Let's denote the width of the rectangular base as w meters. Since the length of the base is twice the width, the length can be represented as 2w meters. The height of the container can be denoted as h meters.
Given that the volume of the container is 24 cubic meters, we have the equation for the volume:
V = lwh = 24
Substitute the values for length and width in terms of w into the equation:
2w• w • h = 24
2w^2h = 24
wh = 12
The cost of materials for the base is given by the area of the base multiplied by the cost per square meter. The cost for the sides is given by the total surface area of the four sides multiplied by the cost per square meter.
The area of the base is 2w^2 square meters. Therefore, the cost for the base is:
10 x 2w^2 = 20w^2 dollars
The total surface area of the four sides is 2lh + 2wh square meters. Substituting the value of l and h in terms of w into the equation, we get:
2(2w)h + 2wh = 4wh + 2wh = 6wh
Substitute the value of wh from the volume equation:
6 x 12 = 72 square meters
Therefore, the cost for the sides is:
6 x 72 = 432 , dollars
The total cost for the container is the sum of the cost for the base and the cost for the sides:
Total cost = 20w^2 + 432 , dollars
To find the minimum cost, we need to minimize this total cost function with respect to w.
hope this helps!
