Volume V in cubic feet of an aquarium modeled by polynomial function V(x)=x^3+2x^2-13x+10, where x is the length of the tank. Volume = 3ft^2
c)Find the dimensions of the aquarium shown

To find the dimensions of the aquarium, we are given the polynomial function for the volume \( V(x) = x^3 + 2x^2 - 13x + 10 \), where \( x \) represents the length of the tank. Additionally, the volume is given as \( 3 \, \text{ft}^3 \).

### Steps:
1. **Set the volume equation equal to 3 cubic feet:**
   \[
   V(x) = x^3 + 2x^2 - 13x + 10 = 3
   \]
   
2. **Simplify the equation by subtracting 3 from both sides:**
   \[
   x^3 + 2x^2 - 13x + 10 - 3 = 0
   \]
   \[
   x^3 + 2x^2 - 13x + 7 = 0
   \]
   
3. **Solve the cubic equation \( x^3 + 2x^2 - 13x + 7 = 0 \):**

   This can be solved either by factoring or using numerical methods such as the Rational Root Theorem or synthetic division.

### Factoring or Finding Real Roots:

Using a numerical approach or factoring techniques, we need to find values of \( x \) that satisfy the equation. We can start by checking possible rational roots like \( \pm 1, \pm 7 \).

Let me solve this equation to find the roots of \( x^3 + 2x^2 - 13x + 7 = 0 \).The roots of the equation \( x^3 + 2x^2 - 13x + 7 = 0 \) are approximately:

\[
x \approx -0.20, \quad x \approx 0.43, \quad x \approx 1.63
\]

Since \( x \) represents the length of the aquarium, we discard the negative root \( x \approx -0.20 \) as it is not physically meaningful. Therefore, the potential valid dimensions for the length are:

\[
x \approx 0.43 \, \text{feet} \quad \text{or} \quad x \approx 1.63 \, \text{feet}.
\]

These lengths correspond to two potential configurations of the aquarium that give a volume of \( 3 \, \text{ft}^3 \).

### Would you like more details on solving this, or have any other questions?

Here are 5 related questions for practice:
1. How do you calculate the volume of a rectangular aquarium?
2. What methods can be used to solve cubic equations?
3. Can a cubic equation have more than one real root?
4. What is the significance of discarding negative roots in practical problems?
5. How do you use the Rational Root Theorem to find potential roots?

**Tip:** When solving polynomial equations, always check if negative or non-real solutions make sense in the context of the problem!

Volume V in cubic feet of an aquarium modeled by polynomial function V(x)=x^3+2x^2-13x+10, where x is the length of the tank. Volume = 3ft^2 c) Find the dimensions of the aquarium shown.

Learn how to find the dimensions of an aquarium modeled by the cubic polynomial V(x) = x^3 + 2x^2 - 13x + 10, where the volume is 3 cubic feet. This detailed solution explores solving cubic equations, factoring techniques, and real root identification. Suitable for high school students in Grades 9-12.

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