Polynomial functions are mathematical expressions that involve variables raised to whole number powers. They can be used in diverse fields, including physics, engineering, and economics. In this article, we will explore three practical examples of working with polynomial functions, each demonstrating different applications and problem-solving methods.
Context:
Imagine you are designing a rectangular garden and want to express the area in terms of the garden’s length and width. This is a common scenario where polynomial functions come into play.
The area (A) of a rectangle can be expressed as a polynomial function:
A = length × width
If we let the length be represented by the polynomial function (x + 2) and the width by (x + 3), we can find the area as follows:
A = (x + 2)(x + 3)
To find the area, we will multiply these two binomials:
Now, combine all the results:
A = x² + 3x + 2x + 6
A = x² + 5x + 6
Notes:
In this example, we’ve expanded the polynomial to express the area as a quadratic polynomial. You can also substitute specific values for x to find the area for different lengths and widths.
Context:
Consider a small business that sells handmade candles. The revenue (R) generated from selling candles can be modeled by a polynomial function based on the number of candles sold (n).
Let’s assume that the revenue is represented by the polynomial:
R(n) = 5n² + 20n
Here, the coefficients represent the price per candle and the fixed costs. We can determine the revenue generated from selling a certain number of candles.
If the business sells 10 candles, we substitute n with 10:
R(10) = 5(10)² + 20(10)
R(10) = 5(100) + 200
R(10) = 500 + 200
R(10) = 700
Thus, the revenue generated from selling 10 candles is $700.
Notes:
You can modify the coefficients in the polynomial to reflect different pricing strategies or fixed costs. This example illustrates how polynomial functions can help businesses forecast revenue based on sales volume.
Context:
In physics, the motion of a projectile can be modeled using polynomial functions. Understanding this concept is crucial for fields like engineering and sports science.
The height (h) of a projectile can be modeled by the polynomial function:
h(t) = -16t² + 64t + 48
Where t represents time in seconds. This function describes the height of the projectile over time, considering the effects of gravity.
To find the height after 2 seconds, substitute t with 2:
h(2) = -16(2)² + 64(2) + 48
h(2) = -16(4) + 128 + 48
h(2) = -64 + 128 + 48
h(2) = 112
So, the height of the projectile after 2 seconds is 112 feet.
Notes:
The coefficients in the polynomial can be adjusted based on the initial velocity and height of the projectile. This example shows how polynomial functions are essential in modeling real-world physical phenomena.