During the spring of 2020, the state of Indiana was on lock down orders due to COVID-19. The state's business sales dropped exponentially and are modeled after the following equation:
Sales = 500 (1 - 0.10)^t
where t = number of days and sales = number of millions of dollars.
When sales have reached $23.5 million, it will be declared a statewide economic crisis. How many days until sales reach the economic crisis?
Answers
♣ Qᴜᴇꜱᴛɪᴏɴ :
During the spring of 2020, the state of Indiana was on lock down orders due to COVID-19. The state's business sales dropped exponentially and are modeled after the following equation:
Sales = 500 (1 - 0.10)^t
where t = number of days and sales = number of millions of dollars.
When sales have reached $23.5 million, it will be declared a statewide economic crisis. How many days until sales reach the economic crisis?
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♣ ᴀɴꜱᴡᴇʀ :
It will take 29 days to declare economic crisis
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♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :
According To Question :
Sales = 500 (1 - 0.10)^t
where t = number of days and sales = number of millions of dollars.
i. Compare it with a model
y = a(1 - b)ˣ
where b is percent change
Hence we get :
b = 0.10 = 10%
ii. What will be the sales after two weeks (14 days) ?
Sales
= 500 (1 - 0.10)¹⁴
= 500 × (0.90)¹⁴
= 500 × 0.228
= 114 million dollars
iii. If sales = $23.5 million , then declare economic crisis
23.5 = 500 (1 - 0.10)^t
Diving both sides by 500 :
23.5/500 = (500 (1 - 0.10)^t )/500
0.047 = (1 - 0.10)^t
0.047 = (0.90)^t
In(0.047) = In(0.90)^t
-3.057 = t In(0.90)
-3.057 = t × (-0.105)
t = 3.057/0.105
t = 29.11
Hence it will take 29 days to declare economic crisis