What is the positive solution to the equation `0=\frac{1}{3}x^{2}-3`

To solve the equation \( 0 = \frac{1}{3}x^{2} - 3 \), we can begin by isolating \( x^{2} \).

First, add \( 3 \) to both sides:

\[ \frac{1}{3}x^{2} = 3 \]

Next, multiply both sides by \( 3 \) to eliminate the fraction:

\[ x^{2} = 9 \]

Now, take the square root of both sides:

\[ x = \pm 3 \]

Since we are looking for the positive solution, we have:

\[ x = 3 \]

Thus, the positive solution to the equation is

\[ \boxed{3} \]

he area of a rectangular trampoline is 112 ft`^{2}`. The length of the trampoline is 6 ft greater than the width of the trampoline. This situation can be represented by the equation `w^{2}+6w-112=0.`

What is the width of the trampoline in feet?

To solve for the width of the trampoline using the equation \( w^2 + 6w - 112 = 0 \), we can use the quadratic formula:

\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation, the coefficients are:

• \( a = 1 \)
• \( b = 6 \)
• \( c = -112 \)

Now, substitute these values into the quadratic formula:

1. Compute the discriminant:

\[ b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot (-112) \] \[ = 36 + 448 \] \[ = 484 \]

2. Find the square root of the discriminant:

\[ \sqrt{484} = 22 \]

3. Now, substitute back into the quadratic formula:

\[ w = \frac{-6 \pm 22}{2 \cdot 1} \]

This gives us two potential solutions:

\[ w = \frac{-6 + 22}{2} = \frac{16}{2} = 8 \]

\[ w = \frac{-6 - 22}{2} = \frac{-28}{2} = -14 \]

Since width cannot be negative, we discard \( w = -14 \).

Thus, the width of the trampoline is

\[ \boxed{8} \text{ ft.} \]

To confirm, we can also find the length:

\[ l = w + 6 = 8 + 6 = 14 \text{ ft.} \]

Finally, check the area:

\[ \text{Area} = w \times l = 8 \times 14 = 112 \text{ ft}^2, \]

which matches the given area of the trampoline. Therefore, the width is indeed

\[ \boxed{8} \text{ ft.} \]

The total number of seats in an auditorium is modeled by `f\left(x\right)=2x^{2}-6x,`where `x` represents the number of rows of seats. How many rows are there n the auditorium if it has a total of 416 seats?

To determine the number of rows in the auditorium given that the total number of seats is 416, we can set the equation:

\[ f(x) = 2x^2 - 6x \]

and set it equal to 416:

\[ 2x^2 - 6x = 416 \]

Now, rearranging this equation, we subtract 416 from both sides:

\[ 2x^2 - 6x - 416 = 0 \]

Next, we can simplify this equation by dividing the entire equation by 2:

\[ x^2 - 3x - 208 = 0 \]

Now, we will use the quadratic formula to solve for \( x \):

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, the coefficients are:

• \( a = 1 \)
• \( b = -3 \)
• \( c = -208 \)

Substituting these values into the quadratic formula:

1. Compute the discriminant:

\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-208) \] \[ = 9 + 832 \] \[ = 841 \]

2. Find the square root of the discriminant:

\[ \sqrt{841} = 29 \]

3. Now substitute back into the quadratic formula:

\[ x = \frac{-(-3) \pm 29}{2 \cdot 1} \] \[ x = \frac{3 \pm 29}{2} \]

This gives us two potential solutions:

\[ x = \frac{3 + 29}{2} = \frac{32}{2} = 16 \]

\[ x = \frac{3 - 29}{2} = \frac{-26}{2} = -13 \]

Since the number of rows cannot be negative, we discard \( x = -13 \).

Thus, the number of rows in the auditorium is

\[ \boxed{16}. \]