KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 03 · Lesson 02

The Number e and Continuous Change

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

The Number e and Continuous Change

U03 · L02
1

A quantity begins at 500 and grows 4% yearly. Write the model.

Exponential visualModel multiplicative change with exponential functions and interpret growth factors, percent rates, doubling time, half-life, and continuous change. xy
Answer: A=500(1.04)^t
Reasoning: Growth factor is 1+0.04.
2

Evaluate 200(0.8)^3.

Exponential visualModel multiplicative change with exponential functions and interpret growth factors, percent rates, doubling time, half-life, and continuous change. xy
Answer: 102.4
Reasoning: Multiply 200 by 0.512.
3

A population doubles every 6 years. Give a base-2 model.

Exponential visualModel multiplicative change with exponential functions and interpret growth factors, percent rates, doubling time, half-life, and continuous change. xy
Answer: P=P₀·2^{t/6}
Reasoning: The exponent counts six-year doubling intervals.
4

Solve 3^x=81.

Exponential visualModel multiplicative change with exponential functions and interpret growth factors, percent rates, doubling time, half-life, and continuous change. xy
Answer: x=4
Reasoning: 81=3⁴.
5

Find the continuous growth rate k if A=A₀e^{0.07t}.

Exponential visualModel multiplicative change with exponential functions and interpret growth factors, percent rates, doubling time, half-life, and continuous change. xy
Answer: 7% per time unit
Reasoning: k=0.07 corresponds to a continuous rate of 7%.
6

Find the half-life for A=A₀e^{-0.2t}.

Exponential visualModel multiplicative change with exponential functions and interpret growth factors, percent rates, doubling time, half-life, and continuous change. xy
Answer: ln2/0.2≈3.47
Reasoning: Set e^{-0.2t}=1/2 and solve.
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