Question: The width of the rectangle is $x$ cm and the length of the rectangle is 1 more than 4 times width.
$x$ cm
a) Find expressions, in terms of $x$, for the perimeter and area of rectangle. Simplify your expressions.
Perimeter=
Area=
b) If the perimeter of rectangle is 32 cm. Work out the width ($x$) of the rectangle in cm.
1. **State the problem:**
We have a rectangle with width $x$ cm and length $4x + 1$ cm.
2. **Find expressions for perimeter and area:**
- The perimeter $P$ of a rectangle is given by the formula:
$$P = 2(\text{length} + \text{width})$$
- The area $A$ of a rectangle is given by:
$$A = \text{length} \times \text{width}$$
3. **Write expressions in terms of $x$:**
- Length $= 4x + 1$
- Width $= x$
4. **Calculate perimeter expression:**
$$P = 2((4x + 1) + x) = 2(5x + 1) = 10x + 2$$
5. **Calculate area expression:**
$$A = x(4x + 1) = 4x^2 + x$$
6. **Given perimeter is 32 cm, find $x$:**
Set perimeter equal to 32:
$$10x + 2 = 32$$
Subtract 2 from both sides:
$$10x + \cancel{2} - \cancel{2} = 32 - 2$$
$$10x = 30$$
Divide both sides by 10:
$$\frac{10x}{\cancel{10}} = \frac{30}{\cancel{10}}$$
$$x = 3$$
**Final answers:**
- Perimeter expression: $10x + 2$
- Area expression: $4x^2 + x$
- Width when perimeter is 32 cm: $x = 3$ cm