Alright, Harley, let's carefully review the calculations step-by-step.
1. Identify the dimensions of the backyard:
- Length (L): [tex]\( L = 5x^2 + 2x + 1 \)[/tex]
- Width (W): [tex]\( W = x + 3 \)[/tex]
2. Calculate the total perimeter excluding the length side by the house:
To find the perimeter of the backyard excluding the side touching the house, we will calculate the perimeter of the full rectangle and then subtract the side touching the house from it.
3. Full rectangle perimeter:
- The perimeter (P) of a rectangle formula is [tex]\( P = 2L + 2W \)[/tex].
4. Calculate [tex]\(2L\)[/tex] and [tex]\(2W\)[/tex]:
- [tex]\( 2L = 2 \times (5x^2 + 2x + 1) = 10x^2 + 4x + 2 \)[/tex]
- [tex]\( 2W = 2 \times (x + 3) = 2x + 6 \)[/tex]
5. Full perimeter:
- Total perimeter (P) = [tex]\( 2L + 2W = 10x^2 + 4x + 2 + 2x + 6 = 10x^2 + 6x + 8 \)[/tex].
From the initial formula of [tex]\( P = 2L + 2W \)[/tex], we are not touching the calculation. Next, we need to subtract the house side like just one length and not both lengths or widths.
6. Subtract the length side touching the house:
From the formula, we have one side of length touching the house.
- [tex]\( One\ length\side = 5x^2 + 2x + 1 \)[/tex]
7. Required perimeter:
- Required perimeter [tex]\( = [2(5x^2 + 2x + 1) + 2(x + 3) - (5x^2 + 2x + 1)] = = (10x^2 + 4x + 2 + 2x + 6) - (5x^2 + 2x + 1) = (10x^2 + 6x + 8) - (5x^2 + 2x + 1) =5x^2 + 4x + 7 \)[/tex].
8. Review Harley’s total perimeter for fencing:
Harley’s final calculation for her total perimeter is [tex]\(10 x^2 + 6 x + 8\)[/tex].
9. Compare Harley's total perimeter with our required perimeter:
- Required perimeter = [tex]\(5x^2 + 4x + 7 \)[/tex]
Since Harley has [tex]\(10 x^2 + 6 x + 8\)[/tex] as her total perimeter, which does not match the required perimeter [tex]\(5x^2 + 4x + 7\)[/tex], Harley’s calculation seems incorrect.
10. Conclusion:
You should inform Harley that the correct perimeter for fencing her backyard is [tex]\(5x^2 + 4x + 7\)[/tex] and not [tex]\(10 x^2 + 6 x + 8\)[/tex]. Therefore, her calculation has a mistake, and she should revise it before proceeding with the fencing.
