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1. The approximate time for an investment to double can be found using the function [tex]n(r) = \frac{72}{r}[/tex], where \(n\) represents the number of years and \(r\) represents the annual interest rate as a percent.
a. How long will it take the investment to double at a 6% interest rate?
b. Find the domain and range.

2. The point \(Q(-2, -5)\) lies on the terminal arm of an angle \(\theta\) in standard position. Determine the values of \(\sin \theta\) and \(\cos \theta\).

3. Emily hits a nail with her hammer, and the nail goes 30 mm into the board. On each successive hit, the nail goes in 40% as far as it did on the previous hit.
a. How far will the nail go in on the sixth hit?
b. After the sixth hit, what is the total distance (to the nearest mm) that the nail has entered into the board?

4. The population of a type of small fish in a particular swamp in Ontario follows a yearly cycle. This cycle can be modeled as a sinusoidal function. The maximum population of 300 fish occurs at the beginning of October. The minimum population of 60 fish occurs 6 months later, at the beginning of April. Determine the equation of this function relating the population, \(P\), of fish over time, \(t\), in months. (Where \(t = 0\) represents October, \(t = 1\) represents November, and so on.)

Answer :

a. On the sixth hit, the nail will go in 12 mm into the board. b. After the sixth hit, distance the nail entered board is 56.718 mm. The equation of the population of fish over time in months is P = 120 * sin((2π/6) * t) + 180.

Question 1:

a. To find how long it will take for an investment to double at a 6% interest rate, we can use the formula n(r) = 72/r. Plugging in r = 6, we get n = 72/6 = 12 years.

b. The domain of the function n(r) is all positive values of r, since interest rates cannot be negative.

The range of the function is all positive values of n, since the number of years it takes for an investment to double can only be positive.

Question 2:

The point Q (-2, -5) lies on the terminal arm of an angle θ in standard position.

To determine the values of sinθ and cosθ, we can use the coordinates of Q.

Since sinθ = y/r and cosθ = x/r, where r is the distance from the origin to Q, we can calculate sinθ = -5/√( (-2)^2 + (-5)^2 )

≈ -5/5

= -1 and

cosθ = -2/√( (-2)^2 + (-5)^2 )

≈ -2/5.

Question 3:

a. To find how far the nail will go in on the sixth hit, we can multiply the distance of the previous hit by 40%, or 0.4.

If the nail went 30 mm into the board on the previous hit, it will go in 30 * 0.4 = 12 mm on the sixth hit.

b. After the sixth hit, the total distance the nail has entered into the board can be calculated by summing up the distances of all the hits.

The distances form a geometric sequence, with a common ratio of 0.4 and a first term of 30 mm.

So, the total distance is 30 * (1 - 0.4^6) / (1 - 0.4) ≈ 56.718 mm.

Question 4:

The population of the fish in the swamp follows a sinusoidal function.

The maximum population occurs at the beginning of October, which is represented by t = 0.

The minimum population occurs 6 months later, at the beginning of April, represented by t = 6.

We can use these two points to determine the equation of the function.

The amplitude of the function is (300 - 60) / 2 = 120, and the period is 6, so the equation is

P = 120 * sin((2π/6) * t) + 180.

Learn more about geometric sequence here:

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