Question

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 $a=\frac{1}{2}bh$. 12 inches 15 inches 30 inches 36 inches

Explanation:

Step1: Substitute into area formula

Given \(A = 180\), \(b=z\), \(h = 2z + 6\), and \(A=\frac{1}{2}bh\). Substitute these into the formula: \(180=\frac{1}{2}(z)(2z + 6)\).

Step2: Simplify the equation

Multiply both sides by \(2\) to get \(360=z(2z + 6)\). Expand: \(360 = 2z^{2}+6z\). Divide by \(2\): \(180=z^{2}+3z\). Rearrange to \(z^{2}+3z - 180=0\).

Step3: Factor the quadratic equation

Find two numbers that multiply to \(- 180\) and add to \(3\). The numbers are \(15\) and \(-12\). So \((z + 15)(z - 12)=0\).

Step4: Solve for \(z\)

Set each factor equal to zero: \(z+15 = 0\) gives \(z=-15\) (rejected as length can't be negative), \(z - 12=0\) gives \(z = 12\).

Step5: Find the height

Since \(h=2z + 6\), substitute \(z = 12\). \(h=2\times12+6=24 + 6=30\).

Answer:

30 inches