Widterm Exam The length of a new rectangular playing field is 4 yards longer than triple the width. If the perimeter of the rectangular playing field is 392 yards, what are its dimensions?

Let's denote the width of the rectangular playing field as \( w \) yards. Given that the length of the playing field is 4 yards longer than triple the width, we can express the length as \( 3w + 4 \) yards. The perimeter of a rectangle is given by the formula: \[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \] Substitute the expressions for length and width into the perimeter formula: \[ 392 = 2 \times ((3w + 4) + w) \] Now, we can solve this equation to find the dimensions of the rectangular playing field. Solve the equation by following steps: - step0: Solve for \(w\): \(392=2\left(\left(3w+4\right)+w\right)\) - step1: Simplify: \(392=2\left(4w+4\right)\) - step2: Swap the sides: \(2\left(4w+4\right)=392\) - step3: Divide both sides: \(\frac{2\left(4w+4\right)}{2}=\frac{392}{2}\) - step4: Divide the numbers: \(4w+4=196\) - step5: Move the constant to the right side: \(4w=196-4\) - step6: Subtract the numbers: \(4w=192\) - step7: Divide both sides: \(\frac{4w}{4}=\frac{192}{4}\) - step8: Divide the numbers: \(w=48\) The width of the rectangular playing field is 48 yards. Now, we can find the length by substituting the value of the width back into the expression for the length: \[ \text{Length} = 3 \times 48 + 4 = 144 + 4 = 148 \text{ yards} \] Therefore, the dimensions of the rectangular playing field are: - Width: 48 yards - Length: 148 yards