Question

There are 400 rice weevils at the beginning of an insect study. The population is expected to grow at a rate of 150% each week.




a.) Write and simplify a formula that can be used to predict the rice weevil population for any week after the beginning of the study.



b.) Use the formula to predict the rice weevil population 10 weeks after the beginning of the study. Show your work.



c.) Use the formula to predict when the rice weevil population will reach 1,000,000,000. Show your work.

Answer 1

Answer:

Part a) $y=400(2.5)^{x}$

Part b) $3,814,697\ rice\ weevil\ population$

Part c)  $16.1\ weeks$

Step-by-step explanation:

Part a) Write and simplify a formula that can be used to predict the rice weevil population for any week after the beginning of the study

we know that

100%+150%=250%=250/100=2.5

In this problem we have an exponential function of the form

$y=a(b)^{x}$

where

a is the initial value

b is the base

y ----> is the rice weevil  population

x ----> the number of weeks

In this problem

a=400 rice weevils

b=2.5

substitute

$y=400(2.5)^{x}$

Part b) Use the formula to predict the rice weevil population 10 weeks after the beginning of the study. Show your work

we have

$y=400(2.5)^{x}$

so

For x=10 weeks

substitute

$y=400(2.5)^{10}=3,814,697\ rice\ weevil\ population$

Part c) Use the formula to predict when the rice weevil population will reach 1,000,000,000.

we have

$y=400(2.5)^{x}$

so

For y=1,000,000,000 rice weevil  population

substitute in the formula and solve for x

$1,000,000,000=400(2.5)^{x}$

Simplify

$2,500,000=(2.5)^{x}$

Apply log both sides

$log(2,500,000)=x*log(2.5)$

$x=log(2,500,000)/log(2.5)=16.1\ weeks$

Answer 2

A function assigns the values. The number of weeks it will take for the population of 400 to be 1,000,000,000 is 16.1 weeks.

What is a Function?

A function assigns the value of each element of one set to the other specific element of another set.

A.) Given there are 400  rice weevils at the beginning. And the population is expected to grow at a rate of 150% each week. Therefore, the function that can represent the population can be written as,

Population after n weeks = 400×(1+1.5)ⁿ = 400×(2.5)ⁿ

where n is the number of weeks.

B.) The population of the rice weevils after 10 weeks will be,

Population after 10 weeks = 400×(2.5)¹⁰ = 3,814,697.266 ≈  3,814,697

C.) The time needed for the population to reach 1,000,000,000 will be,

1,000,000,000 = 400(2.5)ⁿ

2,500,000 = 2.5ⁿ

n = 16.0776 ≈ 16.1 weeks

Hence, the number of weeks it will take for the population of 400 to be 1,000,000,000 is 16.1 weeks.

Learn more about Function:

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