Math  /  Algebra

QuestionOver the past several years, the owner of a boutique on Aspen Avenue has observed a pattern in the amount of revenue for the store. The revenue reaches a maximum of about $56000\$ 56000 in January and a minimum of about $25000\$ 25000 in July. Suppose the months are numbered 1 through 12 , and write a function of the form f(x)=Asin(B[xC])+Df(x)=A \sin (B[x-C])+D that models the boutique's revenue during the year, where xx corresponds to the month. f(x)=f(x)= \square

Studdy Solution

STEP 1

1. The revenue pattern is sinusoidal, following a sine function.
2. The maximum revenue is \$56,000 in January (month 1).
3. The minimum revenue is \$25,000 in July (month 7).
4. The function is of the form \( f(x) = A \sin(B[x - C]) + D \).

STEP 2

1. Determine the amplitude A A .
2. Determine the vertical shift D D .
3. Determine the period and calculate B B .
4. Determine the phase shift C C .
5. Write the final function.

STEP 3

The amplitude A A is half the difference between the maximum and minimum revenue values.
A=56000250002=310002=15500 A = \frac{56000 - 25000}{2} = \frac{31000}{2} = 15500

STEP 4

The vertical shift D D is the average of the maximum and minimum revenue values.
D=56000+250002=810002=40500 D = \frac{56000 + 25000}{2} = \frac{81000}{2} = 40500

STEP 5

The period of the sine function is 12 months (one year). The formula for the period is 2πB=12 \frac{2\pi}{B} = 12 .
Solving for B B :
B=2π12=π6 B = \frac{2\pi}{12} = \frac{\pi}{6}

STEP 6

The phase shift C C is determined by the maximum point. Since the maximum occurs at x=1 x = 1 , and a sine function normally reaches its maximum at π2 \frac{\pi}{2} , we solve for C C in the equation:
B(xC)=π2 B(x - C) = \frac{\pi}{2}
Substitute B=π6 B = \frac{\pi}{6} and x=1 x = 1 :
π6(1C)=π2 \frac{\pi}{6}(1 - C) = \frac{\pi}{2}
1C=3 1 - C = 3
C=2 C = -2

STEP 7

Write the final function using the values for A A , B B , C C , and D D :
f(x)=15500sin(π6(x+2))+40500 f(x) = 15500 \sin\left(\frac{\pi}{6}(x + 2)\right) + 40500
The function that models the boutique's revenue is:
f(x)=15500sin(π6(x+2))+40500 f(x) = 15500 \sin\left(\frac{\pi}{6}(x + 2)\right) + 40500

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