Answer:
Absolutely! Here's how we can find a function to model the perimeter (P) of a rectangle in terms of its length (x):
1. **Relate area and side lengths:** We know the area (A) of the rectangle is 17 square meters (m^2) and it's given by the product of its length (x) and width (w). So we can represent this mathematically as:
A = x * w = 17 m^2
2. **Express width in terms of length:** Since we're looking for the perimeter in terms of length (x), we need to express the width (w) in terms of x. We can rearrange the area equation to solve for w:
w = A / x = 17 m^2 / x
3. **Define the perimeter function:** The perimeter (P) of a rectangle is the total length of all its sides. For a rectangle, it's typically calculated as the sum of twice the length (x) and twice the width (w). We can now express this as a function of x:
P(x) = 2 * x + 2 * w
4. **Substitute width expression:** Now, we can replace the width term (w) with the expression we derived in step 2:
P(x) = 2 * x + 2 * (17 m^2 / x)
5. **Simplify the function:** Distribute the terms and combine them:
P(x) = 2x + 34/x
Therefore, the function that models the perimeter (P) of the rectangle in terms of its length (x) is:
P(x) = 2x + 34/x
This function represents that the perimeter depends on both the length (x) itself and its inverse (1/x).
Let's denote the length of one side of the rectangle as x meters and the width as y meters. Since the area of the rectangle is 17 m², we have the equation:
xy = 17
Now, for a rectangle, the perimeter (P) can be expressed as:
P = 2x + 2y
We can express y in terms of x from the area equation:
y = 17/x
Substitute this expression for y into the perimeter equation:
P = 2x + 2(17/x)
Simplify this expression:
P(x) = 2x + 34/x
So, the function that models the perimeter P in terms of the length x of one of its sides is:
P(x) = 2x + 34/x
