Final answer:
To find the area of the quadrant with a perimeter of 50 cm, we first solve for the radius using the perimeter formula of the quadrant and then use the obtained radius to calculate the area using the area formula for a quadrant of a circle.
Step-by-step explanation:
To find the area of the quadrant when you know the perimeter, we must note that the perimeter (P) of a quadrant of a circle includes the lengths of two radius sides plus the length of the arc that makes up the quarter-circle. The perimeter P is given as P = 2r + (0.5πr). Given P is 50 cm, we can arrange the equation to solve for r:
50 = 2r + (π/2)r
To solve for the radius r, we factor out r from the right-hand side of the equation:
r(2 + π/2) = 50
r = 50 / (2 + π/2)
Once we find the radius, we use it to calculate the area of the quadrant by applying the formula A = (πr²)/4, because the quadrant is one-fourth of the full circle area.
This step-by-step approach leads to the correct area of the quadrant, and from the options provided, we match our result to the correct choice.