Exponential Models Test Answer Key with Scoring – Algebra 2

Question 1 Reading a model (2 points)

a) Initial value 640; factor 0.92, so a decrease of 8% per year.

b) 640(0.92)⁴ ≈ 458.49.

Scoring: 1 point each part.

Question 2 Compound interest (3 points)

n = 4 and nt = 36: A = 3,200(1 + 0.038/4)³⁶ = 3,200(1.0095)³⁶.

A ≈ $4,497.57

Scoring: 1 point for the formula, 1 point for substituting n, t, and r correctly, 1 point for the value.

Question 3 Continuous compounding (3 points)

a) A = 5,000e^{0.04 · 12} = 5,000e^0.48 ≈ $8,080.37.

b) Annual: 5,000(1.04)¹² ≈ $8,005.16. The difference is $8,080.37 − $8,005.16 ≈ $75.21.

Scoring: 1 point for the formula, 1 for the value; 1 point for the difference.

Question 4 Half-life (3 points)

a) N(t) = 240(1/2)^t/8.

b) 24 hours is 3 half-lives: 240/8 = 30 grams.

c) 240(1/2)^0.5 = 240/√2 ≈ 169.7 grams.

Scoring: 1 point per part.

Question 5 Doubling followers (3 points)

a) F(t) = 1,800 · 2^t/4.

b) 12 days is 3 doublings: 1,800 · 8 = 14,400.

c) 1,800 · 2^0.5 ≈ 2,546.

Scoring: 1 point per part.

Question 6 Interpreting parameters (2 points)

28,000 is the initial value. The factor 0.85 means the value is multiplied by 0.85 each year, a decrease of 15% per year.

V(2) = 28,000(0.7225) = $20,230.00.

Scoring: 1 point for the meanings, 1 point for the value.

Question 7 Model from a table (2 points)

Ratios: 275/250 = 1.1, 302.50/275 = 1.1, 332.75/302.50 = 1.1. The ratio is constant, so the growth is exponential.

10% per year; B(t) = 250(1.1)^t

Scoring: 1 point for the constant ratio and percent, 1 point for the model.

Question 8 Decay from a graph (2 points)

a) a = 100 and b = 80/100 = 0.8: y = 100(0.8)^t. Check (2, 64): 100(0.64) = 64 ✓.

b) 100(0.8)³ = 51.20.

Scoring: 1 point each part.