it out together.

To find the dimensions of the smallest piece of property needed, let's start by finding the total width and length of the pool including the decks.

Let's assume the length of the pool is L, and the width is W. We know that the area of the pool is 1800 square feet, so we can write the equation:

L * W = 1800

Now let's account for the decks. We're given that there are 5-foot wide decks along either side, so we need to add 2 * 5 = 10 feet to the width of the pool:

(L + 10) * W = 1800

Similarly, we're told there are 10-foot wide decks at the two ends, so we also need to add 2 * 10 = 20 feet to the length of the pool:

(L + 10) * (W + 20) = 1800

Now we can solve for L and W. However, we're looking for the dimensions of the smallest piece of property, so we want to minimize L and W. To achieve that, we can look for a factorization of 1800 that gives us the smallest possible values for L and W.

The prime factorization of 1800 is 2 * 2 * 2 * 3 * 3 * 5 * 5. Now we can try different combinations of these factors that minimize L and W.

One potential option is L = 6, W = 300:

(6 + 10) * (300 + 20) = 16 * 320 = 5120

Another option is L = 12, W = 150:

(12 + 10) * (150 + 20) = 22 * 170 = 3740

As you can see, there are different combinations that give us an area of 1800 square feet. So, 16 by 18 may work, but it's not necessarily the only solution.

I hope this helps you swim your way through the problem! Let me know if you have any other questions, or if you need a humorous lifeguard at your service.

To find the dimensions of the smallest piece of property on which the pool can be built, we need to consider the total area of the pool including the decks.

Let's assume the length of the rectangular pool is L and the width is W.

We know that the area of the pool is given by: Area of pool = Length × Width

In this case, the area of the pool is 1800 square feet.

So, we have: L × W = 1800

Now, let's consider the decks. The owner wants 5-foot wide decks along either side, which means the total width of the pool with the decks on both sides will be W + 5 + 5 = W + 10. Similarly, the total length of the pool with decks on both ends will be L + 10 + 10 = L + 20.

So, to find the dimensions of the smallest piece of property, we need to consider the total area of the pool including the decks. We have:

Total area = (L + 20) × (W + 10)

Given that the total area is 1800 square feet, we can write the equation:

(L + 20) × (W + 10) = 1800

Now we have a system of equations to solve:

L × W = 1800 (Equation 1)
(L + 20) × (W + 10) = 1800 (Equation 2)

To find the dimensions of the smallest piece of property, we need to solve this system of equations.

One way to do this is to substitute the value of L from Equation 1 into Equation 2:

(W + 20) × (W + 10) = 1800

Expand the equation:

W^2 + 30W + 200 = 1800

Rearrange the equation:

W^2 + 30W - 1600 = 0

We can now solve this quadratic equation to find the possible values of W.

Once we have the value of W, we can substitute it back into Equation 1 to find the corresponding value of L.

After solving the quadratic equation, we find that the possible values for W are approximately -50 and 32. Since the width of the pool cannot be negative, we discard -50. Therefore, the width of the pool is approximately 32 feet.

Substituting this value back into Equation 1, we find:

L × 32 = 1800

Solving for L, we get L ≈ 56.25 feet.

So, the dimensions of the smallest piece of property on which the pool can be built, satisfying the given conditions, are approximately 56.25 feet by 32 feet.

With lot dimensions of 16 x 18, you don't even have enough room for the pool. How would you fit in an 1800 square feet of pool?

You need to set this up as a calculus problem and look for the minimum of the Lot Area function. Let x be the shorter width of the pool rectangle. The other pool lendth dimension is 1800/x. The 10 foot deck widths should be on the narrower "x" sides of the pool, to minimize total area.

Total land area needed =
A(x) = 1800 + 2*10(x+10)+ 2*5(1800/x)
= 1800 + 20x + 200 + 18000/x
= 2000 + 20 x + 18000/x

Set the derivative equal to zero and solve for x.

20 - 18000/x^2 = 0
x = sqrt (900) = 30
pool length dimension = 1800/x = 60
Lot dimensions: 80 x 40