A horse pasture is formed by a large semi-circular grassy area and a triangular-shaped riding area. The base of the
riding area is 50 feet and it extends to a height of 75 feet. What is the total area of the pasture if the height of the riding
area equals the diameter of the grassy area? Round to the nearest tenth.
At first I was going to assume that the base of the triangle would fall on the diameter of the semi-circle but then the data contradicted that, so we have 2 separate areas ?
Height of triangle equals diameter of semi-circle, so r = 75/2 = 37.5 ft
total area = triangle + semicircle
= (1/2)(50)(75) + (1/2) π (37.5^2) = .....
To find the total area of the pasture, we need to find the areas of both the semi-circular grassy area and the triangular riding area, and then add them together.
First, let's find the area of the semi-circular grassy area:
The diameter of the grassy area is equal to the height of the riding area, which is 75 feet. The radius of the grassy area is half of the diameter, so it is 75 feet divided by 2, which is 37.5 feet.
The formula to find the area of a semi-circle is A = (π * r^2) / 2, where A is the area and r is the radius.
Substituting the values, we get A = (π * (37.5 feet)^2) / 2.
Simplifying the equation, we get A = π * (37.5 feet)^2 / 2.
Using the approximation for π as 3.14, we can calculate the area:
A ≈ 3.14 * (37.5 feet)^2 / 2.
A ≈ 3.14 * 1406.25 feet^2 / 2.
A ≈ 4397.0625 feet^2 / 2.
A ≈ 2198.53125 feet^2.
Next, let's find the area of the triangular riding area:
The base of the riding area is given as 50 feet, and the height is also given as 75 feet.
The formula to find the area of a triangle is A = (base * height) / 2.
Substituting the values, we get A = (50 feet * 75 feet) / 2.
Simplifying the equation, we get A = 3750 feet^2 / 2.
A = 1875 feet^2.
Finally, let's find the total area of the pasture by adding the areas of the grassy area and the riding area:
Total Area = Area of grassy area + Area of riding area.
Total Area = 2198.53125 feet^2 + 1875 feet^2.
Total Area ≈ 4073.53125 feet^2.
Rounding to the nearest tenth, the total area of the pasture is approximately 4073.5 square feet.
To find the total area of the pasture, we need to calculate the areas of both the semi-circular grassy area and the triangular riding area, then sum them up.
1. Semi-circular grassy area:
Since the height of the riding area equals the diameter of the grassy area, we can use the height (radius) to find the diameter of the semi-circle.
The diameter is equal to twice the radius, so the diameter is 2 * 75 feet = 150 feet.
To find the area of a circle, we use the formula: A = π * r^2, where r is the radius.
In this case, the radius is 75 feet, so the area of the semi-circle is A = π * (75 feet)^2.
2. Triangular riding area:
To find the area of the triangle, we use the formula: A = 1/2 * base * height.
In this case, the base is given as 50 feet, and the height is given as 75 feet.
Now, we can calculate the area of the triangular riding area using the formula: A = 1/2 * 50 feet * 75 feet.
Finally, to find the total area of the pasture, we sum up the area of the semi-circular grassy area and the triangular riding area.
Total area = Area of semi-circle + Area of triangle
I will now calculate the areas for you.