Advanced Placement (AP), 31.01.2022 01:20 TrueKing184

1. The rate at which people enter a movie theater on a given day is modeled by the function S defined by S(t) = 80 -12 cos 6 The rate at which people leave the same movie theater is modeled by the function R defined by r(t) = 12eKo + 20. Both S (t) and r(t) are measured in people per hour and these functions are t valid for 10 < t < 22. At time t = 10, there are no people in the movie theater.(Problems are in the Picture and I need help)

1. The rate at which people enter a movie theater on a given day is modeled by the function S defin

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Answer from: yah2muchh

Hi there!

To find the total amount of people that have ENTERED by t = 20, we must take the integral of the appropriate function.

$\text{Amount that entered} = \int\limits^{20}_{10} {S(t)} \, dt \\\\ = \int\limits^{20}_{10} {80 - 12cos(\frac{t}{5})} \, dt$

Evaluate using a calculator:

$= 899.97 \approx \boxed{900\text{ people}}$

To solve, we can find the total amount of people that have entered of the interval and subtract the total amount of people that have left from this value.

In other terms:
$\text{Amount of people} = \int\limits^{20}_{10} {S(t)} \, dt - \int\limits^{20}_{10} {R(t)} \, dt$

We can evaluate using a calculator (math-9 on T1-84):

$\text{\# of people} = \int\limits^{20}_{10} {80-12cos(\frac{t}{5})} \, dt - \int\limits^{20}_{10} {12e^{\frac{t}{10}}+20} \, dt$

$= 899.97 - 760.49 = 139.47 \approx \boxed{139 \text{ people}}$

If:
$P(t) = \int\limits^t_{10} {S(t) - R(t)} \, dt$

Then:

$\frac{dP}{dt} = P'(t)= \frac{d}{dt}\int\limits^t_{10} {S(t) - R(t)} \, dt = S(t) - R(t)$

Evaluate at t = 20:

$S(20) = 80 - 12cos(\frac{20}{5}) = 87.844\\\\R(20) = 12e^{\frac{20}{10}} + 20 = 108.669$

S(20) - R(20) = 87.844 - 108.669 = -20.823

This means that at t = 20, there is a NET DECREASE of people at the movie theater of around 20.823 (21) people per hour.

To find the maximum, we must use the first-derivative test.

Set S(t) - R(t) equal to 0:

$80 - 12cos(\frac{t}{5}) - 12e^{\frac{t}{10}} - 20 = 0\\\\60 - 12(cos(\frac{t}{5}) + e^{\frac{t}{10}})= 0$

Graph the function with a graphing calculator and set the function equal to y = 0:

According to the graph, the graph of the first derivative changes from POSITIVE to NEGATIVE at t ≈ 17.78 hours, so there is a MAXIMUM at this value.

Thus, at t = 17.78 hours, the amount of people at the movie theater is a MAXIMUM.

Answer from: Quest

after lydia compare her check register to her bank statement, and the ending balances do not seem to match. then, she must make a bank reconciliation by putting the data all together.

to do this, she need to determine the bank ending balance. after, add the deposits in transit then deduct the outstanding checks. moreover, if the actual checkbook balance does not equal this number, check again for transporting errors by subtracting the balances and add the difference to nine. sometimes, a mathematical error can happen but a bank reconciliation is a great tool to resolve the discrepancies.

Answer from: Quest

If you are talking about typical sand you would find at the beach or in a river bed here on earth, there are some differences. i would say the first biggest difference is in composition and mineral make up. although the moon is largely made of the same materials as earth’s mantle and surface, the moon experiences very little weathering (no liquid water and virtually no atmosphere). “earth” sand is typically composed mostly of quartz grains. this is because quartz is highly resistant to both physical and chemical weathering. most of the other minerals originally with the quartz are broken down into fine clay and carried away, chemically eroded/ dissolved, or some combination of the two. this usually leaves only the quartz behind, and so most of the time when you see/think of sand, you are talking about quartz. without these weathering mechanisms on the moon, the “sand” (which is likely more silt or clay sized grains anyways) would consist of mostly other minerals like anorthosite, olivine, and proxene.

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