A landscape architect is designing a triangular garden to fit in the corner of a lot. The corner of the lot forms an angle of 50°, and the sides of the
garden including this angle are to be 12 feet and 14 feet, respectively. Find, to the nearest integer, the number of square feet in the area of the garden.
To find the area of the triangular garden, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, the base of the triangle is 14 feet, and the corresponding height is the perpendicular distance from the base to the opposite vertex.
To find the height, we can use the concept of trigonometry. Since we have the angles and the lengths of two sides of the triangle, we can use the sine function to find the height.
sin(50°) = height / 12 feet
Solving for height:
height = sin(50°) * 12 feet
Using a calculator, we find that sin(50°) ≈ 0.766.
height ≈ 0.766 * 12 feet ≈ 9.19 feet
Now, we can calculate the area of the triangle:
Area = (1/2) * 14 feet * 9.19 feet
Area ≈ 70.63 square feet
Therefore, the area of the garden is approximately 70 square feet.
To find the area of the triangular garden, we can use the formula for the area of a triangle which is:
Area = (1/2) * base * height
In this case, the base of the triangle is the side of the garden that is 14 feet long, and the height is the length of the perpendicular line from the base to the opposite vertex.
To find the height, we can use the sine of the angle between the base and the height. We know that the angle is 50° and the side opposite the angle is 12 feet. So, we can use the sine function to find the height:
sin(50°) = height / 12
To solve for the height, we multiply both sides of the equation by 12:
12 * sin(50°) = height
Now, we can substitute this height value and the base length into the formula for the area of the triangle:
Area = (1/2) * 14 * (12 * sin(50°))
Calculating the value:
Area ≈ (1/2) * 14 * (12 * 0.766)
Area ≈ (1/2) * 14 * 9.192
Area ≈ 63.924
Rounding to the nearest integer:
Area ≈ 64 square feet
Therefore, the approximate number of square feet in the area of the garden is 64.
Area of triangle
= (1/2)(12)(14)sin50°
= ......