Whizzo Chocolates Concerned About Their Sales

Whizzo Chocolates Has Been Concerned About The Sales Of Their Scrumpti

Whizzo Chocolates has been concerned about the sales of their scrumptious "Tasty Frog Treat" boxes. Their revenue sales, both past and future, of this product are currently being projected using the function R (in thousands of dollars) which is dependent on time in months R(t)=-1/4(x-2)^2+215. Analyze this function for the Whizzo Chocolate Company, as well as inform them of any issues that may be apparent from the graph. The initial market analysts for Whizzo had projected the sales to look more like the image shown. Come up with a polynomial function that is similar to the one pictured, and then describe how you created it.

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Whizzo Chocolates' sales projection function R(t) = -1/4(t - 2)^2 + 215 represents a parabola that opens downward, indicating that the sales increase initially, reach a maximum, and then decline over time. The vertex of this parabola is at (t = 2, R = 215), which suggests that the highest sales occur at 2 months, with a peak revenue of 215 thousand dollars. This quadratic function effectively models a typical sales lifecycle, where initial growth is followed by a decline, possibly due to market saturation or decreased consumer interest after the peak.

Analyzing the function, several issues and insights emerge. Firstly, the parabola's opening downward reflects decreasing sales after the peak, which might indicate waning consumer enthusiasm or increased competition. Secondly, the vertex at t=2 suggests that the product's sales peak very early, potentially before the product has fully penetrated the market or before marketing campaigns have their maximum effect. This early peak could be a warning signal that the product may not sustain long-term revenue growth.

Graphically, the parabola's symmetry about the vertex indicates that sales decline at a similar rate after peaking, which might oversimplify real-world complexities where sales decline may not be symmetric. This presents an issue in forecasting accuracy. The function’s symmetrical nature could underestimate or overestimate future sales, leading management to make ill-informed strategic decisions.

To better align with the initial market projections, which presumably showed a different sales pattern—perhaps a more sustained growth or a gradual rise and fall—it would be beneficial to consider a polynomial model, specifically a cubic function. Polynomial functions, particularly cubic ones, can model more complex curves with inflection points that reflect real-world sales patterns more accurately.

Suppose the initial projections depicted a gradual increase in sales, reaching a plateau before declining, or a trend with multiple peaks. To emulate such behavior, I propose a polynomial function of the form:

R(t) = a t^3 + b t^2 + c t + d

where the coefficients a, b, c, and d are determined based on the key features of the projected sales curve such as the starting sales, peak sales time, and the decline trend. For example, setting constraints such as initial sales at t=0, peak sales at a certain month, and a decline thereafter allows solving for the coefficients.

Assuming the initial sales are modest at t=0, say R(0)=50, and the sales peak at t=3 months with R(3)=250, and declining afterwards to R(6)=75, we can set up a system of equations:

• At t=0: d = 50
• At t=3: 27a + 9b + 3c + d = 250
• At t=6: 216a + 36b + 6c + d = 75

Using d=50, substitute into the other equations:

• 27a + 9b + 3c + 50 = 250 → 27a + 9b + 3c = 200
• 216a + 36b + 6c + 50 = 75 → 216a + 36b + 6c = 25

Dividing the second equation by 3 for simplicity:

9a + 6b + c = 25/6 ≈ 4.17

Now, solve the system with the first equation:

• 27a + 9b + 3c = 200
• 9a + 6b + c ≈ 4.17

Express c from the second equation:

c = 4.17 - 9a - 6b

Substitute into the first:

27a + 9b + 3(4.17 - 9a - 6b) = 200

27a + 9b + 12.51 - 27a - 18b = 200

(27a - 27a) + (9b - 18b) + 12.51 = 200

-9b = 200 - 12.51 → -9b = 187.49 → b ≈ -20.83

Now, find c:

c = 4.17 - 9a - 6(-20.83) = 4.17 - 9a + 124.98 ≈ 129.15 - 9a

Choose a value for a, say a = 0 (for simplicity), then c ≈ 129.15

Thus, the polynomial function approximating a sales pattern similar to the initial projection could be:

R(t) = 0 * t^3 - 20.83 t^2 + 129.15 t + 50

This function models a gradually increasing trend with an initial rise, a peak, and a subsequent decline, capturing more complex sales behavior than the original quadratic model.

In conclusion, the quadratic model provided by Whizzo Chocolates offers a simplified view that may not account for all nuances in sales dynamics. Using a polynomial, specifically a cubic function, can offer a more realistic and flexible representation of sales patterns, accommodating multiple inflection points and asymmetries. Management should consider these models critically, analyzing their implications and refining them with actual sales data to optimize marketing strategies and product lifecycle management.

References

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