Question

1. phil is installing baseboard along four walls in the living room. he measures the length and width of the room to be approximately 14 ft by 11 ft. sketch the floor plan of the room. include a door that is 3 ft wide. label all the given measures.

a) what is the total length of baseboard needed? (remember, baseboard is not installed in doorways)
b) baseboard is sold in lengths of 8 ft and 12 ft. give two combinations of baseboard that phil can buy.

1. diego is calculating the perimeter of his bathroom floor. he plans to install tiles along the base of the walls that do not include the bathtub. he measures the width of the room to be 6 ft and the length to be 9 ½ ft. the door opening measures 2 ½ ft.

a) diego calculates the total length that needs to cover with tile to be 25 feet. is he correct? explain.
b) the tiles diego selects are 6 in by 6 in. how many tiles does he need?

Explanation:

Part 3a

Step 1: Calculate the perimeter of the room

The room is a rectangle with length \( l = 14 \) ft and width \( w = 11 \) ft. The formula for the perimeter of a rectangle is \( P = 2(l + w) \).

$$ P = 2(14 + 11) = 2(25) = 50 \text{ ft} $$

Step 2: Subtract the width of the door

Since baseboard is not installed in the doorway (width = 3 ft), we subtract this from the perimeter.

$$ \text{Total baseboard length} = 50 - 3 = 47 \text{ ft} $$

We need to find combinations of 8 - ft and 12 - ft baseboards that sum to at least 47 ft (since we can't buy a fraction of a baseboard, we need to cover 47 ft).

Combination 1: Let's use \( x \) 12 - ft and \( y \) 8 - ft baseboards.

Try \( x = 4 \) (12 - ft baseboards): \( 4\times12 = 48 \) ft. We can use 4 twelve - ft baseboards (48 ft ≥ 47 ft).

Combination 2: Try \( x = 3 \) (12 - ft baseboards): \( 3\times12 = 36 \) ft. Then we need \( 47 - 36 = 11 \) ft from 8 - ft baseboards. Since \( 2\times8 = 16 \) ft, we can use 3 twelve - ft and 2 eight - ft baseboards: \( 3\times12+2\times8=36 + 16=52 \) ft (≥ 47 ft).

Step 1: Calculate the perimeter of the bathroom floor (excluding bathtub and door)

First, the room has width \( w = 6 \) ft and length \( l=9\frac{1}{2}=\frac{19}{2}\) ft. The perimeter of the rectangle is \( P = 2(l + w)\), but we need to exclude the bathtub and the door. From the diagram, the bathtub takes up \( 6\frac{1}{2}=\frac{13}{2}\) ft and the door is \( 2\frac{1}{2}=\frac{5}{2}\) ft. Wait, actually, the perimeter of the base (excluding bathtub and door) is calculated as follows: The two lengths and two widths, minus the bathtub length and the door width.
The total length around the walls (perimeter - like) without bathtub and door: \( 2\times6+2\times9\frac{1}{2}-\ 6\frac{1}{2}-2\frac{1}{2}\)
First, \( 2\times6 = 12 \), \( 2\times9\frac{1}{2}=19 \), sum of these two: \( 12 + 19=31 \). Then subtract the bathtub (\( 6\frac{1}{2}\)) and door (\( 2\frac{1}{2}\)): \( 31-(6\frac{1}{2}+2\frac{1}{2})=31 - 9 = 22 \) ft. Wait, Diego calculated 25 feet, so he is incorrect. Let's re - calculate properly.
Wait, maybe the bathroom has a different shape. From the diagram, the width is 6 ft, and the length is composed of two parts: \( 6\frac{1}{2}\) ft and 3 ft, so total length \( l=6\frac{1}{2}+3 = 9\frac{1}{2}\) ft (which matches). The perimeter of the rectangle is \( 2(6 + 9\frac{1}{2})=2(15\frac{1}{2}) = 31 \) ft. Now, we need to exclude the bathtub (length \( 6\frac{1}{2}\) ft) and the door (width \( 2\frac{1}{2}\) ft). So the length to tile is \( 31-6\frac{1}{2}-2\frac{1}{2}=31-(6\frac{1}{2}+2\frac{1}{2})=31 - 9 = 22 \) ft. Since 22≠25, Diego is incorrect.

Answer:

The total length of baseboard needed is \( \boldsymbol{47} \) feet.

Part 3b