Problem 10 - Beginning and Intermediate Algebra

Question

Solve. The average number of traffic tickets issued in a city on any given day Sunday - Saturday can be approximated by $$T(x)=-7 x^{2}+70 x+43$$ where \(x\) represents the number of days after Sunday \((x=0 \text { represents Sunday, } x=1\) represents Monday, etc.), and \(T(x)\) represents the number of traffic tickets issued. On which day are the most tickets written? How many tickets are issued on that day?

Step-by-Step Solution

Answer

The most tickets are written on Friday (when x=5), with 218 tickets issued on that day.

Step 1: Identify the coefficient values a and b in the given function

In the function \(T(x) = -7x^2 + 70x + 43\), the coefficients are a = -7 and b = 70.

Step 2: Use the vertex formula to find the x-coordinate of the vertex

By using the vertex formula \(x = -\frac{b}{2a}\), we can find the x-coordinate of the vertex (the day of the week when the maximum number of tickets are issued). In this case, we have: \(x = -\frac{70}{2(-7)}\) \(x = -\frac{70}{-14}\) \(x = 5\) So, the x-coordinate of the vertex is 5, which represents Friday (since x=0 represents Sunday, x=1 represents Monday, etc.).

Step 3: Calculate T(x) for the x-coordinate of the vertex

Now, let's plug the x-coordinate of the vertex (x = 5) back into the function T(x) to find the maximum number of tickets issued: \(T(5) = -7(5)^2 + 70(5) + 43\) \(T(5) = -7(25) + 350 + 43\) \(T(5) = -175 + 350 + 43\) \(T(5) = 218\) So, 218 tickets are issued on that day.

Step 4: Present the final answer

The most tickets are written on Friday (when x=5), with 218 tickets issued on that day.

Key Concepts

Vertex Formula

The vertex formula is a fundamental concept in understanding quadratic functions like the one in this problem. A quadratic function typically looks like this:

• Standard form: \( ax^2 + bx + c \)

The vertex formula is used to find the vertex of a parabola, and it is crucial for determining either the maximum or the minimum value of the function, depending on its orientation. The formula for finding the x-coordinate of the vertex is:

• \( x = -\frac{b}{2a} \)

Here, the terms 'a' and 'b' are the coefficients from the quadratic equation.
For our given function, \( T(x) = -7x^2 + 70x + 43 \), the coefficients are \( a = -7 \) and \( b = 70 \). By applying these values into the vertex formula, we arrive at the x-coordinate of the vertex, which corresponds to the specific day in the week. In this case, the calculation \( x = -\frac{70}{2(-7)} \) yields \( x = 5 \), indicating that the vertex, or the maximum point of our quadratic function, falls on Friday.

Parabola

A parabola is the graphical representation of a quadratic function, and it can open upwards or downwards. The direction of the opening depends on the coefficient 'a' from the standard form of the quadratic equation:

• If \( a > 0 \), it opens upwards.
• If \( a < 0 \), it opens downwards.

In the context of our function, \( T(x) = -7x^2 + 70x + 43 \), the coefficient \( a = -7 \) is negative. Therefore, the parabola opens downward, which is consistent with having a maximum value rather than a minimum.
The vertex of a parabola is a crucial point. For a downward-opening parabola, the vertex represents the peak, or highest point. In real-world applications, such as our traffic tickets scenario, determining this point is essential for solving problems like figuring out the day with the maximum number of tickets issued. Thus, understanding the shape and orientation helps interpret the meaning of the vertex in practical terms.

Maximum Value

In a quadratic function represented by a downward-opening parabola, the maximum value is found at the vertex. This is because, for a function where \( a < 0 \), the vertex is the topmost point of the curve.
To find the maximum value, once we have located the x-coordinate of the vertex, we substitute it back into the original quadratic function. For the function \( T(x) = -7x^2 + 70x + 43 \), substituting \( x = 5 \) (the x-coordinate we found earlier) gives us:

• \( T(5) = -7(5)^2 + 70(5) + 43 \)

Carrying out the calculation, we simplify it to find:

• \( T(5) = -175 + 350 + 43 = 218 \)

This value represents the maximum number of traffic tickets issued, which is 218 on Friday. Understanding the concept of maximum value in quadratic functions helps in identifying peak points, which can be incredibly useful for optimization problems in various domains.