An open rectangular tank (no top) with volume 11 cubic meters has a square base. The base has side x m. Write an equation for the surface area of the tank as a function of the length x.
Surface Area = ?
b. The base costs 1.1 cents/square m and the sides cost 1.6 cents/square m. Write an expression for the cost as a function of x.
Cost = ?
c. The value of x to achieve minimum cost is
x = ?
If the base is square of area x^2, then the height is 11/x^2
area = x^2 + 4(11/x^2)
cost = 1.1 x^2 + (1.6)(4)(11/x^2) = 1.1x^2 + 70.4/x^2
dc/dx = 2.2x - 140.8/x^3 = 2.2x(x^4-64)/x^3
dc/dx=0 when x=2√2
a. To find the surface area of the tank, we need to consider the base and the four sides of the tank.
The base is a square with side length x, so its area is x^2 square meters.
The four sides of the tank are rectangles with length x and height h. The height h can be determined by dividing the volume of the tank by the area of the base, since volume = area * height. In this case, the volume of the tank is 11 cubic meters and the area of the base is x^2 square meters. Therefore, the height h = 11/x^2.
The surface area of each side is given by the formula length * height. So, the total surface area of the four sides is 2 times the sum of the areas of the two long sides plus 2 times the sum of the areas of the two short sides. This can be written as:
Surface Area = 2(l1 * h) + 2(l2 * h).
Given that l1 = l2 = x, we can simplify the equation to:
Surface Area = 2(x * h) + 2(x * h).
Substituting the value of h = 11/x^2, the equation becomes:
Surface Area = 2(x * 11/x^2) + 2(x * 11/x^2).
Simplifying further, we get:
Surface Area = 22/x + 22/x.
Therefore, the equation for the surface area of the tank as a function of the length x is:
Surface Area = 44/x.
b. To find the cost of the tank, we need to consider the cost of the base and the cost of the sides.
The base has an area of x^2 square meters, and it costs 1.1 cents/square meter. So, the cost of the base can be calculated as:
Cost of base = 1.1 * x^2.
The four sides have a combined surface area of 44/x square meters, and they cost 1.6 cents/square meter. So, the cost of the sides can be calculated as:
Cost of sides = 1.6 * (44/x).
The total cost is the sum of the cost of the base and the cost of the sides. Thus, we can write the expression for the cost as a function of x as:
Cost = 1.1 * x^2 + 1.6 * (44/x).
c. To find the value of x that achieves the minimum cost, we need to take the derivative of the cost function with respect to x and set it equal to zero. Then, solve for x.
First, let's simplify the cost function:
Cost = 1.1x^2 + 70.4/x.
Now, let's find the derivative of the cost function:
d(Cost)/dx = 2.2x - 70.4/x^2.
Setting this derivative equal to zero and solving for x:
2.2x - 70.4/x^2 = 0.
Rearranging the equation:
2.2x = 70.4/x^2.
Multiplying both sides by x^2:
2.2x^3 = 70.4.
Dividing both sides by 2.2:
x^3 = 70.4/2.2.
Calculating the cube root of both sides:
x = (70.4/2.2)^(1/3).
Evaluating this expression:
x ≈ 3.599.
Therefore, the value of x that achieves the minimum cost is approximately 3.599.