Question

The blue whale is the largest creature in the world and has been found in every ocean in the world. A marine biologist surveyed the blue whale population in Monterey Bay, off he coast of California, every three years between 1995 and 2010. The figure shows her results. If $w$ is the number of blue whales present in Monterey Bay and $t$ is the number of years since the study began in 1995, which of the following equations best represents the blue whale population of Monterey Bay?

• A. $w=100+2t$
• B. $w=100+\frac{t^2}{4}$
• C. $w=100\times 2^t$
• D. $w=100\times 2^{\frac{t}{4}}$

Answer 1

Answer: (D)

$w=100\times 2^{\frac{t}{4}}$

According to the graph, the population of the whales grew slowly at first and then more quickly. This means that an exponential model is probably the best fit, so you can eliminate A (linear) and B (quadratic). The remaining equations are both exponential, so choose a data point and see which equation is the closest fit. Be careful-the vertical axis represents hundreds of whales, and the question states that t represents the number of years since the study began, so t = $0$ for $1995$, t = $3$ for $1998$, and so on.
Using the data for $1995$, which is the point $(0,100)$, the results are the same for both equations, so choose a different point. Using the data for $2007$, t = $2007-1995$ = $12$, and the number of whales was $800$. Substitute these values into C and D to see which one is true.
Choice C is not true because $800$ is equal to $100\times 2^{12}$.
Choice (D) is correct because
$800=100\times 2^{\frac{12}{4}}$ = $100\times 2^3$ = $100\times 8$.