How to figure out word problems.

Calculus Tutorial

How to figure out word problems.


Many times, while learning calculus, you will run into concepts that you understand, but can’t quite figure out in word problems. I will write a short guide that will give you a couple of things to think about before starting the problem, and a couple of tips in general. Some of these concepts could be applied to other levels of math, but this is what I found helpful while taking calculus in high school.

Sample Problem

Before even writing anything, figure out exactly what the problem is asking. If the problem has units, figure out which units you’re going to need to have in your answer. Find out which pieces of information provided by the question are actually going to be important and in what way. Break apart the problem in the way that is most convenient for you. I will do a practice problem to demonstrate this, and hopefully, it will be of help in future mathematical endeavors.


Sonya can paint at a rate of

v(t) = 150 – 4t

square feet per hour, where t is the number of hours since she started painting. Can Sonya paint the walls of a 12 foot by 12 foot office with an 8 foot high ceiling in 3 hours?

What about the walls of a 14 foot by 14 foot office with an 8 foot high ceiling?


Often times, word problems have some excess information. Most of the time it is useful information, but we must be aware of what the question writers want us to do.
So, first we need to think about what the question is telling us.

We must note what the answer should be. In other words, which units, or what format the answer will be in. In this case its nothing too complicated, but we know we have to answer these two questions:
a) can she paint an office that 12x12x8 in three hours?
b) can she paint an office that is 14x14x8 in three hours?

From this, we know that we will have to find the area that she covers in three hours and compare that to the area of rooms with the dimensions of 12x12x8 and 14x14x8

We are given an equation, so we know that’s going to be an important part of the answer, but what are we supposed to do with this equation?

The question is asking, if Sonya can paint the walls of an office within 3 hours.
The rate at which she paints(the equation) represents the derivative of the amount of space that she would be able to paint. Usually, when finding area from something covered by a certain amount of space, like a to b, or in this case, 0 to 3 hours, we can utilize the integral. Therefore, we can assume that the question wants us to integrate the equation in order to find the area that she will cover with paint.


After integrating, we know that in three hours Sonya can paint 432 square feet. That was the more complicated part, the rest will be easier, since we already know what we are looking for, and we already did the part that actually involves calculus.

Sonya is painting only the walls, so we don’t use the surface area formula, because that would include the ceiling and floor. Instead, what we will have to do is find the area of one wall and multiply it by 4. Thus, we have 12 x 8 x 4 = 384

Since 384 < 432, we know that Sonya would be able to paint the office in less than three hours. And thats the answer for the first part of the question. Let's make sure not to forget that the question had a second part, as that can be a costly mistake. Always go back and make sure you're answering the complete question. The second part asks if Sonya would finish the 14x14x8 office in time. So, we do the same thing we did to find out if she would finish a 12x12x8 office. 14 x 8 x 4 = 448 448 > 432, so Sonya cannot finish it in three hours.

Now that we have finished both parts of the problem, we can either circle the answer or rewrite our answers at the end. This way if you have any doubts about it, you can easily find your answers. Remember to stay calm and not overthink word problems. The word problems you see will always have concepts that you already know. Don’t be scared of the extra words, just take the time to pick apart the problem. Really understand the question before you try to answer it.

Lastly, If you are really stuck on a problem and don’t even know where to start, try to think about it in different ways. I had a teacher who called it the rule of four. The four ways to think about a problem. Verbally, analytically, numerically, and graphically. So rephrase the problem in your own words to yourself. Analyze the problem; see what concepts of math you will have to use and simplify anything that can be simplified. Numerical: if it helps, make a chart with x and y values to see how the function behaves. Graphical: make some sort of visual to help you. Regardless if this visual is a graph or not, just make something that will help you to understand the problem from a different perspective.

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