you want to make a flower bed in your garden. What shape would it be to make the biggest possible peramiter for an area of 214cmsq
To determine the shape that would result in the largest perimeter for an area of 214 cm², you need to consider different geometric shapes.
1. Rectangle:
Let the length of the rectangle be "l" cm and the breadth be "b" cm. Since the area of a rectangle is given by A = l × b, and the perimeter is given by P = 2l + 2b, we have the equations:
l × b = 214 (equation 1)
2l + 2b = P (equation 2)
From equation 2, we can express l or b in terms of P:
l = (P - 2b)/2 (equation 3)
Substituting this value of l into equation 1, we get:
(P - 2b)/2 × b = 214
Simplifying the equation, we have:
Pb - 2b² = 428
Rearranging terms, we get the quadratic equation:
2b² - Pb + 428 = 0
To find the values of b that satisfy this equation, we can use the quadratic formula:
b = (-(-P) ± √(P² - 4×2×428)) / (2×2)
b = (P ± √(P² - 3424)) / 4
By substituting different values for P, we can calculate the possible values for b. Once we have the values of b, we can calculate the corresponding values of l using equation 3.
To find the maximum perimeter, we need to find the maximum value of P. Therefore, choose the value of b that maximizes P.
2. Circle:
Let the radius of the circle be "r" cm. The area of a circle is given by A = πr², and the perimeter (also known as the circumference) is given by P = 2πr.
We want to find the maximum perimeter for an area of 214 cm², so we need to find the maximum value of P. To do this, we can solve for the radius using the formula:
r = √(A/π)
Substituting the given area value, we get:
r = √(214/π)
Once we find the value of r, we can calculate the corresponding perimeter.
By comparing the perimeters obtained from both the rectangle and the circle, you can determine which shape would result in the largest perimeter for an area of 214 cm².