A steel plate has the form of one-fourth of a circle with a radius of 54 centimeters. Two two-centimeter holes are drilled in the plate, positioned as shown in the figure. Find the coordinates of the center of each hole.
To find the coordinates of the center of each hole, we need to first understand the position of the holes in relation to the steel plate.
From the problem description, we know that the steel plate has the shape of one-fourth of a circle with a radius of 54 centimeters. Let's imagine this quarter circle lying in the Cartesian coordinate plane, with its center at the origin (0,0) and the positive x-axis extending to the right and the positive y-axis extending upwards.
Since the plate is a quarter of a circle, we only need to consider the top-right quadrant (Q1) of the circle. The circle's circumference is given by C = 2πr, where r is the radius. In this case, the circumference of the plate is C = 2π(54) = 108π centimeters.
Next, we need to position the two 2-centimeter holes on the plate. The problem states that the holes are positioned as shown in the figure, but since we don't have access to the figure, we'll need to make some assumptions.
Let's assume that one hole is positioned at a distance of x units from the vertical y-axis (or in other words, along the x-axis) and at a distance of y units from the horizontal x-axis (or along the y-axis). The second hole is positioned symmetrically in relation to the first hole, meaning it will have the same x coordinate but a different y coordinate.
Since the radius of the plate is 54 centimeters and the circumference is 108π centimeters, we know that the arc length from the center of the quarter circle to the edge is half the circumference, or 54π centimeters.
From geometry, we know that the arc length of a circle sector with angle θ is given by L = rθ, where L is the length and r is the radius. In this case, we have L = 54π centimeters and θ = 90 degrees (since it's a quarter of a circle).
Thus, we can write the equation for the arc length as 54π = 54(π/2)θ. Solving for θ gives us θ = π/2.
Now, using trigonometry, we can find the coordinates of the first hole. The x coordinate is given by x = r * cos(θ) = 54 * cos(π/2) = 0 centimeters (since cos(π/2) = 0). The y coordinate is given by y = r * sin(θ) = 54 * sin(π/2) = 54 centimeters.
Therefore, the coordinates of the first hole are (0, 54). Since the second hole is positioned symmetrically, its coordinates will be (0, -54).
In summary, the coordinates of the center of each hole are:
First hole: (0, 54)
Second hole: (0, -54)