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Answer :
To solve the problem of finding the maximum revenue from ticket sales and determining how many people should attend the lecture to achieve this maximum, we need to analyze the given revenue function:
[tex]\[ R(x) = -0.05x^2 + 60x - 3125 \][/tex]
This is a quadratic function in the form of [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -0.05 \)[/tex]
- [tex]\( b = 60 \)[/tex]
- [tex]\( c = -3125 \)[/tex]
The curve of this quadratic function is a downward-opening parabola because the coefficient [tex]\( a \)[/tex] is negative. The maximum point of a parabola is its vertex.
### Step 1: Find the Vertex
The x-coordinate of the vertex can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{60}{2 \times -0.05} \][/tex]
[tex]\[ x = -\frac{60}{-0.1} \][/tex]
[tex]\[ x = 600 \][/tex]
This means that 600 people should attend the lecture to maximize the revenue.
### Step 2: Calculate the Maximum Revenue
To find the maximum revenue, substitute [tex]\( x = 600 \)[/tex] back into the revenue function [tex]\( R(x) \)[/tex]:
[tex]\[ R(600) = -0.05(600)^2 + 60(600) - 3125 \][/tex]
Calculate each term:
1. [tex]\( 600^2 = 360000 \)[/tex]
2. [tex]\( -0.05 \times 360000 = -18000 \)[/tex]
3. [tex]\( 60 \times 600 = 36000 \)[/tex]
Now substitute back:
[tex]\[ R(600) = -18000 + 36000 - 3125 \][/tex]
[tex]\[ R(600) = 14875 \][/tex]
So, the maximum revenue of \$14,875 is achieved when 600 people attend the lecture.
[tex]\[ R(x) = -0.05x^2 + 60x - 3125 \][/tex]
This is a quadratic function in the form of [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -0.05 \)[/tex]
- [tex]\( b = 60 \)[/tex]
- [tex]\( c = -3125 \)[/tex]
The curve of this quadratic function is a downward-opening parabola because the coefficient [tex]\( a \)[/tex] is negative. The maximum point of a parabola is its vertex.
### Step 1: Find the Vertex
The x-coordinate of the vertex can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{60}{2 \times -0.05} \][/tex]
[tex]\[ x = -\frac{60}{-0.1} \][/tex]
[tex]\[ x = 600 \][/tex]
This means that 600 people should attend the lecture to maximize the revenue.
### Step 2: Calculate the Maximum Revenue
To find the maximum revenue, substitute [tex]\( x = 600 \)[/tex] back into the revenue function [tex]\( R(x) \)[/tex]:
[tex]\[ R(600) = -0.05(600)^2 + 60(600) - 3125 \][/tex]
Calculate each term:
1. [tex]\( 600^2 = 360000 \)[/tex]
2. [tex]\( -0.05 \times 360000 = -18000 \)[/tex]
3. [tex]\( 60 \times 600 = 36000 \)[/tex]
Now substitute back:
[tex]\[ R(600) = -18000 + 36000 - 3125 \][/tex]
[tex]\[ R(600) = 14875 \][/tex]
So, the maximum revenue of \$14,875 is achieved when 600 people attend the lecture.
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