the length of a new rectangular playing field is 7 years longer than triple the width. If the perimeter of the rectangular playing field is 422 yards, what are the dimensions?

Let's denote the width of the rectangular playing field as \( w \) (in yards). According to the problem, the length \( l \) of the field is 7 yards longer than triple the width, which can be written as:

\[
l = 3w + 7
\]

The perimeter \( P \) of a rectangle is given by the formula:

\[
P = 2l + 2w
\]

We are told that the perimeter is 422 yards, so:

\[
2l + 2w = 422
\]

Now, substitute \( l = 3w + 7 \) into the perimeter equation:

\[
2(3w + 7) + 2w = 422
\]

Simplify this equation:

\[
6w + 14 + 2w = 422
\]

Combine like terms:

\[
8w + 14 = 422
\]

Subtract 14 from both sides:

\[
8w = 408
\]

Divide by 8:

\[
w = 51
\]

Now, substitute \( w = 51 \) into the equation for the length:

\[
l = 3(51) + 7 = 153 + 7 = 160
\]

Thus, the dimensions of the rectangular playing field are:

- Width: 51 yards
- Length: 160 yards

Would you like further details or explanations?

Here are 5 related questions to consider:
1. How would the formula for perimeter change if the shape were a square?
2. What if the perimeter were doubled—how would the dimensions change?
3. How would the equation change if the width were 7 yards longer than triple the length?
4. What is the area of this rectangular field?
5. How do you find the diagonal of a rectangular field?

**Tip:** For any rectangle, the perimeter formula always involves doubling both the length and the width, making it a useful starting point for solving such problems.

The length of a new rectangular playing field is 7 yards longer than triple the width. If the perimeter of the rectangular playing field is 422 yards, what are the dimensions?

Learn how to find the dimensions of a rectangular playing field where the length is 7 yards longer than triple the width, and the perimeter is 422 yards. This problem uses algebraic methods and geometric formulas, suitable for students in Grades 7-9.

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