4. Rachelle is buying panelling for wainscotting for her hall. The panels are 4' by 8'. The wainscotting will be 4' high and the room is 19' by 13'. There are two doors measuring 30" wide, and one 12' window that will only need a 2' panel of wainscotting under it. How many panels will she need, assuming no wastage?

To find out how many panels Rachelle will need for the wainscoting in her hall, we need to calculate the total area to be covered by the panels and then divide it by the area of each panel.

The room is 19 ft × 13 ft, so its perimeter is:

[tex]\sf Perimeter= 2(19 + 13)\\\\Perimeter= 2(32)\\\\Perimeter= 64\; ft[/tex]

There are two doors each measuring 30 inches wide, so a total of 60 inches. As 12 inches equals one foot, then the total width of the two doors can be found by dividing 60 inches by 12 inches per foot:

[tex]\sf \textsf{Total width of two doors} = \dfrac{60}{12} = 5\; ft[/tex]

Now, subtract the width of the two doors from the perimeter of the room and multiply it by the height of the panelling, which is 4 ft:

[tex]\sf Area=(64 - 5) \times 4)\\\\Area=59 \times 4)\\\\Area = 236\;ft^2[/tex]

Given that the area under the 12 ft window is only 2 ft high, we need to subtract an area that is 12 ft × 2 ft from the area 236 ft² to find the total area to be covered by the panels:

[tex]\sf Total\;area=236- (12 \times 2)\\\\Total\;area=236- 24\\\\Total\;area=212 \; ft^2[/tex]

Therefore, the total area to be covered by the panels is 212 ft².

Finally, to find the number of panels needed to cover 212 ft², divide the total area by the area of one panel:

[tex]\sf \textsf{Number of panels}=\dfrac{212}{4 \times 8}\\\\\\\textsf{Number of panels}=\dfrac{212}{32}\\\\\\\textsf{Number of panels}=6.625[/tex]

Since we can't have a fraction of a panel, Rachelle will need to purchase 7 panels to cover the wainscoting in her hall, assuming no wastage.