ANSWERS
a. 863.08 ft²
b. 474 ft²
c. 249 ft²
d. $4,259.24
EXPLANATION
Given:
• Rooms with ,carpet,: Family room and living room
,
• Rooms with ,hardwood,: kitchen and foyer
,
• Rooms with ,tiles,: breakfast nook and mudroom
• Cost ,carpet,: $1.80 per square foot
,
• Cost ,hardwood,: $4.50 per square foot
,
• Cost ,tiles,: $2.30 per square foot
First, we have to find the area of each of the mentioned rooms,
a. The family room is in the shape of a square whose sides are all 20 feet and a semicircle whose diameter is 20 feet. The area of the family room is the sum of the areas of each of these figures,
[tex]A_{family-room}=(20ft\cdot20ft)+(\frac{1}{2}\cdot\pi\cdot(20ft/2)^2)[/tex]
As shown above, the area of the semicircle is half the area of the full circle with the same diameter. Solve,
[tex]A_{family-room}\approx(400ft^2)+(157.08ft^2)=557.08ft^2[/tex]
The living room is in the shape of a rectangle with side lengths 15ft and 22ft, but a right triangle is cut out from the corner. The area of the living room is the area of the mentioned rectangle minus the area of the triangle of the corner,
The side lengths of the triangle are 15ft - 9ft = 6ft and 22ft - 14ft = 8ft. The area of the triangle is,
[tex]A_{triangle}=\frac{6ft\cdot8ft}{2}=24ft^2[/tex]
So, the area of the living room is,
[tex]A_{living-room}=(22ft\cdot15ft)-24ft^2=330ft^2-24ft^2=306ft^2[/tex]
The area that Rachel wants to cover with a new carpet is,
[tex]A_{carpet}=A_{family-room}+A_{living-room}=557.08ft^2+306ft^2=863.08ft^2[/tex]
Hence, the area to be covered with carpet is 863.08 ft².
b. The kitchen is a rectangle with dimensions 24 ft and 15 ft, so its area is,
[tex]A_{kitchen}=24ft\cdot15ft=360ft^2[/tex]
The foyer is in the shape of a rectangle with dimensions 10ft and 15ft. However, assuming that the stairs won't be covered with hardwood and that they cover that part of the foyer's floor, to find the area we have to subtract the area the stairs occupy in that room.
The rectangle where the stairs are has dimensions 9ft and (10ft - 6ft ) = 4ft. The area of the Foyer is,
[tex]A_{foyer}=(10ft\cdot15ft)-(9ft\cdot4ft)=(150ft^2)-(36ft^2)=114ft^2[/tex]
The area Rachel wants to be covered by hardwood is the sum of the two,
[tex]A_{hardwood}=A_{kitchen}+A_{foyer}=360ft^2+114ft^2=474ft^2[/tex]
Hence, the area to be covered with hardwood is 474 ft².
c. The mudroom is in the shape of a rectangle with dimensions 15 ft and 7 ft, so its area is,
[tex]A_{mudroom}=15ft\cdot7ft=105ft^2[/tex]
The breakfast nook is in the shape of a trapezoid, where the height is 8ft and the lengths of the bases are 24ft and 12ft. Its area is,
[tex]A_{breakfast-nook}=\frac{(12ft+24ft)}{2}\cdot8ft=\frac{36ft}{2}\cdot8ft=18ft\cdot8ft=144ft^2[/tex]
The area Rachel wants to cover with tiles is the sum of these two areas,
[tex]A_{tiles}=A_{mudroom}+A_{breakfast-nook}=105ft^2+144ft^2=249ft^2[/tex]
Hence, the area to be covered with tiles is 249 ft².
d. Finally, we have to find the cost of flooring with each material,
[tex]Costs=\begin{cases}C_{carpet}=A_{carpet}\cdot p_{carpet}=863.08ft^2\cdot1.80=1,553.54 \\ C_{hardwood}=A_{hardwood}\cdot p_{hardwood}=474ft^2\cdot4.50=2,133 \\ C_{tiles}=A_{tiles}\cdot p_{tiles}=249ft^2\cdot2.30=572.70\end{cases}[/tex]
The total cost of the project is the sum of the three,
[tex]C_{project}=C_{carpet}+C_{hardwood}+C_{tiles}=1,553.54+2,133+572.70=4,259.24[/tex]
Hence, the total cost of the re-flooring project will be $4,259.24.
