Answer :
Answer:
A 3x3 inches base.
A 3x5 inches sides
Explanation:
As they want the rectangular prism with a square base, you need to make a 3 by 3 base so it is a square and then you just add a 5x3 side to make it 45 in3 and that creates you the minimal cost, because the cheapest would be to make a 1 by 1 but in that one you couldnt fit the pencils efficciently.
To minimize the construction cost, the dimensions of the cup be 3 x 3 x 5 inches.
Given the following data:
- Capacity = 45 [tex]in^3[/tex].
- Cost of material (sides) = 27 ¢[tex]/in^2[/tex]
- Cost of material (base) = 90 ¢[tex]/in^2[/tex]
How to calculate the dimensions of the cup.
Let the height be h.
Let the square base be x.
Therefore, the capacity (volume) of the pencil cup is given by:
[tex]V=x^2h\\\\45=x^2h\\\\h=\frac{45}{x^2}[/tex]
At the minimum cost, we have:
[tex]C(x)=90x^2+27(4xh)\\\\C(x)=90x^2+108xh\\\\C(x)=90x^2+108x(\frac{45}{x^2} )\\\\C(x)=90x^2+\frac{4860}{x}[/tex]
Differentiating wrt x, we have:
[tex]C(x)=180x-\frac{4860}{x^2} =0\\\\180x^3-4860=0\\\\180x^3=4860\\\\x^3=27\\\\x=\sqrt[3]{27}[/tex]
x = 3 inches.
For the height:
[tex]h=\frac{45}{x^2} \\\\h=\frac{45}{3^2}\\\\h=\frac{45}{9}[/tex]
h = 5 inches.
Dimensions = [tex]square \;base \times height[/tex]
Dimensions = 3 x 3 x 5 inches.
Read more on capacity here: brainly.com/question/25248189