We have a rectangular room with a length of 16 feet and a width of 14 ft, and a circular rug inside the room with a radius of 4.5 feet.
The following diagram represents the situation:
What we need to find is the red area.
The first step will be to calculate the area of the rectangle:
[tex]A_{\text{rectangle}}=l\times w[/tex]where l is the length l=16ft and w is the width w=14ft. We substitute l and w and find the area of the complete rectangle:
[tex]\begin{gathered} A_{\text{rectangle}}=16ft\times14ft \\ A_{\text{rectangle}}=224ft^2 \end{gathered}[/tex]The second step is to calculate the area of the circle:
[tex]A_{\text{circle}}=\pi r^2[/tex]Where π=3.1416 and r is the radius, r=4.5 ft. Substituting those values, we can find the area of the circle:
[tex]A_{\text{circle}}=(3.1416)(4.5ft)^2[/tex]Solving the operations:
[tex]\begin{gathered} A_{\text{circle}}=(3.1416)(20.25ft^2) \\ A_{\text{circle}}=63.617ft^2 \end{gathered}[/tex]The third step is to subtract the area of the circle to the area of the rectangle in order to find the amount of floor left uncovered:
[tex]A_{\text{uncovered}}=A_{\text{rectangle}}-A_{\text{circle}}[/tex]Substituting the known areas:
[tex]\begin{gathered} A_{\text{uncovered}}=224ft^2-63.617ft^2 \\ A_{\text{uncovered}}=160.383ft^2 \end{gathered}[/tex]Answer:
[tex]160.383ft^2[/tex]
