To determine the cost of the least expensive tank with a volume of 75 m³ and a depth of 3 m, we first need to establish the dimensions of the tank. Let's denote the length of the base as L and the width as W.
Step 1: Volume Equation
The volume of the tank can be expressed as:
Volume = Length × Width × Depth
Given that the depth is 3 m, we can write:
75 = L × W × 3
From this, we can simplify to:
L × W = 25
Step 2: Surface Area Calculation
The surface area of the tank consists of the base and the four sides. The area of the base is:
Base Area = L × W
The area of the sides is:
- Two sides of area 3 × L
- Two sides of area 3 × W
Thus, the total surface area (A) can be calculated as:
A = L × W + 3L + 3W
Step 3: Cost Function
The cost of constructing the tank can be expressed as:
Cost = Cost of Base + Cost of Sides
Substituting the costs:
Cost = 100(L × W) + 50(3L + 3W)
Replacing L × W with 25 gives:
Cost = 100(25) + 150(L + W)
Cost = 2500 + 150(L + W)
Step 4: Expressing W in Terms of L
From the equation L × W = 25, we can express W as:
W = 25/L
Substituting this into the cost function:
Cost = 2500 + 150(L + 25/L)
Step 5: Minimizing the Cost
To find the minimum cost, we need to differentiate the cost function with respect to L and set the derivative to zero:
Cost = 2500 + 150L + 3750/L
Taking the derivative:
d(Cost)/dL = 150 - 3750/L²
Setting this equal to zero:
150 = 3750/L²
Solving for L gives:
L² = 3750/150 = 25
L = 5 m
Step 6: Finding W
Using L = 5 m in the equation W = 25/L:
W = 25/5 = 5 m
Step 7: Final Cost Calculation
Now substituting L and W back into the cost function:
Cost = 2500 + 150(5 + 5) = 2500 + 150(10) = 2500 + 1500 = 4000
Result
The cost of the least expensive tank is Rs. 4000.
