You have been hired as a marketing consultant to Johannesburg Burger Supply, Inc., and you wish to come up with a unit price for its hamburgers in order to maximize its weekly revenue. To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form q = mp + b, where p is the price per hamburger, q is the demand in weekly sales, and m and b are certain constants you must determine.

(a) Your market studies reveal the following sales figures: When the price is set at $2.00 per hamburger, the sales amount to 7000 per week, but when the price is set at $4.00 per hamburger, the sales drop to zero. Use these data to calculate the demand equation.

q =

(b) Now estimate the unit price that maximizes weekly revenue.
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Predict what the weekly revenue will be at that price.
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(a) Your market studies reveal the following sales figures: When the price is set at $2.00 per hamburger, the sales amount to 7000 per week, but when the price is set at $4.00 per hamburger, the sales drop to zero. Use these data to calculate the demand equation.

If we assume that the demand equation has the linear form

q = mp + b

where p is the price per hamburger, q is the demand in weekly sales then we have the following 2 linear equations:

for price p=2

(1) 7000 = 2m +b

for price p=4

(2) 0 = 4m +b

Multiplying equation (1) by -1 and adding it to equation (2)

-7000 + 0 = -2m + 4m -b + b

and

-7000 = 2m, hence m =- 7000/2 = -3500.

Substitute this value of m in (2) to obtain

0 = 4(-3500) + b, hence b = 14000

and the formula for the demand equation is

q = -3500p + 14000

(b) Now estimate the unit price that maximizes weekly revenue.

Let R(p) the function which gives the revenue. Then

[tex] \bf R(p)=qp=(-3500p +14000)p=-3500p^2+14000p[/tex]

Taking the derivative with respect to p, we get

R'(p) = -7500p + 14000

and R'(p) = 0 when p = 14000/7500 = 1.8666

Since R''(1.8666) < 0 then p=1.8666 is a maximum.

Predict what the weekly revenue will be at that price.

The maximum revenue could be estimated as

[tex] \bf R(1.8666)=-3500(1.8666)^2+14000(1.8666)\approx \$13937.78[/tex]

francocanacari francocanacari · Answer 2 · 2022-03-24T14:10:46+01:00

The unit price that maximizes weekly revenue is $2, and the weekly revenue at that price will be $14,000.

Cost calculation

Given that your market studies reveal that when the price is set at $2.00 per hamburger, the sales amount to 7000 per week, but when the price is set at $4.00 per hamburger, the sales drop to zero, to estimate the unit price that maximizes weekly revenue and predict what the weekly revenue will be at that price, the following calculations must be made:

4 = 0
2 = 7000
2 / 7000 = X
0.000285 = X
2,285 x 6,000 = 13,710
2 x 7,000 = 14,000
1,715 x 8,000 = 13,720

Therefore, the unit price that maximizes weekly revenue is $2, and the weekly revenue at that price will be $14,000.

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