The area of the warehouse is
[tex]A=8000ft^2[/tex]Half of this area stock paint, cans and rims:
[tex]\begin{gathered} A_{\text{stock}}=4000ft^2 \\ \text{then, the volume of the room is} \\ V_{\text{stock}}=4000\times20 \\ V_{\text{stock}}=80000ft^3 \end{gathered}[/tex]thats because the heigth of the stock room is equal to 20 ft.
On the other hand, we know that there are 2 cans in a box which volume
[tex]\begin{gathered} V_{\text{box}}=15\times7\times6inches^3 \\ \text{then for one can, the volume is} \\ V_{\text{can}}=\frac{V_{box}}{2}=\frac{15\times7\times6}{2}=15\times7\times3inches^3 \\ V_{\text{can}}=315in^3 \end{gathered}[/tex]and a rim is inside a box with measures
[tex]\begin{gathered} V_{\text{rim box}}=36\times36\times15inches^3 \\ V_{\text{rim box}}=19440in^3 \end{gathered}[/tex]Then, we need to find the ratio V_total to V_stock in order to find the number of rims in the room.
Then, V_total is the sum of 4 times the volume of one can plus the volume of 1 rim, that is,
[tex]V_{\text{total}}=4\cdot V_{\text{can}}+V_{\text{rim}}[/tex]because we need 4 cans and 1 rim in our room. This total volume is given by
[tex]V_{\text{total}}=4\cdot315+19440inches^3[/tex]which gives
[tex]V_{\text{total}}=20700inches^3[/tex]The last step is convert the V_total from cubic inches to cubic feets. We can do that by means of
[tex]V_{\text{total}}=20700inches^3(\frac{1ft^3}{12^3inches^3})[/tex]because 1 feet is equal to 12 inches. It yields,
[tex]\begin{gathered} V_{\text{total}}=20700(\frac{1}{144}) \\ V_{\text{total}}=143.75ft^3 \end{gathered}[/tex]Finally, we can find the ratio mentioned above:
[tex]\text{ratio}=\frac{V_{stock}}{V_{total}}=\frac{80000}{143.75}=556.52[/tex]By rounding down to the nearest interger, the ratio is 556. This means that we can stock 556 rims in the warehouse.
