Question

a cube has an edge length of 9 centimeters. if the side length is increased by a factor of 3, how much larger is the perimeter of a face of the new cube? 72 cm 72 cm² 108 cm 108 cm²

Explanation:

Step1: Find new edge length

Original edge length is \( 9 \, \text{cm} \). Multiply by 3: \( 9 \times 3 = 27 \, \text{cm} \).

Step2: Perimeter of a square face

A face of a cube is a square. Perimeter of square: \( 4 \times \text{side length} \). For new cube: \( 4 \times 27 = 108 \, \text{cm} \)? Wait, no—wait, original perimeter? Wait, no, the question is "how much larger is the perimeter". Wait, original edge length is 9 cm, original perimeter: \( 4 \times 9 = 36 \, \text{cm} \). New perimeter: \( 4 \times 27 = 108 \, \text{cm} \). Difference: \( 108 - 36 = 72 \, \text{cm} \). Wait, I made a mistake earlier. Let's redo:

Step1: Original edge length \( a = 9 \, \text{cm} \). New edge length \( a' = 9 \times 3 = 27 \, \text{cm} \).

Step2: Perimeter of a face (square) is \( P = 4 \times \text{side} \). Original perimeter \( P_{\text{old}} = 4 \times 9 = 36 \, \text{cm} \). New perimeter \( P_{\text{new}} = 4 \times 27 = 108 \, \text{cm} \). Wait, no—wait, the question is "how much larger is the perimeter of a face of the new cube"—wait, maybe I misread. Wait, maybe it's the difference between new and old? Wait, no, let's check the options. The options include 72 cm, 72 cm², 108 cm, 108 cm². Wait, perimeter is in cm, area in cm². So first, find new perimeter: side is 27, perimeter is 427=108? No, wait original side is 9, new side is 27. Wait, no—wait, the problem says "how much larger is the perimeter of a face of the new cube"—maybe compared to the original? Wait, original perimeter: 49=36. New perimeter: 4*27=108. 108-36=72. So the increase is 72 cm. Wait, but let's check the options. The first option is 72 cm. Let's confirm:

Original edge: 9 cm. New edge: 93=27 cm. Perimeter of a face (square) is 4side. Original perimeter: 49=36 cm. New perimeter: 427=108 cm. Difference: 108-36=72 cm. So the answer is 72 cm.

Wait, but earlier I thought new perimeter is 108, but the question is "how much larger", so it's the difference. So:

Step1: Calculate new edge length: \( 9 \times 3 = 27 \, \text{cm} \).

Step2: Calculate original perimeter: \( 4 \times 9 = 36 \, \text{cm} \).

Step3: Calculate new perimeter: \( 4 \times 27 = 108 \, \text{cm} \).

Step4: Find the difference: \( 108 - 36 = 72 \, \text{cm} \).

Yes, so the perimeter of the new face is 72 cm larger than the original. So the correct option is 72 cm.

Answer:

72 cm (Option: 72 cm)