Final answer:
The perimeter of a rectangular field whose length is four times its width and has an area of 30976 cm² is found by first calculating the width using the area equation and then determining the length since it is four times the width.
Step-by-step explanation:
The student has asked how to find the perimeter of a rectangular field whose length is four times its width and has an area of 30976 cm². To solve this, we need to use the formulas for the area and perimeter of a rectangle. The formula for the area (A) of a rectangle is A = length × width, and the perimeter (P) of a rectangle is P = 2(length + width).
Let's denote the width of the field as w and the length as 4w, since the length is four times the width. With an area of 30976 cm², we can set up the equation 4w² = 30976. Dividing both sides by 4 gives us w² = 7744, and taking the square root of both sides gives us w = 88 cm (since width cannot be negative).
Now that we have the width, we can find the length by multiplying the width by 4, which is 4 × 88 cm = 352 cm. To find the perimeter, we substitute the values of length and width into the perimeter formula: P = 2(352 cm + 88 cm) = 2(440 cm) = 880 cm.
Therefore, the perimeter of the rectangular field is 880 cm.