Problem 99 - College Algebra with Modeling and Visualization

Question

Skin Cancer Approximately \(4.5 \%\) of all cancer cases diagnosed in 2007 were skin cancer. (Source: American Cancer Society.) (a) If \(x\) cases of cancer were diagnosed, how many of these were skin cancer? (b) There were \(65,000\) cases of skin cancer diagnosed in \(2007 .\) Find the total number of cancer cases in 2007

Step-by-Step Solution

Answer

(a) 0.045x; (b) approximately 1,444,444 cases.

Step 1: Understanding the Problem

We are given that approximately 4.5% of all cancer cases diagnosed in 2007 were skin cancer. We need to solve two parts: (a) Determine how many cases of skin cancer occurred if x cases of cancer were diagnosed. (b) Find the total number of cancer cases if there were 65,000 skin cancer cases.

Step 1: Calculate skin cancer cases for part (a)

To find how many cases were skin cancer, we use the given percentage. If 4.5% of cancer cases were skin cancer, then the number of skin cancer cases is 4.5% of x. We calculate this as follows:\[c = 0.045x\]where c is the number of skin cancer cases. This formula represents the solution to part (a).

Step 2: Solve for total cases in part (b)

For part (b), we know there were 65,000 cases of skin cancer, which represents 4.5% of all diagnosed cancer cases. Let the total number of cancer cases be y.So, we set up the equation:\[0.045y = 65,000\]

Step 3: Solve the equation for y

To find the total number of cancer cases, y, we solve the equation from Step 2:\[y = \frac{65,000}{0.045}\]Calculate to find y:\[y \approx 1,444,444.44\]Rounding to the nearest whole number, y = 1,444,444. Thus, there were approximately 1,444,444 total cancer cases.

Key Concepts

Cancer Statistics

Cancer statistics give us insights into the prevalence of different types of cancer within a given population. In this exercise, we focus on skin cancer statistics from the year 2007, where it was reported that skin cancer constituted approximately 4.5% of all cancer cases. This kind of information is crucial for health organizations to allocate resources effectively and prioritize research efforts.
Understanding cancer statistics helps in assessing:

• Public health needs.
• The success of prevention and treatment programs.
• The allocation of funding towards research and treatments.

These statistics are not just numbers; they shape the way we understand and approach the fight against cancer globally.

Percentage Calculations

Percentage calculations are a fundamental part of mathematics, especially when dealing with statistical data. In the context of our problem, it is used to determine the portion of skin cancer cases out of the total cancer diagnoses. To calculate a percentage of a number, follow these steps:

• Convert the percentage into a decimal by dividing by 100. For example, 4.5% becomes 0.045.
• Multiply this decimal by the total number of cases (e.g., total cancer diagnoses) to find the number of instances represented by the percentage.

For example, if you have a total of \(x\) cancer cases, to find 4.5% of these, you use the formula \(c = 0.045x\). This provides a clear, straightforward way to see how percentages apply to real-world data.

Algebraic Equations

Algebraic equations are essential for solving problems where some quantities are unknown. In real-life situations like the exercise here, they turn word problems into solvable mathematical expressions. For example, in finding the total number of cancer cases from a given percentage of skin cancer cases, we set up the equation:

• \[0.045y = 65,000\]
• The goal is to solve for the unknown \(y\), which represents the total cancer cases.

Rearranging and solving the equation involves dividing both sides by 0.045, resulting in \(y = \frac{65,000}{0.045}\), which gives approximately 1,444,444 as the solution. This demonstrates how algebra can help us solve complex real-world problems efficiently.