An Aquarium Has Dimensions $2x$ Feet, $(x-1$\] Feet, And $(x+2$\] Feet. The Aquarium Volume Must Be No More Than 280 Cubic Feet.What Are The Possible Values Of $x$?A. $(1,5$\]B. $(1,5\]C.
Introduction
When it comes to designing an aquarium, one of the most critical factors to consider is its volume. The volume of an aquarium determines the amount of water it can hold, which in turn affects the type of fish and plants that can be kept in it. In this article, we will explore the problem of finding the possible values of x, given that an aquarium has dimensions 2x feet, (x-1) feet, and (x+2) feet, and its volume must be no more than 280 cubic feet.
The Problem
The problem can be stated as follows: Find the possible values of x, given that the volume of the aquarium is no more than 280 cubic feet. The dimensions of the aquarium are 2x feet, (x-1) feet, and (x+2) feet.
The Solution
To find the possible values of x, we need to calculate the volume of the aquarium and set it less than or equal to 280 cubic feet. The volume of a rectangular prism, such as an aquarium, is given by the formula:
V = lwh
where l is the length, w is the width, and h is the height.
In this case, the length, width, and height of the aquarium are 2x feet, (x-1) feet, and (x+2) feet, respectively. Therefore, the volume of the aquarium is:
V = (2x)(x-1)(x+2)
To find the possible values of x, we need to set the volume less than or equal to 280 cubic feet and solve for x:
(2x)(x-1)(x+2) ≤ 280
Expanding and Simplifying the Inequality
To solve the inequality, we need to expand and simplify it. We can do this by multiplying out the terms:
2x(x-1)(x+2) ≤ 280
Expanding the left-hand side of the inequality, we get:
2x(x^2 + x - 2) ≤ 280
Simplifying the left-hand side of the inequality, we get:
2x^3 + 2x^2 - 4x ≤ 280
Rearranging the Inequality
To make it easier to solve the inequality, we can rearrange it by subtracting 280 from both sides:
2x^3 + 2x^2 - 4x - 280 ≤ 0
Factoring the Inequality
To solve the inequality, we can try to factor the left-hand side. Unfortunately, this inequality does not factor easily, so we will need to use other methods to solve it.
Using the Rational Root Theorem
One method we can use to solve the inequality is the rational root theorem. This theorem states that if a rational number p/q is a root of the polynomial, then p must be a factor of the constant term, and q must be a factor of the leading coefficient.
In this case, the constant term is -280, and the leading coefficient is 2. Therefore, the possible rational roots of the polynomial are:
±1, ±2, ±4, ±5, ±7, ±10, ±14, ±20, ±28, ±35, ±70, ±140
Using Synthetic Division
To test these possible rational roots, we can use synthetic division. This is a method of dividing a polynomial by a linear factor, and it can be used to test whether a given number is a root of the polynomial.
Let's try using synthetic division to test the possible rational roots. We will start by testing the positive rational roots.
Testing the Positive Rational Roots
Using synthetic division, we can test the positive rational roots as follows:
- 1: 2x^3 + 2x^2 - 4x - 280 = (x - 5)(2x^2 + 12x + 56)
- 2: 2x^3 + 2x^2 - 4x - 280 = (x - 5)(2x^2 + 12x + 56)
- 4: 2x^3 + 2x^2 - 4x - 280 = (x - 5)(2x^2 + 12x + 56)
- 5: 2x^3 + 2x^2 - 4x - 280 = (x - 5)(2x^2 + 12x + 56)
As we can see, the polynomial is equal to zero when x = 5. Therefore, x = 5 is a root of the polynomial.
Testing the Negative Rational Roots
Now that we have found one root, we can use synthetic division to test the negative rational roots. We will start by testing the negative rational roots.
- -1: 2x^3 + 2x^2 - 4x - 280 = (x + 1)(2x^2 + 10x + 280)
- -2: 2x^3 + 2x^2 - 4x - 280 = (x + 2)(2x^2 + 6x + 140)
- -4: 2x^3 + 2x^2 - 4x - 280 = (x + 4)(2x^2 + 2x + 70)
- -5: 2x^3 + 2x^2 - 4x - 280 = (x + 5)(2x^2 - 12x + 56)
As we can see, the polynomial is not equal to zero when x = -1, -2, -4, or -5. Therefore, these numbers are not roots of the polynomial.
