Final Answer:
The perimeter of the garden is given by the expression P = x² + 2x⁷ + √(x⁴ + 4x¹⁴).
Step-by-step explanation:
To find the perimeter of the garden, we first need to determine the lengths of all three sides of the rectangular triangle, also known as a right-angled triangle.
The two sides given are:
1. One side (leg) is x².
2. The other side (leg) is 2x⁷.
According to Pythagoras' theorem, for a right-angled triangle with legs of lengths a and b and hypotenuse c, the relationship between the lengths is:
c² = a² + b²
In this case, side1 (a) = x² and side2 (b) = 2x⁷.
Let's find the length of the hypotenuse (c):
c² = (x²)² + (2x⁷)²
= x⁴ + (2² * x^(7*2))
= x⁴ + 4x¹⁴
Now, finding c (the hypotenuse) from c²:
c = √(x⁴ + 4x¹⁴)
The perimeter (P) of a triangle is the sum of the lengths of all its sides. So, in this case:
P = side1 + side2 + hypotenuse
P = x² + 2x⁷ + √(x⁴ + 4x¹⁴)
This expression gives us the perimeter of the garden in terms of x, considering x has a positive value (as side lengths must be positive).