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After the drama club sold 100 tickets to a show, it had \[tex]$300 in profit. After the next show, it had sold a total of 200 tickets and had a total of \$[/tex]700 profit. Which equation models the total profit, [tex]$y$[/tex], based on the number of tickets sold, [tex]$x$[/tex]?

A. [tex]$y - 300 = 4(x - 100)$[/tex]

B. [tex][tex]$y - 300 = 2.5(x - 100)$[/tex][/tex]

C. [tex]$y + 300 = 2.5(x + 100)$[/tex]

D. [tex]$y + 300 = 4(x + 100)$[/tex]



Answer :

GinnyAnswer
Sure! Let's find out which equation models the total profit, [tex]\( y \)[/tex], based on the number of tickets sold, [tex]\( x \)[/tex].

We have two known data points:
1. When 100 tickets were sold, the profit was \[tex]$300. 2. When 200 tickets were sold, the profit was \$[/tex]700.

First, we need to determine the rate of change of profit, or the slope, between these points.

The formula for the slope ([tex]\( m \)[/tex]) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:

[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substituting the given points [tex]\((100, 300)\)[/tex] and [tex]\((200, 700)\)[/tex]:

[tex]\[ \text{Slope} = \frac{700 - 300}{200 - 100} = \frac{400}{100} = 4.0 \][/tex]

The slope [tex]\( m \)[/tex] is 4.0.

We can now use the point-slope form of a linear equation to model the total profit [tex]\( y \)[/tex] based on the number of tickets sold [tex]\( x \)[/tex]. The point-slope form equation is:

[tex]\[ y - y_1 = m(x - x_1) \][/tex]

We will use the point [tex]\((100, 300)\)[/tex] and the calculated slope 4.0:

[tex]\[ y - 300 = 4.0(x - 100) \][/tex]

Thus, the equation that models the total profit based on the number of tickets sold is:

[tex]\[ y - 300 = 4(x - 100) \][/tex]

Therefore, the correct equation is:

A. [tex]\( y - 300 = 4(x - 100) \)[/tex]
anuradhadevi1937

Answer:

Let's analyze the problem step by step based on the given information:

1. First Show Details:

- Tickets sold: 100

- Profit: $300

2. Second Show Details:

- Tickets sold: 200 (total, so tickets sold in the second show = 200 - 100 = 100)

- Profit: $700

We need to find an equation that models the total profit [tex]y[/tex] based on the number of tickets sold [tex]x[/tex].

• Let's use the information given:

- For the first show: [tex]( y¹ = 300)\: when\: ( x¹ = 100)[/tex]

- For the second show: [tex]( y²= 700)\: when \: ( x² = 200)[/tex]

• To find the equation that models total profit y:

Step 1: Calculate the profit per ticket for each show:

- For the first show:[tex]({Profit per ticket} = \frac{300}{100} = 3)[/tex]

- For the second show: [tex]{Profit per ticket} = \frac{700 - 300}{200 - 100} = \frac{400}{100} = 4)[/tex]

So, the profit per ticket is consistent across both shows.

Step 2: Construct the equation:

Since profit is directly proportional to the number of tickets sold (each ticket contributes a fixed amount to the profit), the equation should reflect this linear relationship.

Let's use the slope-intercept form of a linear equation [tex](y = mx + c)[/tex], where m is the slope (profit per ticket) and c is the y-intercept (initial profit).

From our calculation, the profit per ticket m=4.

The y-intercept c can be found by substituting in one of the points. Let's use the first show:

[tex](300 = 4 \cdot 100 + c )[/tex]

[tex]( 300 = 400 + c)[/tex]

[tex](c = 300 - 400)[/tex]

[tex](c = -100)[/tex]

So, the equation becomes:

[tex](y = 4x - 100)[/tex]

Now, let's check which option matches this equation:

A. [tex]( y - 300 = 4(x - 100))[/tex]

- Simplifying: [tex]( y - 300 = 4x - 400)[/tex]

- Adding 300 to both sides: [tex]( y = 4x - 100)[/tex]

This matches our equation.

Therefore, the correct equation that models the total profit y based on the number of tickets sold x is:

[tex][ {A. \quad y - 300 = 4(x - 100)}][/tex]

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