Question

linear and exponential growth: medium

w	1	2	4	8	16	32

laxman opened a restaurant 32 weeks ago. the table shows the number of customers, ( c ), during the ( w^{\text{th}} ) week. which of the following correctly explains the growth of ( c ) with respect to ( w )?

choose 1 answer:

a) the number of customers per week grew linearly because the number of customers per week increased by approximately 30 every week.

Explanation:

To determine the growth type, we analyze the relationship between \( w \) (week) and \( C \) (customers):

Step 1: Check for Exponential Growth (Multiplicative Relationship)

• When \( w \) doubles from 1 to 2, \( C \) doubles from 30 to 60.
• When \( w \) doubles from 2 to 4, \( C \) doubles from 60 to 120.
• When \( w \) doubles from 4 to 8, \( C \) doubles from 120 to 240.
• This pattern (doubling \( w \) leads to doubling \( C \)) continues for all given values. This indicates a multiplicative (exponential) relationship where \( C \) is proportional to \( w \) (or \( C = k \cdot w \), but here the ratio \( \frac{C}{w} \) is constant? Wait, no—wait, let’s check the ratio \( \frac{C}{w} \):
• For \( w = 1 \), \( \frac{C}{w} = \frac{30}{1} = 30 \)
• For \( w = 2 \), \( \frac{C}{w} = \frac{60}{2} = 30 \)
• For \( w = 4 \), \( \frac{C}{w} = \frac{120}{4} = 30 \)
• For \( w = 8 \), \( \frac{C}{w} = \frac{240}{8} = 30 \)
• For \( w = 16 \), \( \frac{C}{w} = \frac{480}{16} = 30 \)
• For \( w = 32 \), \( \frac{C}{w} = \frac{960}{32} = 30 \)

Step 2: Analyze the Relationship

The ratio \( \frac{C}{w} = 30 \) is constant for all \( w \). This means \( C = 30w \), which is a linear relationship (since it fits \( y = mx + b \) with \( b = 0 \) and \( m = 30 \)). Wait, but the option A says: "The number of customers per week grew linearly because the number of customers per week increased by approximately 30 every week." Wait, let's check the "increase per week":

• From \( w = 1 \) to \( w = 2 \): \( C \) increases by \( 60 - 30 = 30 \) (over 1 week, so 30 per week).
• From \( w = 2 \) to \( w = 4 \): \( w \) increases by 2 weeks, \( C \) increases by \( 120 - 60 = 60 \), so 30 per week (60 / 2 = 30).
• From \( w = 4 \) to \( w = 8 \): \( w \) increases by 4 weeks, \( C \) increases by \( 240 - 120 = 120 \), so 30 per week (120 / 4 = 30).

Thus, the rate of increase is constant (30 customers per week), which is characteristic of linear growth. Option A correctly describes this: the number of customers per week grew linearly because the number of customers per week increased by approximately 30 every week.

Answer:

A. The number of customers per week grew linearly because the number of customers per week increased by approximately 30 every week.