Question

1. a rectangle has an area of 8x + 16 and a perimeter of 2x + 20. determine the dimensions.

solution:
guess and
2x comes
must have

Explanation:

Step1: Use perimeter formula

The perimeter formula of a rectangle is \(P = 2(l + w)\). Given \(P=2x + 20\), then \(2(l + w)=2x + 20\). Divide both sides by 2: \(l + w=x + 10\).

Step2: Use area formula

The area formula of a rectangle is \(A=l\times w\). Given \(A = 8x+16\). Factor \(A\): \(A = 8(x + 2)\).

Step3: Assume dimensions

Let's assume the length \(l\) and width \(w\) such that \(l + w=x + 10\) and \(l\times w=8(x + 2)\). We can try to express the sum and product. Let \(l=a\) and \(w=b\). We know that \((a - b)^2=(a + b)^2-4ab\). Substitute \(a + b=x + 10\) and \(ab = 8x+16\). \((a - b)^2=(x + 10)^2-4(8x + 16)=x^{2}+20x + 100-(32x + 64)=x^{2}-12x + 36=(x - 6)^2\). Then \(a - b=x - 6\) (assuming \(a>b\)).

Step4: Solve the system of equations

We have the system \(

$$\begin{cases}a + b=x + 10\\a - b=x - 6\end{cases}$$

\). Add the two equations: \(2a=2x + 4\), so \(a=x + 2\). Substitute \(a=x + 2\) into \(a + b=x + 10\), then \(x + 2+b=x + 10\), so \(b = 8\).

Answer:

The dimensions of the rectangle are \(x + 2\) and \(8\).