Lc math 1 Adv – Writing and Solving Exponential Equations (lt 5)

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LC Math 1 Adv – Writing and Solving Exponential Equations (LT 5)
1. Suppose a school has 2,500 students now and the number of students is decreasing by 2% each year.

Year	Number of Students	Ratio	% Change
0		---	---
1
2
3
4

a) Approximate the school population for the next four years in the table to the right.

b) Is the number of students a function of the years that have passed since 2016? Explain.
c) If this rate continues, write a function that models the number of students at the school after x years.
d) If this rate continues, how many students will the school have 10 years from now?
e) How many years will it be until the school has 1800 students? Explain how you solved this problem.

f) How long will it take for the school’s population to be cut in half? Explain.

2. A person with diabetes requires a dose of 15 units of insulin. Assume that 3% of the insulin is lost from the bloodstream each minute.
a) Write a function to model the amount of insulin remaining in the bloodstream after t minutes.

b) How much insulin remains in the blood after 1 hour?

c) How long will it take for the bloodstream to contain 5 units of insulin? Explain how you solved this problem.
3. The amount of a radioactive substance decreases over time. The half-life of a substance is the amount of time it takes for half of the material to decay. Strontium-90 has a half-life of 29 years. This means that in each 29-year period, one half of the strontium-90 decays and one half remains. Suppose you have 2000 grams of strontium-90.
a) How much strontium will remain after 29 years? Explain without using a mathematical equation.

b) How much strontium will remain after 87 years? Explain without using a mathematical equation.

c) Write a function to model the amount of strontium remaining after t years.
d) How much strontium remains after 75 years?

e) How long will it take until there is 700 grams of Strontium-90 remaining? Explain how you solved this problem.

f) Sarah wrote the equation to model the amount of Strontium-90 remaining after t years. Using test points, critique the validity of her equation.

4. The population of Colorado between 1860 and 2000 is shown in the graph below.


a) Would the data be more appropriately represented by a linear or exponential model? Explain.

b) Use Desmos to find a regression equation that models the population of Colorado t years after 1860. Write the equation in the space below.

c) Explain the meaning of the a and b values in your equation in the context of the scenario.

d) According to the model, what was the population of Colorado in 1960? How accurate is this?

e) Use the model to predict the population of Colorado in 2030. Are there any considerations that should be made when using this model to predict the population in the future?
5. The number of deer in the state of Massachusetts is a problem. In 1998, the deer population was estimated to be about 85,000. The state had to decide whether to allow hunting (a limiting factor) or to ban hunting (no limiting factor). They estimated that the population would increase by about 270 deer per year if hunting was allowed but that it would grow by 15% each year if hunting were banned.
a) Write an equation to model the deer population size for each of the two scenarios.

b) Determine the number of deer estimated to be in the population in 2005 if hunting is allowed and if it is banned.

c) Determine how long it will be until the deer population reaches 100,000 if hunting is banned. Then determine how long it will be if hunting is allowed.

6. Fermium-253 is radioactive, meaning that it decays (or loses mass) over time. The half-life of Fermium-253 (time it takes for half of a sample to remain) is 100 days. If the initial amount present is 600 grams,
a) Write an equation to model the amount of Fermium-253 present t days after it is initially measured.
b) Determine the amount of Fermium-253 present after 60 days. Explain.
c) Determine the amount of time it will take for 500 grams of the substance to decay. Explain.