Question

the value of the perimeter (p = 2l + 2w) is equal to the area (a = l·w). find the value of x for the problems below.
write an equation for each of the situations described below, solve it, then check it.

1. one - third of a number x is equal to 22 less than the number. write and solve an equation to find the number.
2. find the perimeter of the regular polygon.

Explanation:

Step1: Set up equation for first rectangle

For the rectangle with length $L = 6$ cm and width $W=x$ cm, since $P = A$, we have $2(6)+2x=6x$.
$12 + 2x=6x$.

Step2: Solve the equation

Subtract $2x$ from both sides:
$12=6x - 2x$.
$12 = 4x$.
Divide both sides by 4:
$x = 3$ cm.

Step3: Set up equation for second rectangle

For the rectangle with length $L = 6$ in and width $W = 2x$ in, since $P=A$, we have $2(6)+2(2x)=6\times(2x)$.
$12 + 4x=12x$.

Step4: Solve the equation

Subtract $4x$ from both sides:
$12=12x - 4x$.
$12 = 8x$.
Divide both sides by 8:
$x=\frac{3}{2}$ in.

Step5: Set up equation for the number problem

The equation for "One - third of a number $x$ is equal to 22 less than the number" is $\frac{1}{3}x=x - 22$.
Multiply through by 3 to clear the fraction:
$x = 3x-66$.
Subtract $x$ from both sides:
$0=3x - x-66$.
$0 = 2x-66$.
Add 66 to both sides:
$2x=66$.
Divide both sides by 2:
$x = 33$.

Step6: Set up equation for the regular hexagon

Since it's a regular hexagon, all sides are equal. So $7x + 4=5(x + 2)$.
Expand the right - hand side:
$7x+4=5x + 10$.
Subtract $5x$ from both sides:
$7x-5x+4=5x-5x + 10$.
$2x+4=10$.
Subtract 4 from both sides:
$2x=10 - 4$.
$2x=6$.
Divide both sides by 2:
$x = 3$.
The perimeter of the hexagon is $6\times(7x + 4)$. Substitute $x = 3$:
$6\times(7\times3+4)=6\times(21 + 4)=6\times25 = 150$.

Answer:

For the first rectangle, $x = 3$ cm.
For the second rectangle, $x=\frac{3}{2}$ in.
For the number problem, $x = 33$.
For the regular hexagon, $x = 3$ and the perimeter is 150.