Question

1. geometry the length of a rectangle is 3 cm greater than its width. the perimeter is 24 cm. what are the dimensions of the rectangle?

Explanation:

Step1: Define variables

Let the width of the rectangle be \( w \) cm. Then the length \( l = w + 3 \) cm.

Step2: Recall perimeter formula

The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \). We know \( P = 24 \) cm, so substitute \( l = w + 3 \) into the formula: \( 24 = 2((w + 3) + w) \).

Step3: Simplify the equation

First, simplify inside the parentheses: \( 24 = 2(2w + 3) \). Then distribute the 2: \( 24 = 4w + 6 \).

Step4: Solve for \( w \)

Subtract 6 from both sides: \( 24 - 6 = 4w \), so \( 18 = 4w \). Then divide by 4: \( w=\frac{18}{4}=4.5 \) cm? Wait, no, wait, let's check again. Wait, 24 = 2(2w + 3) → divide both sides by 2: 12 = 2w + 3 → subtract 3: 9 = 2w → \( w = 4.5 \)? Wait, no, that can't be. Wait, no, length is 3 cm greater than width. Wait, maybe I made a mistake. Wait, let's redo step 2. Perimeter formula: \( P = 2(l + w) \). Given \( l = w + 3 \), \( P = 24 \). So \( 24 = 2((w + 3) + w) \) → \( 24 = 2(2w + 3) \) → divide both sides by 2: \( 12 = 2w + 3 \) → subtract 3: \( 9 = 2w \) → \( w = 4.5 \)? Wait, but 4.5 + 3 = 7.5, then perimeter is 2(4.5 + 7.5) = 2(12) = 24. Oh, right, so width is 4.5 cm, length is 7.5 cm? Wait, but maybe the problem expects integer values? Wait, no, the problem doesn't say it has to be integer. Wait, but let's check again. Wait, maybe I messed up the variable definition. Wait, length is 3 cm greater than width. Let me define width as \( w \), length as \( l = w + 3 \). Perimeter \( 2(l + w) = 24 \). So \( l + w = 12 \). Since \( l = w + 3 \), substitute: \( w + 3 + w = 12 \) → \( 2w + 3 = 12 \) → \( 2w = 9 \) → \( w = 4.5 \) cm, length \( l = 4.5 + 3 = 7.5 \) cm. So the dimensions are width 4.5 cm and length 7.5 cm, or 9/2 cm and 15/2 cm.

Wait, but maybe I made a mistake in calculation. Let's do it again. Perimeter is 2(length + width) = 24. So length + width = 12. Length = width + 3. So (width + 3) + width = 12 → 2width + 3 = 12 → 2*width = 9 → width = 4.5, length = 7.5. Yes, that's correct.

Answer:

The width of the rectangle is \( 4.5 \) cm (or \( \frac{9}{2} \) cm) and the length is \( 7.5 \) cm (or \( \frac{15}{2} \) cm). Wait, but maybe the problem expects in decimal or fraction. Alternatively, maybe I made a mistake. Wait, no, let's check with width 4.5 and length 7.5: perimeter is 2(4.5 + 7.5) = 212 = 24. Correct. So the dimensions are width \( 4.5 \) cm and length \( 7.5 \) cm.