# A zoo wishes to construct an aquarium in the shape of a rectangular prism such that the length is twice the width and 5m greater than the height. If the aquarium must have a volume strictly between 1125m^3 and 3000m^3, determine the restrictions on the length of the aquarium.

## let the length be x

let the width be x/2
let the height be x-5

x(x/2)(x-5) > 1125
x^2(x-5) ≥ 2250
let's solve x^2(x-5) = 2250
with the help of Wolfram

http://www.wolframalpha.com/input/?i=solve+x%5E2(x-5)+%3D+2250

x = 15

x(x/2)(x-5) < 3000
x^2(x-5) = 6000 and that gives me x = 20

so 15 < x < 20

## Well, the zoo really needs to "scale" back their ambitions here. They don't want an aquarium that's too small or too big, they just want something "just right" - like a porridge temperature Goldilocks.

To find the restrictions on the length, let's break this problem down. We know that the length is twice the width, so let's call the width 'w.' That means the length is 2w.

Oh, and the height is 5m greater than the height...I mean, 5m greater than the width. So, the height can be represented as w + 5.

To calculate the volume of a rectangular prism, we multiply the length, width, and height. So the volume of this aquarium can be represented as:

Volume = length (2w) * width (w) * height (w + 5)

We also know that the volume must be between 1125m^3 and 3000m^3. So, we can set up the following inequality:

1125m^3 < 2w * w * (w + 5) < 3000m^3

But hey, clown math here. Let's simplify this inequality a bit:

1125 < 2w^2(w + 5) < 3000

Now, if we divide the whole inequality by 2 to make our lives easier, we get:

562.5 < w^3 + 5w^2 < 1500

So, the length of the aquarium (2w) must satisfy the inequality 1125/2 < 2w^2(w + 5) < 3000/2.

That gives us a range of acceptable lengths for the aquarium!

## Let's assign variables to the dimensions of the rectangular prism:

Width = W (in meters)
Length = 2W (in meters)
Height = H (in meters)

We know that the volume of a rectangular prism is obtained by multiplying the three dimensions, so we can write the equation for the volume:

Volume = Width * Length * Height

Substituting the given values:

Volume = W * (2W) * H
Volume = 2W^2 * H

From the given information, the volume must be strictly between 1125m^3 and 3000m^3. Therefore, we can write the following inequalities:

1125 < 2W^2 * H < 3000

Let's focus on the lower bound first:

1125 < 2W^2 * H

Simplifying, we get:

1125 < 2W^2 * H
1125 / H < 2W^2
562.5 / H < W^2

Now let's move on to the upper bound:

2W^2 * H < 3000

Simplifying, we get:

2W^2 * H < 3000
W^2 * H < 1500
W^2 < 1500 / H

Combining the two inequalities, we have:

562.5 / H < W^2 < 1500 / H

To determine the restrictions on the length, we need to find the possible values of W that satisfy these inequalities.

Taking the square root of both sides, we get:

sqrt(562.5 / H) < W < sqrt(1500 / H)

However, we need to keep in mind that W = Width, so it must be positive. Therefore, the restrictions on the length of the aquarium are:

sqrt(562.5 / H) < W ≤ sqrt(1500 / H)

## To determine the restrictions on the length of the aquarium, we need to analyze the given information about the dimensions and volume.

Let's say:
- Width = w meters
- Length = 2w meters (since it is twice the width)
- Height = h meters (5 meters less than the length)

The volume of a rectangular prism is given by the formula: Volume = Length x Width x Height.

According to the problem, the volume should be strictly between 1125m^3 and 3000m^3. So we have the inequality:

1125 < (Length)(Width)(Height) < 3000

Substituting the dimensions into the inequality gives us:

1125 < (2w)(w)(w-5) < 3000

Simplifying the inequality, we get:

1125 < 2w^2(w-5) < 3000

Now, let's solve this inequality to determine the restrictions on the length:

1125 < 2w^2(w-5) < 3000

First, let's solve the lower bound of the inequality:

1125 < 2w^2(w-5)

Dividing both sides by 2 gives:

562.5 < w^2(w-5)

Taking the cube root of both sides, we get:

(w-5)^(1/3) < w^(2/3)

Simplifying the cube root and square root terms, we have:

(w-5)^(1/3) < (w^2)^(1/3)

w-5 < w^2

Rearranging the equation, we get:

w^2 - w + 5 > 0

Using the quadratic formula, we can find the roots of this quadratic equation:

w = (-(-1) ± √((-1)^2 - 4(1)(5))) / (2(1))
w = (1 ± √(-19)) / 2

Since the discriminant is negative (√(-19)), there are no real solutions for w in this case.

Now, let's solve the upper bound of the inequality:

2w^2(w-5) < 3000

Dividing both sides by 2 gives:

w^2(w-5) < 1500

Expanding the equation gives:

w^3 - 5w^2 - 1500 < 0

We can use a graphing calculator or software to find the approximate solutions, which are:

w ≈ -25.729
w ≈ 0.399
w ≈ 25.33

Since the width of the aquarium cannot be negative, the valid solution for the width is approximately 0.399 meters (rounded to three decimal places).

Now, we can determine the length of the aquarium, which is twice the width:

Length = 2w ≈ 2(0.399) ≈ 0.798 meters

Therefore, the restriction on the length of the aquarium is that it must be approximately 0.798 meters (rounded to three decimal places).