Questions: The profit of a dog toy company x years after 2006 can be modeled as p(x) = 4.222x^2 - 1.829x - 8.969 hundred dollars. Calculator Checkpoint: p(5) = 87.436 a. Numerically estimate the rate of change of the profit in 2008. (Round your answer to three decimal places.) hundred dollars per year b. What is the percentage rate of change of profit in 2008? (Round your answer to three decimal places.) % per year c. Is business getting better or worse for the dog toy company in 2008? In 2008, the profit is and the profit is so business is

Solution Steps

a. To numerically estimate the rate of change of the profit in 2008, we need to find the derivative of the profit function \( p(x) \) and evaluate it at \( x = 2 \) (since 2008 is 2 years after 2006).

b. To find the percentage rate of change of profit in 2008, we calculate the rate of change from part (a) and divide it by the profit in 2008, then multiply by 100 to convert it to a percentage.

c. To determine if the business is getting better or worse, we compare the profit in 2008 with the rate of change. If the rate of change is positive, the business is getting better; if negative, it's getting worse.

To find the rate of change of profit in 2008, we evaluate the derivative of the profit function \( p(x) = 4.222x^2 - 1.829x - 8.969 \) at \( x = 2 \): \[ p'(x) = 8.444x - 1.829 \] Substituting \( x = 2 \): \[ p'(2) = 8.444(2) - 1.829 = 15.059 \]

Next, we calculate the profit in 2008 by evaluating the profit function at \( x = 2 \): \[ p(2) = 4.222(2^2) - 1.829(2) - 8.969 = 4.261 \]

The percentage rate of change of profit is given by: \[ \text{Percentage Rate of Change} = \left( \frac{p'(2)}{p(2)} \right) \times 100 \] Substituting the values: \[ \text{Percentage Rate of Change} = \left( \frac{15.059}{4.261} \right) \times 100 \approx 353.4147 \]

Since the rate of change of profit \( p'(2) = 15.059 \) is positive, we conclude that the business is getting better.

Final Answer