Finding the Other Roots
Now that we have found one root, x = 5, we can use synthetic division to find the other roots. We will start by dividing the polynomial by (x - 5).
2x^3 + 2x^2 - 4x - 280 = (x - 5)(2x^2 + 12x + 56)
Using synthetic division, we can find the other roots as follows:
- 2x^2 + 12x + 56 = (2x + 7)(x + 8)
Therefore, the other roots of the polynomial are x = -7/2 and x = -8.
Conclusion
In conclusion, the possible values of x are x = 5, x = -7/2, and x = -8. These values satisfy the inequality (2x)(x-1)(x+2) ≤ 280.
Discussion
The problem of finding the possible values of x, given that an aquarium has dimensions 2x feet, (x-1) feet, and (x+2) feet, and its volume must be no more than 280 cubic feet, is a classic example of a problem that can be solved using algebraic methods. The solution involves expanding and simplifying the inequality, factoring the polynomial, and using synthetic division to find the roots.
The possible values of x are x = 5, x = -7/2, and x = -8. These values satisfy the inequality (2x)(x-1)(x+2) ≤ 280.
Final Answer
The final answer is .
Introduction
In our previous article, we explored the problem of finding the possible values of x, given that an aquarium has dimensions 2x feet, (x-1) feet, and (x+2) feet, and its volume must be no more than 280 cubic feet. In this article, we will answer some of the most frequently asked questions related to this problem.
Q: What is the formula for the volume of a rectangular prism?
A: The formula for the volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height.
Q: How do I calculate the volume of the aquarium?
A: To calculate the volume of the aquarium, you need to multiply the length, width, and height of the aquarium. In this case, the length, width, and height of the aquarium are 2x feet, (x-1) feet, and (x+2) feet, respectively. Therefore, the volume of the aquarium is:
V = (2x)(x-1)(x+2)
Q: What is the inequality that represents the problem?
A: The inequality that represents the problem is (2x)(x-1)(x+2) ≤ 280.
Q: How do I solve the inequality?
A: To solve the inequality, you need to expand and simplify it, factor the polynomial, and use synthetic division to find the roots.
Q: What are the possible values of x?
A: The possible values of x are x = 5, x = -7/2, and x = -8.
Q: Why are these values of x possible?
A: These values of x are possible because they satisfy the inequality (2x)(x-1)(x+2) ≤ 280.
Q: What is the significance of the possible values of x?
A: The possible values of x represent the possible dimensions of the aquarium that satisfy the given condition.
Q: How do I use the possible values of x to design the aquarium?
A: To design the aquarium, you need to choose a value of x that satisfies the given condition. For example, if you choose x = 5, the dimensions of the aquarium will be 10 feet, 4 feet, and 12 feet.
Q: What are some common mistakes to avoid when solving this problem?
A: Some common mistakes to avoid when solving this problem include:
- Not expanding and simplifying the inequality correctly
- Not factoring the polynomial correctly
- Not using synthetic division correctly
- Not checking the possible values of x carefully
Q: How can I apply this problem to real-world scenarios?
A: This problem can be applied to real-world scenarios such as designing aquariums, swimming pools, and other rectangular prisms.
Q: What are some extensions of this problem?
A: Some extensions of this problem include:
- Finding the maximum volume of the aquarium
- Finding the minimum volume of the aquarium
- Finding the dimensions of the aquarium that maximize the volume
- Finding the dimensions of the aquarium that minimize the volume
Conclusion
In conclusion, the problem of finding the possible values of x, given that an aquarium has dimensions 2x feet, (x-1) feet, and (x+2) feet, and its volume must be no more than 280 cubic feet, is a classic example of a problem that can be solved using algebraic methods. The possible values of x are x = 5, x = -7/2, and x = -8. These values satisfy the inequality (2x)(x-1)(x+2) ≤ 280.
Final Answer
The final answer is .