# Differentiation word problem (basic)

• Taylor_1989
In summary, a rectangle with a varying length x has a constant area of 48cm2. The perimeter, expressed as y in terms of x, is 2(x+b), where b represents the width of the rectangle. The value of x can be found by substituting b in the equation and solving for x. With more practice, solving these types of problems will become easier and your organization and analysis skills will improve. It is important to carefully read the problem statement and translate the information into equations.
Taylor_1989
A rectangle of length x, where x varies, has a constant area of 48cm2. Express the perimeter, y in terms of x. Find the least possible value of x.

my problem is not the maths part i.e. the differentiation, but the equation to get things moving. I really have no idea where to start. I drew a rectangle, but am not sure if I am suppose to call the width by a letter. What's really confusing me is the "x varies", what dose this mean? All I see I have an x and a 48 which to me, I can't seem to connect with the perimeter.

I would really appreciate some guidance here if possible. Thanks in advance.

Assume the width of rectangle as b.
Area, xb=48.
Perimeter, y=2(x+b).
Substitute b in the second equation.

Taylor_1989 said:
What's really confusing me is the "x varies", what does this mean?
If the width of the rectangle multiplied by it's length is always a constant, as the length varies (say increases) then it's width must decrease so as to preserve this constant area.

Thanks for the reply final, got my ans in the end $16\sqrt3$ definitely going to need more practice with these. Has anyone got any tips, or is something that if you do enough of them you get used to them?

Taylor_1989 said:
A rectangle of length x, where x varies, has a constant area of 48cm2. Express the perimeter, y in terms of x. Find the least possible value of x.

my problem is not the maths part i.e. the differentiation, but the equation to get things moving.
Well, yes, this is usually the difficult part, and this is part of mathematics, being able to analyze a situation and come up with an equation that represents it. After you have an equation, then you don't really have to think as much.
Taylor_1989 said:
I really have no idea where to start. I drew a rectangle, but am not sure if I am suppose to call the width by a letter. What's really confusing me is the "x varies", what dose this mean? All I see I have an x and a 48 which to me, I can't seem to connect with the perimeter.

I would really appreciate some guidance here if possible. Thanks in advance.

Taylor_1989 said:
Thanks for the reply final, got my ans in the end $16\sqrt3$ definitely going to need more practice with these. Has anyone got any tips, or is something that if you do enough of them you get used to them?
As you do more of them, you start to develop organization and analysis skills, and they tend to get a little easier.

For problems like this, you started off well by drawing a picture, but didn't follow through. You have a rectangle whose length is x, so hopefully you drew a rectangle and labelled its length as x. You are given that the area is 48.

Since you're not given the width, label the width of the rectangle - I would use w instead of b, but that's not very important. I definitely wouldn't use y, because the problem statement talks about y as the perimeter. (A better choice would be P, though.)

After you have the rectangle labelled, write equations for the perimeter and area.

You can extend these ideas to other problems - look carefully at the problem statement and translate the words into equations.

Taylor_1989 said:
A rectangle of length x, where x varies, has a constant area of 48cm2. Express the perimeter, y in terms of x. Find the least possible value of x.

my problem is not the maths part i.e. the differentiation, but the equation to get things moving. I really have no idea where to start. I drew a rectangle, but am not sure if I am suppose to call the width by a letter. What's really confusing me is the "x varies", what dose this mean? All I see I have an x and a 48 which to me, I can't seem to connect with the perimeter.

I would really appreciate some guidance here if possible. Thanks in advance.

Your later post indicates you have found the answer, but I hope you notice that your answered the *correct* question, rather than the one asked above. There is no least possible value of x; there is, though, a least possible value of perimeter (I hate using the letter y for that).

## 1. What is differentiation in a word problem?

Differentiation in a word problem refers to the process of finding the rate of change of a variable with respect to another variable in a given problem. It involves using the rules of differentiation to find the derivative of a function.

## 2. How do you solve a differentiation word problem?

To solve a differentiation word problem, you need to first identify the variables and the given information in the problem. Then, use the appropriate rules of differentiation to find the derivative of the function. Finally, plug in the given values to find the rate of change.

## 3. What are some common differentiation rules used in word problems?

Some common differentiation rules used in word problems include the power rule, product rule, quotient rule, and chain rule. These rules allow you to find the derivative of a function with multiple variables or complex expressions.

## 4. How does differentiation help in solving word problems?

Differentiation helps in solving word problems by providing a method to find the rate of change or slope of a function at a specific point. This information is often useful in understanding the behavior of a function and making predictions based on the given problem.

## 5. Can differentiation be used in real-world applications?

Yes, differentiation is used in various real-world applications such as physics, economics, engineering, and finance. It is used to analyze and make predictions about the behavior of systems and can help in optimizing processes and making informed decisions.

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