♣ 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