Math

QuestionFind the inequality for hours tt after 6 a.m. when cars passing the tollbooth exceed 4,300, starting with 1,380 and increasing by 46%46\% each hour.

Studdy Solution

STEP 1

Assumptions1. The number of cars that passed through the tollbooth prior to6 a.m. is1,380. The number of cars that pass through the tollbooth from6 a.m. through the morning rush hour increases by46% every hour3. We need to find the number of hours, tt, after6 a.m. when the number of cars that have passed through the tollbooth is over4,300

STEP 2

We can model the number of cars passing through the toll booth after6 a.m. as an exponential growth problem. The formula for exponential growth is=(1+r)t =(1 + r)^tWhere- isthefinalamount is the final amount- is the initial amount (or principal) - rr is the rate of growth (expressed as a decimal) - tt is the time

STEP 3

We are given that the initial number of cars () is1,380, the rate of growth ($r$) is46%, and we want to find the time ($t$) when the number of cars () is over,300. So, we can set up the inequality4300<1380(1+0.46)t4300 <1380(1 +0.46)^t

STEP 4

Convert the percentage to a decimal value.
46%=0.4646\% =0.46So, the inequality becomes4300<1380(1+0.46)t4300 <1380(1 +0.46)^t

STEP 5

implify the inequality4300<1380(1.46)t4300 <1380(1.46)^tThis inequality can be used to determine the number of hours, tt, after a.m. when the number of cars that have passed through the tollbooth is over4,300.

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