Answer:
i) length > 9 ft
breadth > 5 ft
height > 3 ft
ii) Area of floor > 45 ft²
Step-by-step explanation:
Volume of a rectangular prism
[tex]\textsf{V}=lbh[/tex]
where:
- l is the length
- b is the breadth
- h is the height
Given:
- [tex]\sf V=(2a^3+a^2-2a-1)\:\:ft^3, \quad where\:a > 4\:ft[/tex]
Part (i)
To find expressions for the 3 dimensions of the tank, factor the expression for Volume.
Using the Factor Theorem, if V(x) = 0 then (a - p) is a factor:
[tex]\implies \sf V(1)=2(1)^3+(1)^2-2(1)-1=0[/tex]
[tex]\implies \sf V(-1)=2(-1)^3+(-1)^2-2(-1)-1=0[/tex]
Therefore (a - 1) and (a + 1) are factors:
[tex]\implies \sf 2a^3+a^2-2a-1=(a-1)(a+1)(2a+p)[/tex]
(where p is a constant to be found)
To find the value of p, expand:
[tex]\implies \sf 2a^3+a^2-2a-1=2a^3+pa^2-2a-1[/tex]
and compare coefficients:
[tex]\implies \sf a^2=pa^2 \implies p=1[/tex]
Therefore:
[tex]\implies \sf 2a^3+a^2-2a-1=(a-1)(a+1)(2a+1)[/tex]
If l > b and b > h then:
- length (l) = (2a + 1)
- breadth (b) = (a + 1)
- height (h) = (a - 1)
If a > 4 ft then:
- length > 9 ft
- breadth > 5 ft
- height > 3 ft
Part (ii)
The area of the floor can be found by multiplying the found expressions for breadth and length:
[tex]\begin{aligned}\implies \sf Area\:of\:floor & =(a+1)(2a+1)\\& = 2a^2+3a+1\end{aligned}[/tex]
If a > 4 ft then Area of floor > 45 ft²
