A drone is monitoring the atmospheric conditions above a farm field. The drone hovers 5 meters above the crop line. Suddenly, it rises to approximately 5.9 meters (which takes 1.9 seconds) to avoid colliding with the sprinkler system. Based on this information, which equations could model the height, y, of the drone as a function of time, x?

The correct options are;

g(x) = -0.27·x² + x  + 5

h(x) = 2·㏒(x + 1) + 5

Step-by-step explanation:

To answer the question, we substitute x = 1.9 seconds into the given options as follows;

1) For f(x) = √(1.6·x) + 5

When x = 1.9 seconds, we have;

y = f(1.9) = √(1.6×1.9) + 5 = 6.74 which is not equal to the given height of 5.9 meters

Therefore, f(x) = √(1.6·x) + 5 does not model the height of the drone y as a function of time, x

2) For g(x) = -0.27·x² + x  + 5

When x = 1.9 seconds, we have;

y = g(1.9) = -0.27×1.9^2 + 1.9  + 5 = 5.93 meters, which is approximately 5.9 meters to one place of decimal

Therefore, the function, g(x) = -0.27·x² + x  + 5,  approximately models the height of the drone y as a function of time, x

3) For h(x) = 2·㏒(x + 1) + 5

When x = 1.9 seconds, we have;

y = h(1.9) = 2·log(1.9 + 1) + 5 = 5.92 meters,

The function, h(x) = 2·㏒(x + 1) + 5,  approximately models the height of the drone y as a function of time, x

4) For j(x) = -∛(-1.4·x - 1) + 5

When x = 1.9 seconds, we have;

y = j(1.9) = -∛(-1.4×1.9 - 1) + 5 = 6.54 meters

The function j(x) = -∛(-1.4·x - 1) + 5 does not model the height of the drone y as a function of time, x

5) For k(x) = -1.2·x³ + 2.6·x² - 0.5·x + 5

When x = 1.9 seconds, we have;

y = k(1.9) = -1.2×1.9^3 + 2.6×1.9^2 - 0.5×1.9 + 5 = 5.21 meters

Therefore, the function, k(x) = -1.2·x³ + 2.6·x² - 0.5·x + 5, does not model the height of the drone y as a function of time, x

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).

22.83 repeating

step-by-step explanation:

137 ÷ 6 is 22.8333333 therefore we understand that 3 is repeating.

happy to

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