An airline sells all the tickets for a certain route at the same price. If it charges 200 dollars per ticket it sells 10,000 tickets. For every 15 dollars the ticket price is reduced, an extra thousand tickets are sold. This if the tickets are sold for 185 dollars each then 11,000 tickets sell. It costs the airline 100 dollars to fly a person. (a) Express the total profit P in terms of the number n of tickets sold. (b) Express the total profit P in terms of the price p of one ticket.

The function that gives the amount of profit at a given price and number

given are obtained from the relationship between the variables.

a) The profit in terms of the number of tickets sold, P = -0.015·n² + 250·n

(b) The profit in terms of ticket price, is P = -66.·p² + 3000·p + 2,333,333.

Reasons:

Number of tickets sold at x = $200, is y = 10,000 tickets

Number of extra tickets sold for every $15 price reduction = 1,000 tickets

Number of tickets sold at $185 = 11,000

The cost to fly a person = $100

(a) The rate of change, , of the number of ticket sold wit price is given as

follows;

The equation in slope and intercept form is therefore;

n - 11,000 = -66.·(p - 185)

n = -66.·p + 12,333. + 11,000 =  -66.·p + 23,333.

Number of tickets sold, n = -66.·p + 23,333.

The revenue, R = n·x

Therefore;

The profit = Revenue - Cost

The cost = 100·n

(b) The total profit, P, in terms of the price p, for one ticket

Number of tickets sold, n = -66.·p + 23,333.

The revenue, R = n·p

∴ R = -66.·p² + 23,333.·p

Cost = 100·n

∴ Cost = 100·(-66.·p + 23,333.)

Therefore, the profit is given as follows;

Profit = -66.·p² + 23,333.·p - (100·(-66.·p + 23,333.))

Therefore;

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We want to express the total profit in two different ways.

The answers are:

a) P(n) = n*[n - 23,333.3]*(-15/1,000) - n*100

b) P(p) =  n(p)*p - n(p)*100 = [ (-1,000/15)*p + 23,333.3]*(p - 100)

The given information is:

a) We want to get the profit in terms of the number n of tickets sold.

The number of tickets sold depends on the price, we know that it is a linear relation like:

n(p) = a*p + b

where p is the price.

we know that:

n(200) = 10,000

n(185) = 11,000

If this is a linear relation, the slope is known:

a = -1,000/15$

Then the line is something like:

n(p) =  (-1,000/15)*p + b

We know that if k = 200, then n = 10,000

Replacing that we have:

n(200) = 10,000 =  (-1,000/15)*200 + b

               10,000 + (1,000/15)*200 = b = 23,333.3

Then the number of tickets sold as a function of the price is:

n(p) =  (-1,000/15)*p + 23,333.3

Then the revenue is given by the number of tickets sold times the price, this is:

r(n) = n*p

We can write k in terms of n as:

k = [n - 23,333.3]*(-15/1,000)

Then the revenue as a function of the number of tickets sold is:

r(n) = n*[n - 23,333.3]*(-15/1,000)

The profit is the difference between the revenue and the cost, then the profit function is:

P(n) = n*[n - 23,333.3]*(-15/1,000) - n*100

Where n*100 is the cost.

b) now we want to write the same equation, but this time as a function of the price.

now the revenue is:

r(p) = n*p = [ (-1,000/15)*p + 23,333.3]*p

Then the profit is:

P(p) =  n(p)*p - n(p)*100 = [ (-1,000/15)*p + 23,333.3]*(p - 100)

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I'm assuming you want an equation with a slope of -5 that also passes through the point (1, -7).

An equation that fits this is

Hope this helps!