The problem states that a triangular neon billboard has an area of 125 square feet. The base of the triangle is 5 feet longer than twice the length of the altitude. We need to find: (a) The dimensions of the triangular billboard in feet (altitude and base). (b) The dimensions of the triangular billboard in yards.

1. Problem Description

The problem states that a triangular neon billboard has an area of 125 square feet. The base of the triangle is 5 feet longer than twice the length of the altitude. We need to find:

(a) The dimensions of the triangular billboard in feet (altitude and base).

(b) The dimensions of the triangular billboard in yards.

2. Solution Steps

(a) Let $h$ be the length of the altitude (height) of the triangle in feet, and $b$ be the length of the base in feet.

We are given that the base is 5 feet longer than twice the altitude, so we can write this as:

$b = 2h + 5$

The area of a triangle is given by the formula:

$Area = \frac{1}{2} \times base \times height$

We are given that the area is 125 square feet. Therefore, we have:

$125 = \frac{1}{2} \times b \times h$

Substitute $b = 2h + 5$ into the area equation:

$125 = \frac{1}{2} \times (2h + 5) \times h$

Multiply both sides by 2:

$250 = (2h + 5)h$

$250 = 2h^2 + 5h$

$2h^2 + 5h - 250 = 0$

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

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

In this case, $a = 2$, $b = 5$, and $c = -250$.

$h = \frac{-5 \pm \sqrt{5^2 - 4(2)(-250)}}{2(2)}$

$h = \frac{-5 \pm \sqrt{25 + 2000}}{4}$

$h = \frac{-5 \pm \sqrt{2025}}{4}$

$h = \frac{-5 \pm 45}{4}$

We have two possible values for $h$:

$h = \frac{-5 + 45}{4} = \frac{40}{4} = 10$

$h = \frac{-5 - 45}{4} = \frac{-50}{4} = -12.5$

Since the height cannot be negative, we have $h = 10$ feet.

Now, we can find the base:

$b = 2h + 5 = 2(10) + 5 = 20 + 5 = 25$ feet.

So, the altitude is 10 feet and the base is 25 feet.

(b) To convert the dimensions from feet to yards, we use the conversion factor:

1 yard = 3 feet.

Altitude in yards: $h_{yards} = \frac{10}{3}$ yards $\approx 3.33$ yards

Base in yards: $b_{yards} = \frac{25}{3}$ yards $\approx 8.33$ yards

3. Final Answer

(a) The length of the altitude of the triangular billboard is 10 feet.

(a) The dimensions of the triangular billboard in feet are: altitude = 10 feet, base = 25 feet.

(b) The dimensions of the triangular billboard in yards are: altitude = $\frac{10}{3}$ yards, base = $\frac{25}{3}$ yards.