how do you find ...weekly sales in thousands of items of a product has a seasonal sales record approximated by y=18.1sin(piex/24)+52.9 where x is time in weeks and 1 is the first week in janurary. during which weeks will the sales equal 61,950 items this year? list all the weeks...
evaluate y for x from 1 to 52 (weeks).
For example, for the first week, x=1,
so
y=18.1*sin(π*1/24)+52.9=55.26
Do it for all 52 weeks, plot it if possible.
Since the maximum over the year is around y=70, I suppose the sales of 61,950 means 61.950 where the "," is used as a decimal point in many countries.
To find the weeks during which the sales equal 61,950 items, we can solve the given equation for x. The equation is:
y = 18.1sin(pi*x/24) + 52.9
We want to find the value of x when y (weekly sales) is equal to 61,950. Rearranging the equation, we have:
61,950 = 18.1sin(pi*x/24) + 52.9
Now, let's solve this equation step-by-step to find the corresponding weeks:
Step 1: Subtract 52.9 from both sides of the equation
61,950 - 52.9 = 18.1sin(pi*x/24)
61,897.1 = 18.1sin(pi*x/24)
Step 2: Divide both sides of the equation by 18.1
(61,897.1) / 18.1 = sin(pi*x/24)
3,418.4 = sin(pi*x/24)
Step 3: Take the inverse sine (sin^(-1)) of both sides of the equation
sin^(-1)(3,418.4) = sin^(-1)(sin(pi*x/24))
x = (24/pi) * sin^(-1)(3,418.4)
Step 4: Calculate the value of x using a calculator
x ≈ 24.293
So, the value of x is approximately 24.293. However, since x represents the time in weeks, we need to round this value to the nearest whole number.
Therefore, the sales will be equal to 61,950 items approximately during week 24.
To find the weeks during which the sales will equal 61,950 items, we need to solve the equation y = 61,950 for x using the given seasonal sales record equation y = 18.1sin(piex/24) + 52.9.
Here is the step-by-step solution:
Step 1: Substitute the given value of y into the equation:
61,950 = 18.1sin(piex/24) + 52.9
Step 2: Rearrange the equation to isolate the sine term:
18.1sin(piex/24) = 61,950 - 52.9
Step 3: Simplify the right-hand side:
18.1sin(piex/24) = 61,897.1
Step 4: Divide both sides by 18.1 to isolate the sine term:
sin(piex/24) = 61,897.1 / 18.1
Step 5: Now, we can take the inverse sine (arcsine) of both sides to solve for x:
piex/24 = arcsin(61,897.1 / 18.1)
Step 6: Multiply both sides by 24/pi to isolate x:
x = ( 24 / pi ) * arcsin(61,897.1 / 18.1)
After solving this equation using a calculator or computer software, you will obtain the value of x.
The solution for x will give you the number of weeks. To determine which weeks correspond to the x-values you found, add 1 to each value since week 1 is the first week in January.
For example, if one value of x is 5, then the corresponding week would be 5 + 1 = week 6.
Repeat this process for each x-value you obtained to find all the weeks during which the sales will equal 61,950 items this year.