(1 point) Suppose P = f(t) is the population (in thousands) of town t years afte

Question

(1 point) Suppose P = f(t) is the population (in thousands) of town t years after 1990, and that f(2) = 14 and f(8) = 19, (a) Find a formula for f(t) assumingf is exponential: P = f() 0 (b) Find a formula for f-1(P) = (C) Evaluate f(45) = (Round your answer to the nearest whole number.) (d) f-1(45) = (Round your answer to at least one decimal place.) Write out sentences to explain the practical meaning of your answers to parts (c) and (d). Consider the seven numbered statements in the list below: 1. The town's population will reach 45,000 people in f-1(45) years from now. 2. The town's population will reach 45 people inf-l(45) years after 1990. 3. The town's population in 2035 is f(45) people. 4. The town's population will reach 45,000 people in f-1(45) years after 1990. 5. The town's population in 2035 is f(45) thousand people. 6. The town's population in 2045 is f(45) people. 7. The town's population has grown by f(45) people over a 45 year period. (e) which statement above explains the meaning of your answer to (c)? (enter the number 1-7 of the correct statement). (1) Which statement above explains the meaning of your answer to (d)? (enter the number 1-7 of the correct statement).

Explanation / Answer

f(2 ) = 14

f(8) = 19

we have two points

(2 , 14) and (8, 19 )

so exponential function can be written as

14 = ab^2

19 = ab^8

dividing the two equations we get

b^6 = 19/14

b = 1.0522

a = 14/ b^2

a = 14/1.107155

a = 12.645

hence , the function is

f(t) = 12.645 ( 1.0522)^t

f^-1 (P )

x / 12.645 = (1.0522)^y

ln ( x / 12.645 ) / ln (1.0522) = y

f^-1 (P) = ln ( P / 12.645 ) / ln (1.0522)

c) f(45) = 124.837

d) f^-1 (45) = 25

e) option 5 is correct

f) option 4 is correct

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