Final answer:
To complete his patch design, Miguel will need to buy 3 rolls of gold fabric. He must calculate the perimeter and the total diagonal length to determine the total length of fabric required, and since each roll is 9 inches, he needs more than 2 but less than 3 rolls.
Step-by-step explanation:
Miguel is working on a project that requires him to design a patch with a stripe around the perimeter and diagonal stripes crossing from corner to corner. The patch is a square, and each side is 3 inches long. To calculate the total length of the gold fabric he needs, we first calculate the perimeter, which is the total distance around the patch.
The formula for the perimeter (P) of a square is P = 4 × side, so:
P = 4 × 3 inches
P = 12 inches
Next, we calculate the length of one diagonal using the Pythagorean theorem, where the diagonal (d) is the hypotenuse of a right-angled triangle with both the other sides equal to the side of the square.
d² = side² + side²
d² = 3² + 3²
d² = 9 + 9
d² = 18
d = √18
d ≈ 4.24 inches
Since there are two diagonals in a square, we need twice the length of one diagonal:
Total diagonal length = 2 × 4.24 inches
Total diagonal length ≈ 8.48 inches
Now, we sum up the perimeter and the total diagonal length to find the total fabric length Miguel needs:
Total fabric length = Perimeter + Total diagonal length
Total fabric length ≈ 12 inches + 8.48 inches
Total fabric length ≈ 20.48 inches
Miguel has rolls of gold fabric with 9 inches each. To find the number of rolls he needs, we divide the total fabric length by the length of one roll:
Number of rolls = Total fabric length ÷ Length of one roll
Number of rolls ≈ 20.48 inches ÷ 9 inches
Number of rolls ≈ 2.27
Since Miguel cannot buy a fraction of a roll, he will need to purchase 3 rolls of gold fabric to have enough for his patch.