Respuesta :
Answer:
B. Width: 1.8 in. Length: 3.6 in. Height: 2.6 in.
Step-by-step explanation:
Let x represent width of the rectangle and y represent height of the prism.
We have been given that the length must be 2 times the width. So length of the prism would be [tex]2x[/tex].
We are also told that the sum of its length, width, and height 8 in. We can represent this information in an equation as:
[tex]x+2x+y=8...(1)[/tex]
[tex]3x+y=8...(1)[/tex]
[tex]y=8-3x...(1)[/tex]
We know that volume of rectangular prism is product of length, width and height.
[tex]V=lwh[/tex]
Upon substituting the values of length, width and height, we will get:
[tex]V=2x\cdot x\cdot y[/tex]
Upon substituting equation (1) in volume formula, we will get:
[tex]V=2x\cdot x\cdot (8-3x)[/tex]
[tex]V=2x^2\cdot (8-3x)[/tex]
[tex]V=16x^2-6x^3[/tex]
Now, we will take the derivative of volume function using power rule as:
[tex]V'=16\cdot 2x^{2-1}-6\cdot 3x^{3-1}[/tex]
[tex]V'=32x-18x^{2}[/tex]
Now, we will equate derivative with zero and solve for x as:
[tex]32x-18x^{2}=0[/tex]
[tex]2x(16-9x)=0[/tex]
[tex]2x=0, (16-9x)=0[/tex]
[tex]x=0, x=\frac{16}{9}[/tex]
[tex]x=0, x\approx 1.8[/tex]
Since width cannot be 0, therefore, the width of prism would be 1.8 inches.
Length of the prism would be [tex]2x\Rightarrow 2(1.8)=3.6[/tex].
Therefore, the length of prism would be 3.6 inches.
Upon substituting [tex]x=1.8[/tex] in equation (1), we will get:
[tex]y=8-3(1.8)[/tex]
[tex]y=8-5.4[/tex]
[tex]y=2.6[/tex]
Therefore, the height of prism would be 2.6 inches and option B is the correct choice.
