Question

The population of sloths in an area has been increasing by 5% each year since 2000. Let P model the population p(t), in thousands, of sloths t years after the year 2000. The graph of p(t)=1.05t has a general shape that fits the data. Find a scale factor k so that P(t)=kp(t) fits the data.

Answer 1

The scale factor k that makes P(t) = kp(t) fit the data is k = 1.05.

Explanation:

To find the scale factor k that makes P(t) = kp(t) fit the data, we need to compare the given equation p(t) = 1.05t with the desired equation P(t) = kp(t).

The given equation p(t) = 1.05t represents the population of sloths in thousands (p) as a function of time in years (t) since 2000. It assumes that the population increases by 5% each year.

To make P(t) = kp(t) fit the data, we need to find a scale factor k that adjusts the equation to match the given data. Since the population is increasing by 5% each year, the scale factor k should be equal to 1.05.

Therefore, the scale factor k that makes P(t) = kp(t) fit the data is k = 1.05.