Final answer:
To write a function modeling the surface area in terms of x for a closed box, we need to determine the dimensions of the box. Given that the base of the box is twice as long as its width, we can represent the width as x. Therefore, the length of the base would be 2x. The volume of the box is 10 cubic ft, and the function for the surface area is 4x^2 + 6hx. To minimize the surface area, we can take the derivative of the function and set it equal to 0.
Step-by-step explanation:
To write a function modeling the surface area in terms of x for a closed box, we need to determine the dimensions of the box. Given that the base of the box is twice as long as its width, we can represent the width as x. Therefore, the length of the base would be 2x.
The height of the box is not given, so we can use a variable, h, to represent it.
The volume of the box is given as 10 cubic ft, so we can set up the following equation:
Volume = Length x Width x Height
10 = (2x) * x * h
Now, we can solve this equation for h in terms of x:
h = 10 / (2x^2)
Next, we can write the function for the surface area:
Surface Area = 2(Length x Width) + 2(Length x Height) + 2(Width x Height)
Replacing the values with the variables, the function becomes:
Surface Area = 2(2x*x) + 2(2x*h) + 2(x*h)
Surface Area = 4x^2 + 4hx + 2hx
Surface Area = 4x^2 + 6hx
To minimize the surface area, we can take the derivative of the function with respect to x and set it equal to 0:
d(Surface Area)/dx = 8x + 6h = 0
Solving for x:
8x = -6h
x = -6h/8
Therefore, the surface area function in terms of x is: Surface Area = 4x^2 + 6hx