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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.

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mysticchacha

Answer:

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

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