Real Life Situations Involving Functions (Part 1) | General Mathematics

Summary

In this video, Teacher Joanne explains the real-life applications of functions in various fields, particularly in business, physics, and mathematics. She demonstrates how to model situations using cost, revenue, and profit functions, such as fundraising for a school event through cupcake sales. The video also covers real-world scenarios, including calculating the height of a golf ball and the velocity of a free-falling object. By the end of the lesson, viewers should appreciate the importance of functions in everyday life and understand how to solve related problems.

Takeaways

• 😀 Functions model real-world situations, showing the relationship between quantities like cost, revenue, and profit.
• 😀 Cost functions include both variable and fixed costs, with fixed costs remaining constant regardless of production volume.
• 😀 The cost function formula is given by C(x) = variable cost × x + fixed cost, where x is the number of units produced.
• 😀 Revenue functions are calculated by multiplying the price per unit by the number of units sold: R(x) = price × x.
• 😀 Profit functions are determined by subtracting the cost function from the revenue function: P(x) = R(x) - C(x).
• 😀 The concept of break-even points helps to determine how many units need to be sold to avoid a loss.
• 😀 In the cupcake example, a loss occurs when fewer than 61 cupcakes are sold, and 61 cupcakes must be sold to break even.
• 😀 The height of a golf ball shot into the air is modeled by a quadratic function, showing the relationship between time and height.
• 😀 To find the time it takes for a golf ball to hit the ground, solve the height function for t when the height is zero.
• 😀 Free-falling objects' velocity can be modeled as a function of height, and we can solve for the height to achieve a desired velocity.
• 😀 Functions are vital in various real-life contexts, including business, physics, and many other everyday applications.

Q & A

What is the main focus of this lesson in the video?

The main focus of this lesson is on real-life applications of functions, demonstrating how functions are used to model various situations in everyday life.

What are some of the key real-life applications of functions mentioned in the video?

Some key applications of functions include water bills depending on consumption, speed being a function of time and distance, and revenue depending on product price and sales volume.

What are the components of a cost function as explained in the video?

A cost function consists of variable cost, which depends on the quantity produced, and fixed cost, which remains constant regardless of production volume.

How do you calculate the revenue function?

The revenue function is calculated by multiplying the price per unit by the number of units sold, which is expressed as r(x) = price × x.

How is the profit function derived?

The profit function is derived by subtracting the cost function from the revenue function, represented as p(x) = r(x) - c(x).

What does a negative profit indicate in the example about selling cupcakes?

A negative profit indicates a loss. In the example, selling 60 cupcakes results in a loss of 13 pesos, meaning the revenue from selling 60 cupcakes is not enough to cover the cost.

How is the total cost calculated in the example with 150 cupcakes?

The total cost is calculated by using the cost function, c(x) = 12x + 793. For 150 cupcakes, this results in a total cost of 2593 pesos.

What does the break-even point represent, and how is it determined?

The break-even point represents the point where there is no profit or loss. It is determined by setting the profit function equal to zero and solving for the number of units, which in the cupcake example is 61 cupcakes.

In the golf ball problem, how is the time it takes for the ball to hit the ground calculated?

The time is calculated by setting the height function h(t) equal to zero (representing the ball hitting the ground), solving for t, and finding that it takes 4 seconds for the ball to hit the ground.

How do you solve the problem of determining the height from which a watermelon must be dropped to hit the ground at 100 meters per second?

To solve this, substitute the velocity function v(h) = √(9.8 × 2 × h) with v = 100, square both sides, and solve for h, resulting in a height of 510.2 meters.