Part 1:
Given that the expression [tex]x^3+3x^2+x-5[/tex] represents the total length across the front of the mansion. Let the length of side I be a, then
[tex]x^4-2x^3+x-10+a-x^3+5x^2+2=x^3+3x^2+x-5 \\ \\ \Rightarrow a=x^3+3x^2+x-5-x^4+2x^3-x+10+x^3-5x^2-2 \\ \\ =\bold{-x^4+4x^3-2x^2+3}[/tex]
Part 2:
From the given figure it can be seen that the mansion has 12 sides labelled A - L.
Side F will be of the same length as side L.
Side B will be of the same length as side D.
The perimeter of figure is the sum of all the lengths of the outlines of the figure.
The perimeter is given by:
[tex]P=2(x^3+3x^2+x-5)+2(-2x^4+24x^2-10)+2(4x)+x^2-x+6 \\ \\ =2x^3+6x^2+2x-10-4x^4+48x^2-20+8x+x^2-x+6 \\ \\ =\bold{-4x^4+2x^3+55x^2+9x-24}[/tex]
Part 3:
The volume of an object is given by: V = Area of base x height.
Given that the
first floor of the mansion has a base, [tex]15x^4+20x^2+45x+590\ square\ feet[/tex]. If the height of the ceilings on the first floor is 5x-3, then, the total volume of air on the first floor is given by:
[tex]V=(5x-3)(15x^4+20x^2+45x+590) \\ \\ =75x^5+100x^3+225x^2+2950-45x^4-60x^2-135x-1770 \\ \\ =\bold{75x^5-45x^4+100x^3+165x^2-135x+1180}[/tex]
Part 4:
Given that the
area of the Gothic Room in the mansion’s layout is [tex]2x^3+25x^2+169[/tex], with the length of one of the sides of the room as x + 13. Then, the length of the other side is given by:
-13 | 2 25 0 169
|
| -26 13 -169
|______________
2 -1 13 0
Therefore, the length of the other side of the length is [tex]-26x^2+13x-169[/tex]
