Question

1. The expressionx^3+3x^2+x-5 represents the total length across the front of the mansion. Find the length of side I. Show all work. X will stay a variable and your answer will be a polynomial
2. 2. The owners of the mansion want to install a new security system to protect the outside of their house. To get an estimate for the cost of the new security system they need to know the total distance around the outside of the mansion.

a. How many sides does the Mansion have?
b. What side would be the same length as side L?
c. What side would be the same length as side D?
d. Find an expression for the perimeter of the mansion. Show all work. X will stay a variable and your answer will be a polynomial.

3. The owners of the mansion also want to install new AC units. You must know the amount of air on the first floor of the mansion to determine the size of AC units necessary to function properly. The first floor of the mansion covers, or has a base, 15x^4+20x^2+45x+590 square feet. If the height of the ceilings on the first floor is 5x-3, find the total volume of air on the first floor. Use the formula V=Bh where Bis the area of the base and his the height. Show all work. X will stay a variable and your answer will be a polynomial.

4. The area of the Gothic Room in the mansion’s layout is2x^3+25x^2+169, what are the dimensions (length and width) for the room? Side E is x + 13. Use synthetic division to find the other side width. Show all work. X will stay a variable and your answer will be a polynomial.

Answer 1

Part 1:

Given that the expression $x^3+3x^2+x-5$ represents the total length across the front of the mansion. Let the length of side I be a, then

$x^4-2x^3+x-10+a-x^3+5x^2+2=x^3+3x^2+x-5 \\ \\ \Rightarrow a=x^3+3x^2+x-5-x^4+2x^3-x+10+x^3-5x^2-2 \\ \\ =\bold{-x^4+4x^3-2x^2+3}$



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:

$P=2(x^3+3x^2+x-5)+2(-2x^4+24x^2-10)+2(4x)+x^2-x+6 \\ \\ =2x^3+6x^2+2x-10-4x^4+48x^2-20+8x+x^2-x+6 \\ \\ =\bold{-4x^4+2x^3+55x^2+9x-24}$



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, $15x^4+20x^2+45x+590\ square\ feet$. 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:

$V=(5x-3)(15x^4+20x^2+45x+590) \\ \\ =75x^5+100x^3+225x^2+2950-45x^4-60x^2-135x-1770 \\ \\ =\bold{75x^5-45x^4+100x^3+165x^2-135x+1180}$



Part 4:

Given that the area of the Gothic Room in the mansion’s layout is $2x^3+25x^2+169$, 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 $-26x^2+13x-169$