Questions: The length of a rectangle vegetable garden is 7 yards more than twice its width. a. Write an expression for the perimeter of the garden in terms of its width, w. P= b. Ann bought 50 yards of fence to enclose the garden. What are its dimensions? Length: yards Width: yards

The length of a rectangle vegetable garden is 7 yards more than twice its width. a. Write an expression for the perimeter of the garden in terms of its width, w. P= b. Ann bought 50 yards of fence to enclose the garden. What are its dimensions? Length: yards Width: yards
Transcript text: The length of a rectangle vegetable garden is 7 yards more than twice its width. a. Write an expression for the perimeter of the garden in terms of its width, $\boldsymbol{w}$. \[ P= \] b. Ann bought 50 yards of fence to enclose the garden. What are its dimensions? Length: $\square$ yards Width: $\square$ yards
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Solution

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Solution Steps

To solve the problem, we need to express the perimeter of the rectangle in terms of its width and then use the given perimeter to find the dimensions of the garden.

a. The perimeter \( P \) of a rectangle is given by the formula \( P = 2 \times (\text{length} + \text{width}) \). We know the length is 7 yards more than twice the width, so we can express the length as \( 2w + 7 \). Substitute this expression into the perimeter formula to get the expression in terms of \( w \).

b. Given that the perimeter is 50 yards, we can set up an equation using the expression from part (a) and solve for the width \( w \). Once we have the width, we can find the length using the relationship between length and width.

Step 1: Expression for the Perimeter

The length \( L \) of the rectangle vegetable garden is given by the expression: \[ L = 2w + 7 \] The perimeter \( P \) of a rectangle is calculated using the formula: \[ P = 2(L + w) \] Substituting the expression for \( L \) into the perimeter formula, we have: \[ P = 2((2w + 7) + w) = 2(3w + 7) = 6w + 14 \]

Step 2: Setting Up the Equation

We know that Ann bought 50 yards of fence to enclose the garden, so we set the perimeter equal to 50: \[ 6w + 14 = 50 \]

Step 3: Solving for Width

To find the width \( w \), we solve the equation: \[ 6w = 50 - 14 \] \[ 6w = 36 \] \[ w = 6 \]

Step 4: Calculating the Length

Now that we have the width, we can find the length using the expression for \( L \): \[ L = 2(6) + 7 = 12 + 7 = 19 \]

Final Answer

The dimensions of the garden are:

  • Width: \( \boxed{w = 6} \) yards
  • Length: \( \boxed{L = 19} \) yards
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