Final answer:
The area of the unpainted region on the four-inch board is approximately 3.46 square inches.
Step-by-step explanation:
To find the area of the unpainted region on the four-inch board, we first need to understand the given scenario. Two boards, one four inches wide and the other six inches wide, are nailed together to form an X. The angle at which they cross is 60 degrees. This means that the four-inch board is at an angle of 60 degrees with the six-inch board.
Now, when these boards are separated, the four-inch board will have an unpainted region that is formed in the shape of a triangle. To find the area of this triangle, we will use the formula for the area of a triangle which is ½ * base * height. In this case, the base of the triangle is the length of the four-inch board which is 4 inches.
To find the height, we will use trigonometry. Since the angle between the two boards is 60 degrees, we can use the sine function to find the height of the triangle. The sine of 60 degrees is √3/2. Therefore, the height of the triangle is 4 * √3/2 = 2√3 inches.
Now, we can plug in these values in the formula for the area of the triangle. ½ * 4 * 2√3 = 4√3/2 = 3.46 square inches. This is the final answer for the area of the unpainted region on the four-inch board.
Step-by-step explanation:
To find the area of a triangle, we use the formula ½ * base * height, where the base and height are perpendicular to each other. In this scenario, the four-inch board is at an angle of 60 degrees with the six-inch board. This means that the height of the triangle is the perpendicular distance from the four-inch board to the six-inch board.
To find this height, we use trigonometry. In a right-angled triangle, the sine of an angle is equal to the ratio of the opposite side to the hypotenuse. In this case, the opposite side is the height of the triangle and the hypotenuse is the length of the four-inch board. Therefore, we can write the equation as sin(60°) = height/4.
Solving for height, we get height = 4 * sin(60°) = 4 * √3/2 = 2√3 inches. This is the height of the triangle formed on the four-inch board.
Now, we can plug in these values in the formula for the area of the triangle to get the final answer. ½ * 4 * 2√3 = 4√3/2 = 3.46 square inches. This is the area of the unpainted region on the four-inch board.
In conclusion, the area of the unpainted region on the four-inch board is approximately 3.46 square inches. This answer is found by using the formula for the area of a triangle and trigonometry to find the height of the triangle. It is important to understand the given scenario and use the appropriate formulas and calculations to arrive at the correct answer.