Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

A cheese manufacturer produces 18,000 pounds of cheese from January 1 through March \(24 .\) Suppose that this rate of production continues for the remainder of the year. (a) Express the number \(y\) of pounds of cheese produced in terms of the number \(x\) of the day in a 365 -day year. (b) Predict, to the nearest pound, the number of pounds produced for the year.

Step-by-Step Solution

Answer

(a) \(y = 216.87x\); (b) Approximately 79,155 pounds.

Step 1: Calculate Daily Production Rate

First, calculate the rate of cheese production per day by dividing the total pounds of cheese produced from January 1 to March 24 by the number of days in that period. There are 31 days in January, 28 in February, and 24 in March, totaling 83 days.\[ \text{Daily Production Rate} = \frac{18,000 \text{ pounds}}{83 \text{ days}} \approx 216.87 \text{ pounds per day} \]

Step 2: Express Production in Terms of Days

Using the daily production rate, express the total production \(y\) in terms of \(x\), the number of days in the year. The formula is multiplying the daily rate by the number of days:\[ y = 216.87x \]

Step 3: Calculate Total Annual Production

Calculate the total production for a 365-day year by substituting \(x = 365\) into the equation from Step 2:\[ y = 216.87 \times 365 \]

Step 4: Compute the Result

Perform the multiplication to find the approximate total annual production:\[ y = 216.87 \times 365 = 79,154.55 \] Since the number of pounds needs to be a whole number, round to the nearest pound:\[ y \approx 79,155 \text{ pounds} \]

Key Concepts

Daily Production Rate

In the given exercise, determining the daily production rate is key to understanding how much cheese is produced consistently each day. To find this rate, you take the total amount of cheese produced over a set time span and divide it by the number of days in that span. Here, the cheese manufacturer produced 18,000 pounds from January 1 to March 24, which is a total of 83 days when you add up all the days in January, February, and March.

The daily production rate formula is:

• Daily Production Rate = Total Production / Total Days

Applying this here, \[\text{Daily Production Rate} = \frac{18,000}{83} \approx 216.87 \, \text{pounds per day}\] This rate of approximately 216.87 pounds per day is significant because it can be used to predict future production.

Linear Equation

To express the production of cheese over a year, we use a linear equation. A linear equation is an algebraic expression that describes a straight line when graphed, and it usually has the form \(y = mx + b\). However, in this case, because there is no starting amount of cheese to consider at day zero (\(b = 0\)), our equation simplifies to:

• Production Equation: \(y = 216.87x\)

Here, \(y\) represents the total pounds of cheese produced, \(216.87\) is the production rate per day, and \(x\) is the number of days. This equation is a tool for calculating how much cheese will be produced over any given number of days by multiplying the daily production rate by the number of days.

Rate of Change

In this scenario, the rate of change refers to how much the production of cheese increases with each passing day. A constant rate of change, such as our daily production rate of 216.87 pounds, signifies that the same amount of cheese is produced every day.

• Constant Rate of Change: Daily consistency in production.

This idea of a constant rate is essential in linear equations and other areas of algebra because it helps predict outcomes over time. Since the cheese production continues at the same daily rate, the total production increases linearly over time, simplifying predictions about future cheese production.

Total Production Calculation

Finally, calculating the total production for the entire year involves using our linear equation to find how much cheese is produced in 365 days. By substituting 365 for \(x\) in our linear equation, each day’s production is summed up for the entire year.

The equation looks like this:

• Total Production: \(y = 216.87 \times 365\)

Performing the multiplication gives:\[y = 216.87 \times 365 = 79,154.55\]Since production needs to be a whole number, we round 79,154.55 to the nearest pound, which results in approximately 79,155 pounds for the whole year. This simple calculation process lets us understand the total output from a consistent daily production pace.