Final answer:
To find the smallest sheet of metal to make a square bottomed pan with a volume of 250cm3, we can minimize the surface area by minimizing the length of the sides of the square base. The smallest sheet of metal will have dimensions close to 0 cm by 31.62 cm, and the pan's height will be approximately 31.62 cm.
Step-by-step explanation:
To find the smallest sheet of metal needed to make a square bottomed pan with a volume of 250cm3, we need to determine the dimensions of the square base. Let's call the length of each side of the square base 'x'. When the four corners are cut out and the remaining sheet is folded up to form the sides, the pan's volume can be calculated using the formula for the volume of a rectangular prism: V = x^2 * h, where V is the volume, x is the length of each side of the square base, and h is the height of the pan.
Since the volume is given as 250cm3, we have the equation: 250 = x^2 * h.
To find the smallest sheet of metal, we want to minimize the amount of material used. This means minimizing the surface area. The surface area of the pan (excluding the base) is given by the formula: A = 4xh. We want to minimize A while still satisfying the volume requirement. We can rewrite A in terms of h using the equation for the volume: A = 4 * 250 / h.
To minimize A, we need to take the derivative of A with respect to h, set it equal to zero, and solve for h:
dA/dh = -1000 / h^2 = 0.
Solving for h: h^2 = 1000 and h ≈ 31.62 cm (rounded to two decimal places).
Substituting the value of h back into the equation for A, we get A ≈ 31.62x cm².
Since we want to minimize the surface area, we need to minimize x. Since A ≈ 31.62x and we want A to be as small as possible, we want x to be as small as possible. This means x should be close to zero, but not zero since the pan needs to have a base.
Therefore, the smallest sheet of metal needed to make the pan has dimensions of approximately 0 cm by 31.62 cm. The height of the pan will be approximately 31.62 cm.