Question

In hockey, a player's shooting percentage is given by dividing the player's total goals scored by the player's total shots taken on goal. So far this season, Rachel has taken 28 shots on net but scored only 2 goals. She has set a target of achieving a \(30 \%\) shooting percentage this season. a) Write a function for Rachel's shooting percentage if \(x\) represents the number of shots she takes from now on and she scores on half of them. b) How many more shots will it take for her to bring her shooting percentage up to her target?

Short answer

Rachel needs to take 32 more shots.

Step-by-step solution

Define the initial knowns

Rachel has taken 28 shots and scored 2 goals. Her initial shooting percentage is \( \frac{2}{28} \).

Formulate the function for additional shots and goals

Let \( x \) represent the number of additional shots Rachel takes. She will score on half of these shots, so she will have \( \frac{x}{2} \) additional goals.

Create the function for total goals and shots

The total number of goals Rachel will have after taking \( x \) more shots is \( 2 + \frac{x}{2} \). The total number of shots she will have taken is \( 28 + x \).

Write the function for shooting percentage

The shooting percentage function \( P(x) \) can be written as: \[ P(x) = \frac{2 + \frac{x}{2}}{28 + x} \]

Set up the equation for the target shooting percentage

Rachel's target shooting percentage is 30%, or 0.3. Set up the equation to find \( x \): \[ \frac{2 + \frac{x}{2}}{28 + x} = 0.3 \]

Solve the equation for \( x \)

Clear the fraction by multiplying both sides by \( 28 + x \): \[ 2 + \frac{x}{2} = 0.3(28 + x) \] Simplify and solve for \( x \): \[ 2 + \frac{x}{2} = 8.4 + 0.3x \] Multiply everything by 2 to eliminate the fraction: \[ 4 + x = 16.8 + 0.6x \] Rearrange to isolate \( x \): \[ x - 0.6x = 16.8 - 4 \] \[ 0.4x = 12.8 \] Divide both sides by 0.4: \[ x = 32 \]

Key concepts

Rational Functions

A rational function is a ratio of two polynomial functions. In simpler terms, it can be thought of as a fraction where both the numerator and the denominator are polynomial expressions. In our hockey shooting percentage problem, Rachel's shooting percentage can be modeled as a rational function. Rachel's initial scoring attempts and future attempts form the numerator, while the total number of shots forms the denominator.

Understanding rational functions is important because they appear in various real-world contexts, like rates, averages, and percentages. In this exercise, the shooting percentage's function is written as:
\[ P(x) = \frac{2 + \frac{x}{2}}{28 + x} \]
This function helps us understand how Rachel's shooting percentage changes as she takes more shots.

Solving Equations

Solving equations involves finding the value of variables that satisfy the given mathematical statement. In our problem, we need to solve for the variable \(x\), which represents the additional shots Rachel needs to take.

We start by setting up the shooting percentage equation:
\[ \frac{2 + \frac{x}{2}}{28 + x} = 0.3 \]
Next, we clear the fraction by multiplying both sides by \(28 + x\) to get:
\[ 2 + \frac{x}{2} = 0.3(28 + x) \]
By simplifying step-by-step, we find the value of \(x\). Solving equations like this involves careful algebraic manipulation and ensuring each step is correct.

Percentage Calculations

Percentage calculations represent a portion of a whole as a fraction of 100. In Rachel's case, we are dealing with the percentage of goals she scores from her total shots. Her target is a 30% (or 0.3) shooting percentage.

To calculate this, we set up the equation:
\[ \frac{2 + \frac{x}{2}}{28 + x} = 0.3 \]
The numerator represents her total goals, and the denominator represents her total shots. To meet her 30% target, the value of the entire fraction should equal 0.3. Understanding how to form and solve these equations is crucial for handling percentage-related problems effectively.

Problem-Solving Steps

Approaching a problem systematically can make it easier to solve. Here’s how we broke down Rachel’s shooting percentage problem:

• Step 1: Identify known values – Rachel’s initial shots and goals.
• Step 2: Define the variable – Let \(x\) represent additional shots.
• Step 3: Formulate expressions for total goals and shots.
• Step 4: Write the shooting percentage function.
• Step 5: Set up the equation using the target percentage.
• Step 6: Solve the equation step-by-step to find \(x\).

By following these steps, we ensure each part of the problem is addressed logically and clearly, making it easier to reach the solution.