To determine which expressions are equivalent to the expression for the perimeter of a larger garden, let's break down the problem step by step.
1. Original Dimensions:
- Length: 15 feet
- Width: 5 feet
2. New Dimensions After Increase:
- Let [tex]\(x\)[/tex] be the number of feet added to each dimension.
- New Length: [tex]\(15 + x\)[/tex] feet
- New Width: [tex]\(5 + x\)[/tex] feet
3. Perimeter of the Larger Garden:
The perimeter [tex]\(P\)[/tex] of a rectangle is given by:
[tex]\[
P = 2 \times (\text{Length} + \text{Width})
\][/tex]
For the larger garden:
[tex]\[
P = 2 \times ((15 + x) + (5 + x))
\][/tex]
4. Simplify the Expression:
Combine the terms inside the parentheses:
[tex]\[
P = 2 \times (20 + 2x)
\][/tex]
Then, distribute the 2:
[tex]\[
P = 40 + 4x
\][/tex]
So, the expression for the perimeter of the larger garden simplifies to [tex]\(4x + 40\)[/tex].
Now, let's match this expression against the given options:
A. [tex]\(4x + 40\)[/tex]
- This matches directly with our simplified perimeter expression.
B. [tex]\(2(2x + 20)\)[/tex]
- Simplify this expression:
[tex]\[
2(2x + 20) = 4x + 40
\][/tex]
This matches our perimeter expression.
C. [tex]\(2(x + 15)(x + 5)\)[/tex]
- Expand this expression:
[tex]\[
2((x + 15)(x + 5)) = 2(x^2 + 20x + 75) = 2x^2 + 40x + 150
\][/tex]
This is not equivalent to our simplified perimeter expression [tex]\(4x + 40\)[/tex].
D. [tex]\(4(x + 15)(x + 5)\)[/tex]
- Expand this expression:
[tex]\[
4((x + 15)(x + 5)) = 4(x^2 + 20x + 75) = 4x^2 + 80x + 300
\][/tex]
This is not equivalent to our simplified perimeter expression [tex]\(4x + 40\)[/tex].
E. [tex]\(2(x + 15) + 2(x + 5)\)[/tex]
- Simplify this expression:
[tex]\[
2(x + 15) + 2(x + 5) = 2x + 30 + 2x + 10 = 4x + 40
\][/tex]
This matches our perimeter expression.
Conclusion:
The expressions that are equivalent to the perimeter of the larger garden are:
- [tex]\(4x + 40\)[/tex] (Option A)
- [tex]\(2(2x + 20)\)[/tex] (Option B)
- [tex]\(2(x + 15) + 2(x + 5)\)[/tex] (Option E)
So, the correct answers are:
A, B, and E.
