Question

The ceiling of victorias living room is a square that is 15⟌2 ft long on each side. to decorate for a party, she plans to hang crepe paper around the perimeter of the ceiling and then from each corner to the opposite corner. Victoria can buy rolls that each contain 30 ft of crepe paper. What is the minimum number of rolls she should buy? Show your work.

Answer 1

5 rolls.

Explanation:

The perimeter of a two-dimensional shape is the distance all the way around the outside.

Given the square ceiling has side lengths of 15√2 feet, and the side lengths of a square are equal, the perimeter of the ceiling is:

`\implies 4 * 15√(2)=60√(2)\;\; \sf ft`

Diagonal of a square

`d=√(2s^2)`

where s is the side length.

Therefore, the length of the diagonal of the ceiling is:

`\implies d=\sqrt{2(15√(2))^2}`

`\implies d=√(2\cdot 450)`

`\implies d=√(900)`

`\implies d=30\;\; \sf ft`

Therefore, the total amount of crepe paper Victoria needs is:

`\begin{aligned}\implies \sf Total\;length&=\sf Perimeter+2\;diagonals\\&=60√(2)+30+30\\&=144.8528137...\;\; \sf ft\end{aligned}`

If each roll contains 30 ft, to calculate the minimum number of rolls Victoria should buy, divide the total amount of crepe paper by 30 and round up to the nearest whole number:

`\implies (144.8528137...)/(30)=4.828427125...`

Therefore, Victoria should buy a minimum of 5 rolls of crepe paper.

Answer 2

5 rolls

Explanation:

We are here given that the length of a ceiling which is square in shape is 15√2 ft .

She wants to decorate her ceiling with crepe paper around the perimeter and from each corner to the opposite corners ( this means along the diagonals of the wall ) For figure see attachment.

So from the figure th total length of crepe paper required will be ,

l = AB + BC + CD + DA + BD + AC . . . . . (i)

and here AC = BC = CD = DA = 15√2 ft .

Also BD = AC [ as the diagonals of a square are equal]

So , we can write (i) as ,

l = 4*AB + 2*AC

In right angled triangle BDC ,

BC² + CD² = BD² [ Pythagoras theorem]

(15√2ft)² + (15√2ft)² = BD²

450ft2 + 450ft² = BD²

900ft² = BD²

√900ft² = BD²

30ft = BD

So total length of crepe paper required will be,

l = 4*15√2 ft + 2*30ft

l = 60*1.4141ft + 60ft

l = 84.84ft + 60ft

l = 144.84ft

Now we are here given that the length of each roll is 30ft . So total number of such roles that will be required is ,

n = 144.84ft / 30ft

n = 4.828 rolls

n ≈ 5 rolls

and we are done!