Answer :
Let the sides of the rectangle be [tex]$x$[/tex] and [tex]$y$[/tex]. We are given:
1. The area of the rectangle:
[tex]$$xy = 176 \quad \text{(cm}^2\text{)}.$$[/tex]
2. The perimeter of the rectangle:
[tex]$$2(x + y) = 54.$$[/tex]
From the perimeter, we have:
[tex]$$x + y = \frac{54}{2} = 27.$$[/tex]
We can express [tex]$y$[/tex] in terms of [tex]$x$[/tex]:
[tex]$$y = 27 - x.$$[/tex]
Substitute this expression for [tex]$y$[/tex] into the area equation:
[tex]$$x(27 - x) = 176.$$[/tex]
Expanding the expression gives:
[tex]$$27x - x^2 = 176.$$[/tex]
Rearrange the equation to form a standard quadratic equation:
[tex]$$x^2 - 27x + 176 = 0.$$[/tex]
To solve the quadratic equation, we calculate the discriminant:
[tex]$$\Delta = (-27)^2 - 4(1)(176) = 729 - 704 = 25.$$[/tex]
Since [tex]$\Delta = 25$[/tex], the square root of the discriminant is:
[tex]$$\sqrt{\Delta} = 5.$$[/tex]
Now, solve for [tex]$x$[/tex] using the quadratic formula:
[tex]$$x = \frac{-(-27) \pm \sqrt{25}}{2(1)} = \frac{27 \pm 5}{2}.$$[/tex]
This yields the two possible solutions:
[tex]$$x = \frac{27 + 5}{2} = 16 \quad \text{or} \quad x = \frac{27 - 5}{2} = 11.$$[/tex]
Thus, the two dimensions of the rectangle are [tex]$16$[/tex] cm and [tex]$11$[/tex] cm. The order does not matter; one side is [tex]$16$[/tex] cm and the other is [tex]$11$[/tex] cm.
1. The area of the rectangle:
[tex]$$xy = 176 \quad \text{(cm}^2\text{)}.$$[/tex]
2. The perimeter of the rectangle:
[tex]$$2(x + y) = 54.$$[/tex]
From the perimeter, we have:
[tex]$$x + y = \frac{54}{2} = 27.$$[/tex]
We can express [tex]$y$[/tex] in terms of [tex]$x$[/tex]:
[tex]$$y = 27 - x.$$[/tex]
Substitute this expression for [tex]$y$[/tex] into the area equation:
[tex]$$x(27 - x) = 176.$$[/tex]
Expanding the expression gives:
[tex]$$27x - x^2 = 176.$$[/tex]
Rearrange the equation to form a standard quadratic equation:
[tex]$$x^2 - 27x + 176 = 0.$$[/tex]
To solve the quadratic equation, we calculate the discriminant:
[tex]$$\Delta = (-27)^2 - 4(1)(176) = 729 - 704 = 25.$$[/tex]
Since [tex]$\Delta = 25$[/tex], the square root of the discriminant is:
[tex]$$\sqrt{\Delta} = 5.$$[/tex]
Now, solve for [tex]$x$[/tex] using the quadratic formula:
[tex]$$x = \frac{-(-27) \pm \sqrt{25}}{2(1)} = \frac{27 \pm 5}{2}.$$[/tex]
This yields the two possible solutions:
[tex]$$x = \frac{27 + 5}{2} = 16 \quad \text{or} \quad x = \frac{27 - 5}{2} = 11.$$[/tex]
Thus, the two dimensions of the rectangle are [tex]$16$[/tex] cm and [tex]$11$[/tex] cm. The order does not matter; one side is [tex]$16$[/tex] cm and the other is [tex]$11$[/tex] cm.
