A triangular traffic island has a base half as long as it's height. The island has an area of 144m^2. Find the base and the height.
(1/2) b h = 144
b = h/2
(1/2) h^2/2 = 144
h^2 = 144 * 4
so
h = 12 * 2 = 24
To solve this problem, we can use the formula for the area of a triangle, which is given by the formula:
Area = (base * height) / 2.
Let's denote the base of the triangular traffic island as 'b' and its height as 'h'. According to the given information, we have:
base = (1/2) * height.
We are also given the area, which is 144 m^2. Substituting these values into the formula, we get:
144 = (b * h) / 2.
Multiplying both sides of the equation by 2, we have:
288 = b * h.
Next, we substitute the expression for the base in terms of the height:
288 = ((1/2) * h) * h.
Simplifying the equation, we obtain:
288 = (1/2) * h^2.
Now, we can solve for the height (h). First, multiply both sides of the equation by 2 to remove the fraction:
576 = h^2.
Taking the square root of both sides, we find:
h = √576.
Calculating the square root, we get:
h ≈ 24.
Finally, substitute this value back into the expression for the base:
base = (1/2) * 24 = 12.
Thus, the base of the triangular traffic island is 12 meters, and the height is 24 meters.
To find the base and height of the triangular traffic island, we can use the formula for the area of a triangle:
Area = (base * height) / 2
Given that the area of the traffic island is 144m², we can substitute this value into the formula:
144 = (base * height) / 2
Next, let's use the given information that the base is half as long as the height. We can represent this relationship as:
base = (1/2) * height
Now, substitute this value for the base in terms of the height in the equation:
144 = ((1/2) * height * height) / 2
To further simplify, we can multiply both sides of the equation by 2:
288 = height * height
Now, we can solve for the height by taking the square root of both sides:
√288 = √(height * height)
Simplifying further:
height = √288
Calculating the square root of 288, we get:
height ≈ 16.97
Finally, since the base is half as long as the height, we can calculate the base:
base = (1/2) * height
base = (1/2) * 16.97
base ≈ 8.48
Therefore, the height of the triangular traffic island is approximately 16.97m, and the base is approximately 8.48m.