Question

Miguel is designing a patch to sew on his jacket. The patch is a square that is 3 in long on each side. Miguel plans to add a gold fabric stripe around the perimeter of the patch and then from each corner to the opposite corner. Miguel can buy rolls of the gold fabric that each contain 9 in of fabric. How many rolls of fabric will Miguel need to buy? Show all your work.

Answer 1

Miguel needs to buy three rolls of gold fabric for his patch. This includes fabric for the perimeter and diagonals of the square, taking into account the length of each roll.

Step-by-step explanation:

The question involves solving a basic geometry problem related to finding the perimeter of a square and the length of its diagonals. Miguel has a square patch with a side length of 3 inches, and he wants to add a gold fabric stripe around the perimeter and from each corner to the opposite corner (diagonals).

First, we calculate the perimeter of the square patch. The perimeter P of a square is given by P = 4 × side length. For Miguel's patch:

• P = 4 × 3 in = 12 in

Next, we calculate the length of one diagonal using the Pythagorean theorem. Since the square's sides are equal, the diagonal forms two 45-45-90 right triangles. Hence, the diagonal d is given by:

• d = side length × √2
• d = 3 in × √2 ≈ 4.24 in

Since there are two diagonals in the square, the total length of the diagonals is:

• Total diagonal length = 2 × d = 2 × 4.24 in ≈ 8.48 in

The total length of gold fabric needed is the sum of the perimeter and the total diagonal length:

• Total fabric needed = Perimeter + Total diagonal length ≈ 12 in + 8.48 in ≈ 20.48 in

Since each roll contains 9 inches of fabric, Miguel will need to buy:

• Rolls needed = Total fabric needed / 9 in per roll
• Rolls needed = 20.48 in / 9 in ≈ 2.28

Miguel must purchase 3 rolls because he cannot buy a fraction of a roll. Hence, Miguel needs to buy three rolls of gold fabric.

Answer 2

Step-by-step explanation:

so, he needs a much good fabric as the sum of the lengths of the perimeter and the 2 diagonals.

the perimeter (the sum of all 4 sides) = 4×3 = 12 in.

a single diagonal is calculated via Pythagoras (because a square has only 90° angles, a diagonal splits the square into 2 right-angled triangles, with the diagonal being the Hypotenuse) :

c² = a² + b²

c being the Hypotenuse (the side opposite of the 90° angle), a and b are the legs (enclosing the 90° angle).

so, in our case we have

diagonal² = 3² + 3² = 9 + 9 = 18

diagonal = sqrt(18) = 4.242640687... in

both diagonals are equally long, of course.

so both diagonals are

4.242640687... × 2 = 8.485281374... in

together with the perimeter Miguel needs therefore

12 + 8.485281374... = 20.48528137... in

but he can buy the gold fabric only in rolls of 9 in.

how many rolls does he need to buy to cover

20.48528137... in ?

20.48528137... / 9 = 2.276142375... rolls

now, as he cannot buy only a part of such roll, we need to round up to the next whole number : 3

Miguel needs to buy 3 rolls of gold fabric.