Function Word Problems

Try a word problem!

The Problem:

The square patch of grass above needs to be seeded around the $4$ identical circular ponds. Write a function for the area ($A$) of grass that needs to be seeded in terms of the radius of the ponds ($r$).

Solve it!

To solve a word problem, the first step is always to translate the writing into math. We know that the important pieces are:

• The area of the grass ($A$)
• The radius of the ponds ($r$)
• There are $4$ ponds
• The sides of the square are 4 times the ponds' radius.

Awesome! Now we have to figure out the relationship that they have to each other.

Since this problem involves area, your variables will probably be squared. That means that this function is most likely going to describe a nonlinear relationship--the area of the grass won't increase or decrease steadily alongside the radius of the ponds.

We know that the outline of all the grass is in a square around the edges of the circular ponds. Think about how the ponds lay on the grass, and try to imagine what operation we could use to figure out the exact area (if not the equation itself). When you think you have an idea, it continue.

(if you can't figure it out, that's okay too! Just hit the button and we'll explain.)

Since the ponds fit neatly inside the square of grass, and the grass covers all the remaining area, we can subtract the area of the circles (we'll call that $A_C$) from the area of the square (we'll call that $A_S$). You can find the equations for these areas in our explanation about geometric equations!

In this problem, we need the area of a circle ($A_C=\pi r^2$) and the area of a square ($A_S=(s)^2$). Since there are $4$ ponds, the total area of the ponds will be $4A_C$, and since one side of the square is equal to $4$ times a pond's radius, we can say that $s = 4r$. To find the area surrounding the ponds, we simply take the difference between the area of the big grass square and the four smaller circles.

So now we know all our basic equations:

$$\begin{align} A &= A_S - 4(A_C)\\ A_C &= \pi r^2\\ A_S &= (s)^2\\ s &= 4r \end{align}$$

And we can combine them and simplify to get one equation we can use for the whole problem:

$$\begin{align} A &= (s)^2 - 4(\pi r^2)\\ A &= (4r)^2 - 4(\pi r^2)\\ A &= 16r^2 - 4\pi r^2\\ \end{align}$$

Done! The area of the grass is the area of the whole square, minus the area of all four ponds. Since we use the radius of the ponds to figure this out, and the radius is being squared, we know that this is a nonlinear relation.

Word Problems with Functions

Solving word problems with functions is basically the same thing as evaluating a function for a given value. If you need a refresher on evaluating functions, click here.

Once you've got the function (whether it's given to you or whether you have to come up with it yourself), it's a pretty quick trip to solving the word problem.

Let's look at an example.

$$P(b) = 25b + 7\\ b = 6$$ $$\begin{align} P(6) &= 25(6) + 7\\ P(6) &= 150 + 7\\ P(6) &= 157 \end{align}$$

$$P(b) = 25b + 7\\ b = 1$$ $$\begin{align} P(6) &= 25(1) + 7\\ P(6) &= 25 + 7\\ P(6) &= 32 \end{align}$$

For this one, we have to find the input using the output:

$$\begin{align} P(b) &= 25b + 7\\ P(b) &= 107\\\ \\ 107 &= 25b + 7\\ 100 &= 25b\\ 4 &= b\\ \end{align}$$

Let's look at one more example.

$$h(s) = 10s^2 - 3s + 11\\ s = 7$$ $$\begin{align} h(7) &= 10(7)^2 - 3(7) + 11\\ h(7) &= 10(49) - 21 + 11\\ h(7) &= 490 - 21 + 11\\ h(7) &= 480 \end{align}$$

Here we have to work backwards from the output again, and the factoring gets nasty. In the case of a multiple choice problem, you can just test inputs and see what matches! In this case, $5$ works:

$$\begin{align} h(s) &= 10s^2 - 3s + 11\\ 246 &= 10s^2 - 3s + 11\\\ \\ 246 &= 10(5)^2 - 3(5) + 11\\ 246 &= 10(25) - 15 + 11\\ 246 &= 10(25) - 15 + 11\\ 246 &= 250 - 15 + 11\\ 246 &= 246\\ \end{align}$$

Example Walk Through: Word Problem Strategy $\\$

Laurel is a huge fan of hats, she likes all kinds of hats and recently started collecting them. She already owns $3$ hats. She plans on buying a hat according to the season every month.

Let $x$ be how many months have past since Laurel started collecting hats. Let $y$ by how man hats Laurel owns. Can you write a linear function in terms of $x, y$?

Let's break down the problem! $\\$

Read the problem again and get a better understanding of what is given and what is asked.

We know

• How many hats Laurel owns already
• How many hats Laurel collects every month

We are asked

• After $x$ months, how many hats does Laurel own?

Imagine the scenario! $\\$

If you are ever unsure about how to proceed with a math problem, one thing I find very helpful is to imagine the scenario and sub in the numbers.

At the very beginning, when $x=0$, Laurel hasn't bought any new hats yet, she owns $3$ hats, so $y=3$.

After the first month, when $x=1$, laurel owns $3$ hats, plus the $1$ hat she just bought. She now owns $y=3+1$ hats.

After the second month, when $x=2$, laurel owns $3$ hats, plus the $2$ new hats she bought after she started collecting hats. She now owns $y=3+2$ hats.

Write the function! $\\$

Now, we can write the function. $$y=3+x$$

You can always check! $\\$

If you are unsure if you function is correct, you can always put in real numbers to check. For example, when $x=3$, $y=6$. That makes sense, because Laurel would have bought $3$ hats after $3$ months and she now has $6!$