Simplify each expression, and then arrange them in increasing order based on the coefficient of n2. -5(n3 – n2 – 1) + n(n2 – n) (n2 – 1)(n + 2) – n2(n – 3) n2(n – 4) + 5n3 – 6 2n(n2 – 2n – 1) + 3n2

Step-by-step explanation:

-5(n^3 - n^2 - 1) + n(n^2 - n)

-5n^3 + 5n^2 + 5 + n^3 - n^2

-4n^3 + 4n^2 + 5 <===

(n^2 - 1)(n + 2) - n^2(n - 3)

n^2(n + 2) - 1(n + 2) - n^3 + 3n^2

n^3 + 2n^2 - n - 2 - n^3 + 3n^2

5n^2 - n - 2 <===

n^2(n - 4) + 5n3 - 6

n^3 - 4n^2 + 5n^3 - 6

6n^3 - 4n^2 - 6 <===

2n(n^2 - 2n - 1) + 3n^2

2n^3 - 4n^2 - 2n + 3n^2

2n^3 -n^2 - 2n <===

putting them in order based on the coefficient n^2 is :

6n^3 - 4n^2 - 6

2n^3 - n^2 - 2n

-4n^3 + 4n^2 + 5

5n^2 - n - 2

1/3

step-by-step explanation:

to find this, you can divide 12 by 4 which is 3 or you can multiply 4 until you reach 12 (4x3=12). both should give you 3, meaning theres three times less bananas than there are oranges.

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).