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.
Why did the fish blush? Because it saw the aquarium's volume range and thought it was a scale limit!
Let's solve this prism puzzle for the length.
Given that the length is twice the width, we can call the width "w". So, the length would be "2w".
Also, given that the length is 5m greater than the height, we can call the height "h". So, the length would be "h + 5".
Now, the volume of a rectangular prism is given by V = length × width × height.
Substituting the values we have, we get V = (2w) × w × (h + 5).
Simplifying, we have V = 2w²(h + 5).
To satisfy the volume range condition, we can write the inequality as 1125 < 2w²(h + 5) < 3000.
Let's focus on the left side first:
1125 < 2w²(h + 5).
Divide both sides by 2(h + 5):
562.5 < w²(h + 5).
For the right side:
2w²(h + 5) < 3000.
Divide both sides by 2(h + 5):
w²(h + 5) < 1500.
Now, we have two inequalities: 562.5 < w²(h + 5) and w²(h + 5) < 1500.
To find the restriction on the length, we need to solve these inequalities simultaneously.
However, without values for "w" and "h," we cannot determine the specific range for the length. So, it's like a comedy show without a punchline!
If you can provide specific values for "w" and "h," I'd be happy to help you solve this rectangular prism riddle!
Let's solve this step-by-step:
Step 1: Represent the unknowns
Let's represent the width as "w" meters.
Since the length is twice the width, we can represent it as "2w" meters.
The height is 5 meters less than the length, so we can represent it as "2w - 5" meters.
Step 2: Express the volume of the aquarium
The volume of a rectangular prism is given by the formula V = length × width × height. In this case, the volume is given to be between 1125m³ and 3000m³, so we can write the inequality as:
1125 < (2w) × w × (2w - 5) < 3000
Step 3: Simplify the inequality
To simplify the inequality, let's expand and combine like terms:
1125 < 4w² - 10w < 3000
Step 4: Solve the inequality
To solve the inequality, we'll isolate the quadratic equation:
4w² - 10w - 1125 > 0 and 4w² - 10w - 3000 < 0
Now, let's factorize the quadratic equation:
(2w - 45)(2w + 25) > 0 and (2w + 75)(2w - 40) < 0
Step 5: Analyze the factors
Analyzing the factors, we can determine the restrictions on the length:
For greater than zero (positive):
2w - 45 > 0 and 2w + 25 > 0
2w > 45 and 2w > -25
w > 22.5 and w > -12.5
Since width cannot be negative, we can discard w > -12.5.
Thus, the restriction on the width is w > 22.5
For less than zero (negative):
2w + 75 < 0 and 2w - 40 < 0
2w < -75 and 2w < 40
w < -37.5 and w < 20
These restrictions are irrelevant in the context of our problem because we are dealing with physical dimensions, where the width (and any other dimension) cannot be negative.
Step 6: Conclusion
Therefore, the only restriction for the length of the aquarium is that it must be greater than 22.5 meters.
To determine the restrictions on the length of the aquarium, we need to use the given information to set up equations and solve for the variables.
Let's assign variables:
Let's say the width of the rectangular prism is "w" meters.
Then, the length of the aquarium would be "2w" meters (since it is stated that the length is twice the width).
The height of the aquarium would be "h" meters.
The volume of a rectangular prism is given by the formula: V = length * width * height.
Given that the volume of the aquarium must be strictly between 1125m^3 and 3000m^3, we can write the following inequality:
1125 < (2w)(w)(h) < 3000
Let's simplify the inequality:
1125 < 2w^2h < 3000
Now, let's look at the additional information stated in the problem:
The length (2w) is also 5 meters greater than the height (h). So we can write another equation: 2w = h + 5.
We can now substitute this equation into the inequality:
1125 < 2w^2(h + 5) < 3000
Let's simplify further:
1125 < 2w^2h + 10w^2 < 3000
To determine the restrictions on the length, we need to isolate the length term (2w) in the inequality.
First, let's subtract 10w^2 from all sides:
1125 - 10w^2 < 2w^2h < 3000 - 10w^2
Now, let's divide each term by 2h:
(1125 - 10w^2) / (2h) < w^2 < (3000 - 10w^2) / (2h)
To further simplify the inequality, let's address the relationship between the width and the height. We know that the width (w) will always be positive, so we can assume it's greater than zero.
If the height (h) is positive, then the equation w^2 > 0 holds true.
The inequality can now be simplified to:
(1125 - 10w^2) / (2h) < w^2
To determine the restrictions on the length (2w), we need to find the values of w that make this inequality true. Unfortunately, since we don't know the value of h, we can't solve for the specific restrictions on the length. We need more information about the relationship between w and h to determine the exact restrictions on the length of the aquarium.