SAT Saturday Week 14: Geometry & Trigonometry Prep

A wildlife biologist monitors the population of a specific bird species in a state park. The model below estimates the population, P, of the bird species t years after the initial observation.

P = 4,200(1.045)^t

Which of the following statements is the best interpretation of the number 1.045 in this context?

• A) The bird population is estimated to increase by 4.5% each year.
• B) The bird population is estimated to increase by 1.045% each year.
• C) The initial population of the bird species was 1,045.
• D) The bird population is estimated to increase by 45 birds each year.

Correct Answer: A) The bird population is estimated to increase by 4.5% each year.

The Concept:

The SAT regularly tests the standard exponential model structure:

P = a(b)^t

•  represents the initial value (4,200 birds).
•  represents the growth or decay factor (1.045).

The Breakdown:

Since b = 1.045 is greater than 1, it represents exponential growth. To find the annual percentage growth rate, express the base b as (1 + r):

1.045 = 1 + 0.045

Converting the decimal 0.045 into a percentage gives 4.5%. Thus, the bird population grows by 4.5% each year.

A technician places a sample of a certain radioactive isotope into a secure chamber. The mass of the sample, in grams, decreases exponentially over time. The sample loses half of its mass every 6 hours. If the initial mass of the sample is 320 grams, which equation models the mass, M, in grams, remaining after h hours?

• A) M = 320(0.5)^6h
• B) M = 320(0.5)^h/6
• C) M = 320(2)^h/6
• D) M = 320(0.5)^{h – 6}

Correct Answer: B)

The Concept:

Harder SAT module questions often feature time intervals that do not match a single unit of time (e.g., “half-life every 6 hours” instead of “every 1 hour”).

The Breakdown:

1. Identify the Decay Factor: The sample loses half its mass, meaning 50% (0.5) remains after each interval. This immediately eliminates Option C (which uses a base of 2).
2. Scale the Exponent: The halving occurs once every 6 hours. This means when h = 6, the exponent must equal 1 (0.5¹). When h = 12, the exponent must equal 2 (0.5²).

3. Test the Fraction: The fractional exponent h/6 satisfies this condition. Plugging in h = 6 yields M = 320(0.5¹) = 160 grams, which perfectly models the problem.

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