Question

A food truck caters an event attended by 100 guests. Every guest orders one of two possible dishes: a salad or a turkey plate. The price of each meal decreases as more of that particular type are ordered. The price of a salad is $ 10.00 minus $ 0.04 for each salad ordered. The price of a turkey plate is $ 12.00 minus $ 0.02 multiplied by the square of the number of turkey plates ordered. Guests pay for their meal only after everyone has placed their order. Using differentiation, find the maximum revenue for the food truck. Remember that the number of meals is a positive integer. Round revenue to the nearest cent.

Answer 1

To find the maximum revenue, differentiate the revenue functions of salads and turkey plates with respect to the number of salads sold, considering a total of 100 guests, find critical points, and use the second derivative test or sign changes to identify the maximum revenue. Ensure that the number of meals are integers.

Step-by-step explanation:

To determine the maximum revenue for the food truck, we need to derive the revenue functions for salad and turkey plates and then find the total revenue function. Assuming s salads and t turkey plates are sold, the price functions are Ps(s) = 10 - 0.04s for salads and Pt(t) = 12 - 0.02t2 for turkey plates. The revenue functions would be Rs(s) = s × Ps(s) and Rt(t) = t × Pt(t). Because there are 100 guests, s + t = 100, hence t = 100 - s. We substitute t in Rt and add Rs and Rt for the total revenue R(s). To find the maximum revenue, we differentiate R(s) with respect to s, find critical points, and check these for the maximum value using the second derivative test or analyzing the sign changes of R'(s). Remember to check if the critical points result in s and t being positive integers, as per the conditions given.

Answer 2

Max revenue: R = $679.73

Step-by-step explanation:

total people = 100

each person orders 1 of 2 dishes

salad price = $10 - 0.04x

turkey price = $12 - 0.02*y^2

so x + y = 100

s = 10 - 0.04x

t = 12 - 0.02*y^2

Revenue = s*x + t*y

Revenue = (10 - 0.04x)*x + (12 - 0.02y^2)*y

y = 100 - x

so

Revenue = (10 - 0.04x)*x + (12 - 0.02*(100 - x)^2 )*(100 - x)

R =

R = (10 - 0.04x)*x + (12 - 0.02*(100 - x)^2 )*(100 - x)

R = 10x - 0.04x*x + (12 - 0.02*(10000 - 200x + xx) )*(100 - x)

R = 10x - 0.04x*x + (12 - 200 + 4x -0.02 xx )*(100 - x)

R = 10x - 0.04x*x + (-188 + 4x -0.02 xx )*(100 - x)

R = 10x - 0.04x*x + (-188 + 4x -0.02 xx )*100 -x (-188 + 4x -0.02 xx )

R = 10x - 0.04x*x + -18800 + 400x -2 xx -x (-188 + 4x -0.02 xx )

R = 10x - 0.04x*x + -18800 + 400x -2 xx + 188x - 4xx +0.02 xxx

R = 10x - 0.04x*x + -18800 + 588x -6 xx + 0.02 xxx

R = -18800 + 598x -6.04 xx + 0.02 xxx

dR/dx = 598 - 12.08x + 0.06 x^2

set = 0

598 - 12.08x + 0.06xx = 0

299 - 6.04x + 0.03xx = 0

x = -(-6.04)/(2*0.03) + root((-6.04)^2 - 4*0.03*299) / 2*0.03

x = 100.6667 - root(36.4816 - 35.88) / 0.06

x = 100.6667 - 12.927

x = 87.739

so that is where you get the maximum revenue, when you sell 87.7 salad plates and 12.2605 turkey dishes

Revenue = (10 - 0.04*87.739)*87.739 + (12 - 0.02(12.2605)^2)*12.2605

Revenue = 569.464715 + 110.266

R = $679.7307

R = $679.73