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Kyla makes a triangular school pennant. The area of the triangle is 180 square inches. The base of the pennant is z inches long. The height is 6 inches longer than twice the base length.
What is the height of the pennant? Recall the formula
A = bh.
12 inches
15 inches
30 inches
36 inches

We are given a triangular pennant, with the following details:

- The base is \( z \) inches long.
- The height is \( 2z + 6 \) inches, which is 6 inches longer than twice the base.
- The area \( A \) of the triangle is 180 square inches.

We need to find the height.

### Formula:
The area of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Substituting the given values:

\[
180 = \frac{1}{2} \times z \times (2z + 6)
\]

### Solve for \( z \):

1. Multiply both sides of the equation by 2 to eliminate the fraction:
   \[
   360 = z \times (2z + 6)
   \]

2. Distribute \( z \):
   \[
   360 = 2z^2 + 6z
   \]

3. Rearrange the equation into standard quadratic form:
   \[
   2z^2 + 6z - 360 = 0
   \]

4. Divide the entire equation by 2 to simplify:
   \[
   z^2 + 3z - 180 = 0
   \]

### Solve the quadratic equation using the quadratic formula:
The quadratic formula is:
\[
z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( z^2 + 3z - 180 = 0 \), we have \( a = 1 \), \( b = 3 \), and \( c = -180 \).

Now, let's calculate the value of \( z \).The solutions for \( z \) are \( z = 12 \) and \( z = -15 \). Since the base of the triangle cannot be negative, we select \( z = 12 \).

### Find the height:
The height of the pennant is \( 2z + 6 \), so:
\[
\text{Height} = 2(12) + 6 = 24 + 6 = 30 \text{ inches}.
\]

Thus, the height of the pennant is **30 inches**.

Would you like more details on the process or have any questions?

Here are 5 related questions to expand on this problem:
1. What is the perimeter of the triangular pennant if the third side is unknown?
2. How would the area change if the base was increased by 5 inches?
3. Can you verify the solution by plugging \( z = 12 \) into the area formula again?
4. How do we derive the quadratic formula from completing the square?
5. What is the geometric significance of a negative solution in this context?

**Tip:** When solving quadratic equations, always verify if both solutions are feasible in the context of the problem.

Kyla makes a triangular school pennant. The area of the triangle is 180 square inches. The base of the pennant is z inches long. The height is 6 inches longer than twice the base length. What is the height of the pennant?

Learn how to solve for the height of a triangular school pennant where the area is 180 square inches, the base is z inches, and the height is 6 inches longer than twice the base. This problem involves using the area of a triangle formula and solving a quadratic equation. Suitable for Grades 7-9.

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