A company's sales are seasonal with the peak in mid-December and the lowest point in mid-June. The company makes $110000 in sales in December, and only $30000 in June.
A) Find a trigonometric function, s=f(t) , representing sales at time t months after mid-January.
40000 +cos(pi/6 +pi /6)+ 70000
B) What would you expect the sales to be for mid-April?
Expected April sales are $??. Round your answer to the nearest whole number.
C) Find the t -values for which s= 70000.
t = 14 ?
t = 2?
A) To find a trigonometric function representing sales at time t months after mid-January, we can consider the cyclic nature of the sales pattern. Let's assume that the peak of sales occurs at t = 0 (mid-December) and the lowest point occurs at t = 6 (mid-June). We can use the cosine function to model this pattern.
s(t) = A + B*cos(C*t + D)
Since sales in mid-December are $110,000, we can set A = $110,000.
Since sales in mid-June are $30,000, we can set A + B = $30,000.
Substituting these values into the equation:
s(t) = 110000 + B*cos(C*t + D)
B) To find the sales for mid-April (t = 3), we need to solve for B.
30000 = 110000 + B*cos(C*3 + D)
Simplifying the equation and solving for B:
B = -80000/cos(3C + D)
We do not have enough information to determine the exact value of B, so we cannot calculate the sales for mid-April.
C) To find the t-values for which s = 70,000:
70000 = 110000 + B*cos(C*t + D)
Simplifying the equation and solving for t:
cos(C*t + D) = (70000 - 110000)/B
We do not have enough information to solve for C, B, or D, so we cannot determine the values of t when s = 70,000.
To find a trigonometric function representing the sales at time t months after mid-January, we can use a cosine function.
A) Let's assume that t=0 corresponds to mid-January, t=6 corresponds to mid-July, and t=12 corresponds to mid-January of the following year.
Since the peak sales occur in mid-December and the lowest point in mid-June, we can shift the cosine function by 6 months to the right. This means the peak sales will occur at t=6 and the lowest point will occur at t=0.
Let's use the cosine function:
s = A + B*cos(C*t + D)
where A is the average value (estimated as $70,000), B is the amplitude (half the difference between peak and lowest sales, estimated as (110,000-30,000)/2 = 40,000), C is the period (equal to 2π divided by 12 months, estimated as 2π/12 = π/6), and D is the phase shift (equal to π/2 since the function is shifted to the right by 6 months).
So, the trigonometric function representing the sales is:
s = 40,000 + 40,000*cos(π/6*t + π/2)
B) To find the expected sales for mid-April (t=3), we substitute t=3 into the function:
s = 40,000 + 40,000*cos(π/6*3 + π/2)
s = 40,000 + 40,000*cos(π/2 + π/2)
s = 40,000 + 40,000*cos(π)
s = 40,000 - 40,000
s = 0
Therefore, we would expect the sales to be $0 for mid-April.
C) To find the t-values for which s=70,000, we set the equation equal to 70,000 and solve for t:
70,000 = 40,000 + 40,000*cos(π/6*t + π/2)
Let's rearrange the equation:
cos(π/6*t + π/2) = (70,000 - 40,000)/40,000
cos(π/6*t + π/2) = 3/4
Now, we want to find the values of t that satisfy this equation. One solution can be found by taking the inverse cosine of both sides:
π/6*t + π/2 = ±arccos(3/4)
Solving for t:
π/6*t = ±arccos(3/4) - π/2
t = [±arccos(3/4) - π/2] / (π/6)
Using a calculator, we find:
t ≈ 14.62 or t ≈ 1.38
So, the t-values for which s=70,000 are approximately t = 14.62 and t = 1.38.
A) To find a trigonometric function representing sales at time t months after mid-January, we can use a sinusoidal function.
Given that the peak sales of $110,000 occurs in mid-December, which is 11 months after mid-January, and the lowest sales of $30,000 occurs in mid-June, which is 5 months after mid-January, we can form an equation for sales in terms of time (t) as follows:
s = A + B * sin(2π/P * t + C)
where A is the average sales value, B is the amplitude (half the difference between peak and lowest sales), P is the period of the function (in this case, 12 months), and C is the phase shift (the amount of time the function is shifted to the right).
Given that the average sales A is (peak sales + lowest sales) / 2 = ($110,000 + $30,000) / 2 = $70,000, the amplitude B is (peak sales - lowest sales) / 2 = ($110,000 - $30,000) / 2 = $40,000, and the phase shift C is -π/2 (since mid-January corresponds to t = 0), we can now form our equation:
s = 70,000 + 40,000 * sin(2π/12 * t - π/2)
B) To find the expected sales for mid-April, we substitute t = 3 (since mid-April is 3 months after mid-January) into the equation:
s = 70,000 + 40,000 * sin(2π/12 * 3 - π/2)
Evaluating this expression will give us the expected sales value for mid-April.
C) To find the t-values for which s = 70,000, we can set the equation equal to 70,000 and solve for t:
70,000 = 70,000 + 40,000 * sin(2π/12 * t - π/2)
This simplifies to:
sin(2π/12 * t - π/2) = 0
Now, we solve for t. The angle whose sine is 0 is 0 and π, so we have:
2π/12 * t - π/2 = 0 or 2π
Solving these equations for t will give us the t-values for which s = 70,000.
t = 0 + π/2 or π/6 (for 0)
t = 2π + π/2 or π/6 (for 2π)