In the statistic department of UCI, there are 60 graduate students in total, 45 PhD students and 15 master students. Among master students, 40% of them are international students. Among PhD students, 20% are international students. (a) What is the proportion of international students in the statistic department? (b) What is the proportion of master students among all domestic students.

(a) The proportion of international students in the statistic department is 25%.

(b) The proportion of master students among all domestic students is 20%.

Step-by-step explanation:

The following can first be obtained from the question:

Total number of graduate students = 60

Number of PhD students = 45

Number of Master students = 15

Percentage of international students among the master students = 40%

Percentage of international students among the PhD students = 20%

Number of international master students = Number of Master students * Percentage of international students among the master students = 15 * 40% = 6

Number of international PhD students = Number of PhD students * Percentage of international students among the PhD students = 45 * 20% = 9

The questions can now be answered as follows:

(a) What is the proportion of international students in the statistic department?

This can be calculated as follows:

Total number of international students = Number of international master students + Number of international PhD students = 6 + 9 = 15

Total number of graduate students = 60

Proportion of international students in the statistic department = Total number of international students / Total number of graduate students = 15 / 60 = 0.25, or 25%

Therefore, the proportion of international students in the statistic department is 25%.

(b) What is the proportion of master students among all domestic students.

This can be calculated as follows:

Total number of graduate students = 60

Total number of international students = Number of international master students + Number of international PhD students = 6 + 9 = 15

Total number of domestic students = Total number of graduate students - Total number of international students = 60 - 15 = 45

Number of Master students = 15

Number of international master students = Number of Master students * Percentage of international students among the master students = 15 * 40% = 6

Number of domestic master students = Number of Master students - Number of international master students = 15 - 6 = 9

Proportion of master students among all domestic students = Number of domestic master students / Total number of domestic students = 9 / 45 = 0.20, or 20%

Therefore, the proportion of master students among all domestic students is 20%.

-6 is the correct option from the above given options

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).