Answer :
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]
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]
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]
