Question

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

Answer 1

We can use many types of functions to solve real-world situations. The functions may include exponential, logarithmic, polynomials etc. We can also use differentiation and integration to solve some real-world situations. We will see some of the real-world examples in which we will use the functions.

Complete step by step answer:
We will see an example in which we will use the exponential and logarithmic functions to solve a real-world problem.
Let’s say the population of X at the end of 2010 was 584,513. The population increases at an annual rate of 0.2 %. (a) find the population of X in 17 years. (b) how many years it will take for the population to exceed 600,000.
For the (a) part,
The function will be of the form $p=a{{r}^{n}}$. Here a is the initial population, r is the rate of increase, n is the time in years, and p is the population.
$\Rightarrow p=584,513{{(1.002)}^{n}}$
For population after 17 years, $n=17$.

& \Rightarrow p=584,513{{(1.002)}^{17}} \\\ & \Rightarrow p=604,708 \\\ \end{aligned}$$ For the (b) part, We made the equation for the population after n years, we will use the equation to solve this part $$p=584,513{{(1.002)}^{n}}$$, here $$p=600,000$$. Substituting the values, we get $$\Rightarrow 600,000=584,513{{(1.002)}^{n}}$$ Dividing both sides of the above equation by $$584,513$$, we get $$\begin{aligned} & \Rightarrow \dfrac{600,000}{584,513}=\dfrac{584,513{{(1.002)}^{n}}}{584,513} \\\ & \Rightarrow 1.026495561={{(1.002)}^{n}} \\\ \end{aligned}$$ Taking $$\log $$ both sides of the above equation, we get $$\Rightarrow \log \left( 1.026495561 \right)=\log \left( {{(1.002)}^{n}} \right)$$ We know the property of logarithm that states that, $$\log {{a}^{n}}=n\log a$$. Using the property in the above equation, we get $$\Rightarrow \log \left( 1.026495561 \right)=n\times \log \left( 1.002 \right)$$ On calculating the values of the above logarithm, we get $$n=13$$. We used the exponential and logarithmic functions to predict the population, which is a real-world problem. **Note:** Similarly, we can use different types of functions for different real-world problems. For example, using functions we can predict profit or loss, average cost, marginal cost for a business. Of course, if we can express it as a function, we can also plot these quantities on the graph.