Alice cuts a ribbon of length 40cm into two pieces to form two squares as shown in th figure (there was no figure, but from my assumption the two squares are probably different ). Let xcm be the length of a side of one of the two squares
a)express the total area of the two squares in terms of x
b) Find the minimum total ar a of the two squares and the corresponding value of x
If x is one side, let y be the other. Then
4x+4y = 40
x+y = 10
y = 10-x
The total area is thus
a = x^2+y^2 = x^2 + (10-x)^2
= 2x^2-20x+100
a has a minimum at x = 20/4 = 5
As expected, minimum area occurs when the two squares are in fact equal.
To solve this problem, let's break it down step by step:
(a) Expressing the total area of the two squares in terms of x:
Let's assume that one piece of the ribbon is used to form a square with side length x cm. Since the ribbon was originally 40 cm long and we used one piece, the remaining piece will be 40 - x cm long.
The first square formed by the x cm piece of ribbon will have an area of x * x = x^2 square cm.
The second square formed by the (40 - x) cm piece of ribbon will have an area of (40 - x) * (40 - x) = (40 - x)^2 square cm.
Therefore, the total area of the two squares is x^2 + (40 - x)^2 square cm.
(b) Finding the minimum total area of the two squares and the corresponding value of x:
To find the minimum total area, we need to minimize the expression we obtained in part (a), which is x^2 + (40 - x)^2.
One way to find the minimum is to take the derivative of the expression with respect to x and set it equal to zero. However, since this might involve calculus, I will show you an alternative approach.
Expanding the expression, we have x^2 + (40 - x)^2 = x^2 + 1600 - 80x + x^2.
Combining like terms, we get 2x^2 - 80x + 1600.
To find the minimum, we can factor the quadratic expression: 2(x^2 - 40x + 800).
Now, to find the minimum value, we need to find the vertex of the quadratic equation x^2 - 40x + 800.
The x-coordinate of the vertex, given by x = -b / (2a), is x = -(-40) / (2 * 1) = 40 / 2 = 20.
Substituting this value back into the expression, we find the minimum total area: 2(20^2 - 40(20) + 800) = 2(400 - 800 + 800) = 2(400) = 800 square cm.
Therefore, the minimum total area of the two squares is 800 square cm, and the corresponding value of x is 20 cm.