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A garden is 15 feet long by 5 feet wide. The length and width of the garden will each be increased by the same number of feet. This expression represents the perimeter of the larger garden:

[tex]\[ (x+15) + (x+5) + (x+15) + (x+5) \][/tex]

Which expression is equivalent to the expression for the perimeter of the larger garden? Select all that apply.

A. [tex]\[4x + 40\][/tex]
B. [tex]\[2(2x + 20)\][/tex]
C. [tex]\[2(x + 15)(x + 5)\][/tex]
D. [tex]\[4(x + 15)(x + 5)\][/tex]
E. [tex]\[2(x + 15) + 2(x + 5)\][/tex]

Sagot :

GinnyAnsweridn
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.
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