Tour cost A charter bus company has determined that the cost...

Tour cost A charter bus company has determined that the cost of providing $x$ people a tour is $$C(x)=180+2.50 x$$ A full tour consists of 60 people. The ticket price per person is $\$ 15$ plus $\$ 0.25$ for each unsold ticket. Determine a. the revenue function b. the profit function c. the company's maximum profit d. the number of ticket sales that yields the maximum profit

Tour cost A charter bus company has determined that the cost of providing $x$ people a tour is
$$C(x)=180+2.50 x$$
A full tour consists of 60 people. The ticket price per person is $\$ 15$ plus $\$ 0.25$ for each unsold ticket. Determine
a. the revenue function
b. the profit function
c. the company's maximum profit
d. the number of ticket sales that yields the maximum profit

Key Concepts

Cost Function

A cost function represents the total cost incurred by a business to produce a certain number of goods or provide a service, typically expressed as a function of the quantity produced. It includes fixed costs, which do not change with the level of production, and variable costs, which vary directly with the number of units produced. This provides a basis for analyzing expenses and determining break-even points and profitability.

Revenue Function

A revenue function calculates the total income generated from sales, defined as the product of the number of units sold and the price per unit. This function may incorporate factors such as variable pricing strategies, discounts, or price increments that depend on the number of units sold, allowing businesses to model and predict their earning behavior in different scenarios.

Profit Function

A profit function is derived by subtracting the cost function from the revenue function, representing the net income or profit generated. This function is crucial for determining how changes in sales volume, pricing strategies, or cost structures impact the overall profitability of a business, and helps in identifying the optimal conditions for maximum profit.

Quadratic Optimization

Quadratic optimization involves finding the maximum or minimum value of a quadratic function, typically represented in vertex form. In many business problems, the profit function becomes a quadratic equation due to combined linear cost and revenue functions. Identifying the vertex of this quadratic function is key to determining the optimal number of units that either maximize profit or minimize cost.

Transcript

00:01 So we know we have a bus, a tour bus, and we know that the maximum number of people that can go on this tour is 60.

00:07 And we know that the cost to this company is $180 plus $2 .50 per person.

00:18 So that's what their cost is.

00:19 And again, this is for a full bus.

00:23 And we know that the revenue that they make is going to equal, and that's going to be for part a, the revenue.

00:31 One i'll call that r of x is the cost of the ticket.

00:37 And the cost of the ticket they say is $15 plus 25 times however many tickets are not sold.

00:46 So if they sell x, that means that 60 minus x will be the empty seats.

00:56 And so we don't know what the cost is for a person.

00:59 It all depends on how many people they have going.

01:01 So it's not automatically $15 a person.

01:03 It's $15 plus 25 cents times the number of empty seats.

01:09 So that's how much it's going to cost per person.

01:14 But now we need to multiply that by how many people are going to get the revenue.

01:18 So again, this is what the selling price is going to be per person times the number of tickets that are sold.

01:25 So let's clean that up.

01:27 So inside we have 15 plus 0 .25 times that.

01:33 So that's going to be one -fourth of 60, which is going to be 15.

01:37 Whoops, that should be a subtraction sign, minus, there we go, minus 0 .25x.

01:46 And then we'll distribute that, add these two to get 30, and then distribute this through.

01:52 So i'll put the x squared in front.

01:54 So negative 0 .25x squared by distributing here, and then plus 30x.

02:03 So there's our revenue equation.

02:07 Now we want to find our profit.

02:09 I'll call that p of x.

02:11 Our profit equation is we know that if we take how much money you bring in, which is your revenue, and subtract away the cost, we'll get what our profit is.

02:23 And we know our revenue equation.

02:25 We just found that out.

02:26 So this is how much money we're bringing in.

02:30 And then we need to subtract away the cost equation...

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