An open topped taking tin with a volume of 864 cm is to be constructed with a square base and vertical sides. Find the least amount of tin plate required.
if the base has side x and the tin has height y, then
x^2 y = 864
y = 864/x^2
The surface area a is
a = x^2 + 4xy = x^2 + 4x(864/x^2)
= x^2 + 3456/x
da/dx = 2x - 3456/x^2
= (2x^3-3456)/x^2
da/dx=0 when x=12.
Now find the area.
To solve this problem, we need to find the dimensions of the open-top tin that minimize the surface area, and then calculate the amount of tin plate required.
Let's start by identifying the variables involved:
Let's call the length of each side of the square base "x."
Let's call the height of the tin "h."
Now, let's define the volume of the tin:
Volume = (Area of the base) * Height
864 cm³ = (x²) * h
Next, we need to define the surface area of the tin in terms of "x" and "h" to minimize it:
Surface Area = (Area of the base) + (Area of the sides)
Surface Area = x² + 4(x * h)
To find the minimum surface area, we can use the volume equation to eliminate one variable. Rearranging the volume equation:
h = 864 cm³ / x²
Substituting this value of "h" into the surface area equation:
Surface Area = x² + 4(x * (864 cm³ / x²))
Surface Area = x² + (3456 cm³ / x)
Now, we have the surface area equation in terms of a single variable, "x." To find the minimum surface area, we can differentiate the equation with respect to "x" and set it equal to zero:
d(Surface Area) / dx = 2x - (3456 cm³ / x²) = 0
Multiplying through by x²:
2x³ - 3456 cm³ = 0
Simplifying further:
2x³ = 3456 cm³
x³ = 1728 cm³
x = ∛1728 cm
x = 12 cm
Now that we have the value of "x," we can substitute it back into the volume equation to find the value of "h":
h = 864 cm³ / (12 cm)²
h = 6 cm
So, the dimensions of the tin that minimize the surface area are a square base with each side measuring 12 cm and a height of 6 cm.
Finally, we can calculate the amount of tin plate required, which is the surface area of the tin. Substituting the values of "x" and "h" into the surface area equation:
Surface Area = (12 cm)² + 4(12 cm * 6 cm)
Surface Area = 144 cm² + 288 cm²
Surface Area = 432 cm²
Therefore, the least amount of tin plate required is 432 cm².