Question

How can functions be used to solve real-world situations?

Answer 1

I'm going to use a few examples.

The population of Wyoming at the end of 2014 was `584,513`. The population increases at an annual rate of `0.2%`.

a) Determine an equation for the population with respect to the number of years after `2014`, assuming that the population increases at a constant rate.

Solution

The function will be of the form `p = ar^n`, where `a` is the initial population, `r` is the rate of increase, `n` is the time in years and `p` is the population.

`p = 584,513(1.0002)^n`

b) Using this model, estimate the population of Wyoming in `17` years.

solution

Here, `n = 17`.

`p = 584,513(1.002)^17`

`p = 604,708`

c) Using this model, determine in how many years it will take for the population to exceed `600,000`.

Solution

Here, `p = 600,000`

`600,000 = 584,513(1.002)^n`

`1.026495561 = (1.002)^n`

`ln(1.026495561) = ln(1.002)^n`

`ln(1.026495561) = nln(1.002)`

`n = 13`

It will take `13` years for the population to exceed `600,000`.

A vending machine currently sells packages of chips that sell for $2.00 and they sell 100 packages. They find that for each 25 cent increase, they lose five customers.

a) Write a function, in standard form, to represent this problem. Make sure that the dependant variable is "Profit".

Solution

We can use the following formula:

`P = "number of packages sold" xx "price/package"`

`P = (100 - 5x)(2.00 + 0.25x)`

`P = 200 - 10x + 25x - 5/4x^2`

`P = -5/4x^2 + 15x + 200`

b) Determine the profit after `10` increases in price.

Solution

Here, `x= 10`.

`P = -5/4(10)^2 + 15(10) + 200`

`P = -5/4(100) + 90 + 200`

`P = -125 + 90 + 200`

`P = 165`

Hence, the profit after `10` increases in price will be `$165`.

c) Find the optimum price for the chips to maximize the profit for the company.

Solution

This will be the vertex of the function. We will need to complete the square.

`P = -5/4x^2 + 15x + 200`

`P = -5/4(x^2 - 12x) + 200`

`P = -5/4(x^2 - 12x + 36- 36) + 200`

`P = -5/4(x - 6)^2 + 45 + 200`

`P = -5/4(x - 6)^2 + 245`

The optimum price is `$6` per bag.

Hopefully this proves that functions can effectively model real world problems, and can be used to solve them!

Hopefully this helps!