Question

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

Answer 1

Hint: 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, \[n=17\].
\[\begin{align}
  & \Rightarrow p=584,513{{(1.002)}^{17}} \\
 & \Rightarrow p=604,708 \\
\end{align}\]
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
\[\begin{align}
  & \Rightarrow \dfrac{600,000}{584,513}=\dfrac{584,513{{(1.002)}^{n}}}{584,513} \\
 & \Rightarrow 1.026495561={{(1.002)}^{n}} \\
\end{align}\]
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.