The problem states that a vacuum cleaner depreciates in value each month. We are given the values of the vacuum cleaner for the first three months and need to calculate: a) The multiplier used to calculate the value each month. b) The percentage of value lost each month. c) Whether the vacuum cleaner will be worth less than $100 after 1 year (12 months).

1. Problem Description

The problem states that a vacuum cleaner depreciates in value each month. We are given the values of the vacuum cleaner for the first three months and need to calculate:

a) The multiplier used to calculate the value each month.

b) The percentage of value lost each month.

c) Whether the vacuum cleaner will be worth less than $100 after 1 year (12 months).

2. Solution Steps

a) To find the multiplier, we can divide the value of the vacuum cleaner in one month by its value in the previous month.

Multiplier from month 0 to month 1:

$multiplier_1 = \frac{1288}{1400} = 0.92$

Multiplier from month 1 to month 2:

$multiplier_2 = \frac{1184.96}{1288} = 0.92$

Multiplier from month 2 to month 3:

$multiplier_3 = \frac{1090.16}{1184.96} = 0.92$

Since the multipliers are approximately the same, the multiplier to calculate the value each month is 0.

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2.

b) To find the percentage of value lost each month, we can subtract the multiplier from 1 and multiply by

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0. $percentage\_lost = (1 - multiplier) * 100$

$percentage\_lost = (1 - 0.92) * 100 = 0.08 * 100 = 8\%$

The vacuum cleaner loses 8% of its value each month.

c) To determine if the vacuum cleaner will be worth less than $100 after 1 year, we can use the formula for exponential decay:

$Value(t) = InitialValue * (multiplier)^t$

Where t is the number of months. In this case t=12, InitialValue = $1400 and multiplier = 0.

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$Value(12) = 1400 * (0.92)^{12}$

$Value(12) = 1400 * 0.36766671 = 514.7334$

Since $514.73 > 100$, the vacuum cleaner will not be worth less than $100 after 1 year.

3. Final Answer

a) The multiplier to calculate the value each month is 0.

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2. b) The vacuum cleaner is losing 8% of its value each time.

c) No, in 1 year the vacuum cleaner will not be worth less than $

1

0

0. Its approximate value is $514.

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3.