Questions: A charter flight charges a fare of 300 per person plus 20 per person for each u holds 100 passengers. Let x represent the number of unsold seats. a. Find an expression for the total revenue received for the flight R(x). R(x)=(100-x)(300+20x) b. Choose the correct graph of the function, R(x), below. c. The number of unsold seats that will produce the maximum revenue is (Round to the nearest whole number as needed.)

A charter flight charges a fare of 300 per person plus 20 per person for each u holds 100 passengers. Let x represent the number of unsold seats.
a. Find an expression for the total revenue received for the flight R(x).
R(x)=(100-x)(300+20x)
b. Choose the correct graph of the function, R(x), below.

c. The number of unsold seats that will produce the maximum revenue is
(Round to the nearest whole number as needed.)

Transcript text: A charter flight charges a fare of $\$ 300$ per person plus $\$ 20$ per person for each $u$ holds 100 passengers. Let x represent the number of unsold seats. a. Find an expression for the total revenue received for the flight $R(x)$. \[ R(x)=(100-x)(300+20 x) \] b. Choose the correct graph of the function, $\mathrm{R}(\mathrm{x})$, below. c. The number of unsold seats that will produce the maximum revenue is $\square$ (Round to the nearest whole number as needed.)

Solution

a. Find an expression for the total revenue received for the flight $ R(x) $.

Define variables

Let $x$ be the number of unsold seats. The number of sold seats is $100 - x$. The fare per person is $$300 + $20x$.

Find the expression for $R(x)$

The total revenue $R(x)$ is the product of the number of sold seats and the fare per person. $R(x) = (100-x)(300+20x)$

$ \boxed{R(x) = (100-x)(300+20x)} $

b. Choose the correct graph of the function, $R(x)$, below.

Analyze $R(x)$

$R(x) = (100-x)(300+20x) = 30000 + 2000x - 300x - 20x^2 = -20x^2 + 1700x + 30000$ This is a quadratic function with a negative leading coefficient, so its graph is a parabola opening downwards. When $x = 0$, $R(0) = 30000$. When $x = 100$, $R(100) = 0$. The x-coordinate of the vertex is $x = -\frac{b}{2a} = -\frac{1700}{2(-20)} = \frac{1700}{40} = 42.5$. $R(42.5) = -20(42.5)^2 + 1700(42.5) + 30000 = -20(1806.25) + 72250 + 30000 = -36125 + 72250 + 30000 = 66125$

Determine the graph

The graph should be a parabola opening downwards, passing through the points $(0, 30000)$ and $(100, 0)$, with a vertex close to $(42.5, 66125)$. This corresponds to the first graph.

$ \boxed{\text{First graph}} $

c. The number of unsold seats that will produce the maximum revenue is $\square$

Find the x-coordinate of the vertex

The x-coordinate of the vertex represents the number of unsold seats that maximize the revenue. $x = -\frac{b}{2a} = -\frac{1700}{2(-20)} = \frac{1700}{40} = 42.5$

Round to the nearest whole number

Rounding to the nearest whole number gives 43.

$ \boxed{43} $

$ R(x) = (100-x)(300+20x) $ First graph 43

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