The area of a triangular billboard is 76 square feet. The base of the triangle is 3 feet longer than twice the length of the altitude. We need to find the dimensions of the triangular billboard in feet. Specifically, we need to find the length of the altitude.

1. Problem Description

The area of a triangular billboard is 76 square feet. The base of the triangle is 3 feet longer than twice the length of the altitude. We need to find the dimensions of the triangular billboard in feet. Specifically, we need to find the length of the altitude.

2. Solution Steps

Let $h$ be the length of the altitude in feet, and $b$ be the length of the base in feet.

We are given that the base is 3 feet longer than twice the length of the altitude. So,

$b = 2h + 3$.

We are also given that the area of the triangle is 76 square feet. The formula for the area of a triangle is

$A = \frac{1}{2}bh$.

Substituting the given area and the expression for $b$ in terms of $h$, we have:

$76 = \frac{1}{2}(2h + 3)h$

Multiplying both sides by 2, we get

$152 = (2h + 3)h$

$152 = 2h^2 + 3h$

Rearranging the terms, we get a quadratic equation:

$2h^2 + 3h - 152 = 0$

We can solve this quadratic equation for $h$ using the quadratic formula:

$h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 2$, $b = 3$, and $c = -152$. Plugging these values into the quadratic formula, we get:

$h = \frac{-3 \pm \sqrt{3^2 - 4(2)(-152)}}{2(2)}$

$h = \frac{-3 \pm \sqrt{9 + 1216}}{4}$

$h = \frac{-3 \pm \sqrt{1225}}{4}$

$h = \frac{-3 \pm 35}{4}$

We have two possible values for $h$:

$h = \frac{-3 + 35}{4} = \frac{32}{4} = 8$

$h = \frac{-3 - 35}{4} = \frac{-38}{4} = -9.5$

Since the altitude must be positive, we take the positive value: $h = 8$ feet.

The base is $b = 2h + 3 = 2(8) + 3 = 16 + 3 = 19$ feet.

So the length of the altitude is 8 feet.

3. Final Answer

The length of the altitude of the triangular billboard is 8 feet.