(10 The length of a rectangle, $$ x \mathrm{~cm} $$, is three times its width. The perimeter of the rectangle is not more than 120 cm . (a) Write an inequality in $$ x $$. (b) Hence find the maximum possible value of its length.

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### Known Conditions:
1. The length of the rectangle, $$ L $$, is given as $$ x $$ cm.
2. The width of the rectangle, $$ W $$, is one-third of the length: $$ W = \frac{x}{3} $$.
3. The perimeter of the rectangle is not more than 120 cm.

### Step (a): Write an inequality in $$ x $$.

The formula for the perimeter $$ P $$ of a rectangle is given by:
$$
P = 2L + 2W
$$
Substituting the expressions for $$ L $$ and $$ W $$:
$$
P = 2x + 2\left(\frac{x}{3}\right)
$$
Now, we can express the perimeter inequality:
$$
2x + 2\left(\frac{x}{3}\right} \leq 120
$$

### Step (b): Solve the inequality to find the maximum possible value of its length.

Let's simplify the inequality:
1. Combine the terms:
   $$
   2x + \frac{2x}{3} \leq 120
   $$
2. To combine $$ 2x $$ and $$ \frac{2x}{3} $$, we can express $$ 2x $$ as $$ \frac{6x}{3} $$:
   $$
   \frac{6x}{3} + \frac{2x}{3} \leq 120
   $$
   $$
   \frac{8x}{3} \leq 120
   $$
3. Multiply both sides by 3 to eliminate the fraction:
   $$
   8x \leq 360
   $$
4. Divide both sides by 8:
   $$
   x \leq 45
   $$

Thus, the maximum possible value of the length $$ x $$ is 45 cm.

### Summary:
(a) The inequality in $$ x $$ is $$ \frac{8x}{3} \leq 120 $$.

(b) The maximum possible value of the length is $$ 45 $$ cm.
(a) The inequality is $$ \frac{8x}{3} \leq 120 $$. (b) The maximum possible length is 45 cm.

(10 The length of a rectangle, , is three
times its width. The perimeter of the
rectangle is not more than 120 cm .
(a) Write an inequality in .
(b) Hence find the maximum possible value
of its length.

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Solución

Known Conditions:

Step (a): Write an inequality in .

Step (b): Solve the inequality to find the maximum possible value of its length.

Summary:

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(10 The length of a rectangle, , is three times its width. The perimeter of the rectangle is not more than 120 cm . (a) Write an inequality in . (b) Hence find the maximum possible value of its length.