(a) To find the width of the rectangular field, we can use the formula for the perimeter of a rectangle:
\[ P = 2(L + W) \]
where \(P\) is the perimeter, \(L\) is the length, and \(W\) is the width. Given that the perimeter is 314 m and the length is 98 m, we can substitute these values into the formula:
\[ 314 = 2(98 + W) \]
Now, divide both sides by 2:
\[ 157 = 98 + W \]
Next, subtract 98 from both sides:
\[ W = 157 - 98 \] \[ W = 59 , \text{m} \]
So, the width of the rectangular field is 59 m.
(b) To find the length of the rectangular painting, we can use the formula for the area of a rectangle:
\[ A = L \times W \]
where \(A\) is the area, \(L\) is the length, and \(W\) is the width. Given that the area is 4488 cm² and the width is 51 cm, we can rearrange the formula to solve for length:
\[ L = \frac{A}{W} \]
Substituting the values we have:
\[ L = \frac{4488}{51} \]
Calculating this gives:
\[ L = 88 , \text{cm} \]
So, the length of the rectangular painting is 88 cm.