a rectangular metal sheet of length 30cm and breadth 25cm is to be made into an open box of base area 300cm by cutting out equal squares from each of the four corners and then bending up the edges find the length and the side cut from each corner
(25-2x)(30-2x) = 300
To find the length and side cut from each corner, we need to consider the properties of the rectangular metal sheet and the desired dimensions of the open box.
Given:
Length of the rectangular metal sheet = 30 cm
Breadth of the rectangular metal sheet = 25 cm
Base area of the open box = 300 cm²
Let's assume the side length of each square cut from the corners is "x" cm.
When we cut out squares from each corner, the length of the base of the open box will be reduced by twice the value of "x" (since each corner has two edges being folded). Similarly, the breadth of the base will be reduced by twice the value of "x".
So, the length of the base of the open box will be (30 - 2x) cm, and the breadth will be (25 - 2x) cm.
To find the dimensions of the open box, we need to solve the equation for the base area of the open box:
Base area = Length × Breadth
300 cm² = (30 - 2x) cm × (25 - 2x) cm
Expanding this equation:
300 = 750 - 60x - 50x + 4x²
0 = 4x² - 110x + 450
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most convenient method.
Factoring the quadratic equation:
0 = (2x - 30)(2x - 15)
Setting each factor equal to zero:
2x - 30 = 0 or 2x - 15 = 0
Solving for "x":
2x = 30 or 2x = 15
x = 15/2 or x = 15/4
Since length cannot be negative, we discard the negative value of x.
Therefore, the side length cut from each corner is x = 15/2 cm.
To find the length of the open box:
Length = 30 cm - 2 * (15/2) cm
Length = 30 cm - 15 cm
Length = 15 cm
So, the length of the open box is 15 cm, and the side cut from each corner is 15/2 cm.