A caterer is quoting a charge for producing a dinner proposes the following terms.?

For a group of 60 people, he will charge $30 per person. For every extra 10 people, he will lower the price bu $1.50 per person for the whole group. What size group does the caterer want to maximize his income?A caterer is quoting a charge for producing a dinner proposes the following terms.?
Call the number of people n and

Call C(n) the cost per person



Then C(60) = 30 and

C(60+10i) = 30 - 1.5i but I take it it goes in jumps of 10 people (rounded down)

If so we can rewrite that by using the integer floor ?...? operators

C(n) = 30 - 1.5 * ? (n-60)/10 ?

C(n) = 1.5 ( 20 - ? (n/10) - 6 ?

C(n) is a decreasing staircase fn. (At n=300, the food becomes free, and for 310 people or more, they pay you to eat it, sounds good...)



Anyway Total Income I(n) = n*C(n)

I(n) = n* 1.5 ( 20 - ? (n/10) - 6 ? )

I(n) = 1.5n * ( 14 - ? n/10 ? )



Note that C(n) and I(n) are not differentiable, so you can find their extrema empirically or graphically.

(Alternatively it's not hard to see the upper and lower bounding fns for I(n) and C(n), which are differentiable, so you could find the neighborhood of the maximum of I(n) then pinpoint it empirically:

Upper bound: replace ? n/10 ? simply by n/10

Iupper(n) = 1.5n * (14 - n/10)

= 3/20 ( n(140 - n) ) = 3/20 (140n - n2)

Lower bound: replace ? n/10 ? by (n-9)/10



Anyway here's the empirical trial-and-error approach:

Iupper(n) = 3/20 (140n - n2)

Iupper'(n) = 3/20 (140 - 2n)

Iupper'(n) = 0 when 3/20 (140 - 2n) = 0 =%26gt; n =70



So now you heuristically use the precise function I(n) = 1.5n * ( 14 - ? n/10 ? ) in the region of n≈70



[For 70≤n≤79] C(n) = 28.5

I(n) = 1.5n (14 - 7) = 10.5n



[For 80≤n≤89] C(n) = 27.0

I(n) = 1.5n (14 - 8) = 9n



[For 60≤n≤69] C(n) = 30

I(n) = 1.5n (14 - 6) = 12n



Try n=69 =%26gt; I(69) = 12*69 = 828

Try n=79 =%26gt; I(79) = 10.5*79 = 829.5 [MAXIMUM]

Try n=89 =%26gt; I(89) = 9*89 = 801



So the group size which maximizes income is n=79A caterer is quoting a charge for producing a dinner proposes the following terms.?
The more people you get to come, the more he can give you price breaks because he buys in bulk. His motive is not to lower his income, but to have you purchase more totally, which really increases his income. The incentive to get you to invite more people is to lower the per-person price.