To find the dimensions of the rectangular storage container, let’s denote the width of the base as w meters. Since the length of the base is twice the width, we can express the length as l = 2w meters. The volume V of the container is given by the formula:
V = length × width × height
Given that the volume is 10 m³, we can write:
10 = (2w) × w × h
From this, we can simplify it:
10 = 2w²h
Now, we can express the height in terms of width:
h = rac{10}{2w^2} = rac{5}{w^2}
The next step is to calculate the cost of the material for the base. The area of the base is:
Area = length × width = 2w × w = 2w²
If the cost of the material for the base is $15 per square meter, then the total cost C of the material for the base can be calculated as:
C = Area × cost per square meter = 2w² × 15 = 30w²
We need to minimize this cost while maintaining the volume constraint. Substituting the expression for h back into the volume equation gives us:
10 = 2w² × rac{5}{w^2}
Through solving the equation, we find that:
w = 1
So, Length: l = 2w = 2
Height: h = rac{5}{1^2} = 5
Finally, the dimensions of the container are:
Width: 1 m
Length: 2 m
Height: 5 m
In conclusion, the dimensions of the rectangular storage container with a volume of 10 m³, where the length of the base is twice the width, are a width of 1 meter, a length of 2 meters, and a height of 5 meters, resulting in a material cost of $30.