A resturant offers a buffet dinner at group prices.It costs $10 for one person,$20 for two,$29 for three, $37 for four,$44 for five and so on.
A)how much does a buffet dinner for 8 cost?how much does a group of eight save if it's members eat together rather than alone.
B)the buffet costs the restaurant $6 per person.
How large a group can the restaurant serve without losing money?
P.S I came up with A.
Please and thank-you
Ok we have some kind of sequence that goes
10,20,29,37,44 for the first 5 terms.
It looks like the n-th term is given by
(1) n*10 -((n-2)(n-1))/2
At least this would be one expression to generate the sequence. I don't know if your text/teacher made you aware that through any finite set of data points there are an infinite number of functions. The expression (1) I gave would probably be the most 'natural' one we could use. To answer part A) put in 8 for n in (1) and calculate.
For part B) we know that the amount of revenue per group is given by (1). The cost is 6/person, thus use expression (1) - 6n = net profit. Since we want this greater than 0 set (1) - 6n => 0 and solve for n. Use the last n for which it is positive.
Hint: make a table and calculate n for the first dozen terms or so. I did not get twelve, so don't think this a short-cut.
A) To find out how much a buffet dinner for 8 people costs, we can use the expression (1) given: n*10 - ((n-2)(n-1))/2.
Substituting n = 8 into the expression, we get:
8*10 - ((8-2)(8-1))/2
= 80 - (6*7)/2
= 80 - 21
= $59
Therefore, a buffet dinner for 8 people would cost $59.
To calculate the savings if the members eat together rather than alone, we need to find the difference between the total cost if they eat together and the total cost if they eat individually.
If each person eats alone, the total cost would be 8*$10 = $80.
But if they eat together, the total cost is $59.
So, the group of 8 people would save $80 - $59 = $21 by eating together.
B) The cost for the restaurant per person is given as $6.
To find out how large a group the restaurant can serve without losing money, we need to determine the net profit. The net profit is calculated as the revenue (given by expression (1)) minus the cost per person ($6n).
Setting (1) - 6n greater than 0, we have:
n*10 - ((n-2)(n-1))/2 - 6n > 0
Now, we need to solve this inequality to find the largest value of n for which it holds true.
One way to approach this is by creating a table and calculating the net profit for various values of n. Start with small values of n and increment it gradually until the net profit becomes negative. The last value of n for which the net profit is positive is the maximum group size the restaurant can serve without losing money.
Note: Unfortunately, without having the values of the additional terms in the sequence, we cannot determine the exact group size the restaurant can serve without losing money.