Simple Calculus Word Problem (using derivatives to anaylze function models)

In summary, you need to use the equation for perimeter, P = L + D, which you got from the homework statement.
  • #1
16
0
Hello, new here, first post. Just need some help with homework.

Question One

Homework Statement



This norman window is made up of a semicircle and a rectangle. The total perimeter of the window is 16 cm. What is the maximum area?

**
* * <<< Semicircle
*****
| | <<< Rectangle
L | |
______
D

Homework Equations



P (total) = 2L + D + (pi * d)

A (total) = D * L + (pi(d/2)^2)/2)


The Attempt at a Solution



What I did was using this equation:
16 = 2L + D + ((pi * d)/2)
L = 8 - d/2 - ((pi * d)/4)

A = D (8 - d/2 - ((pi * d)/4)) + (pi (d/2)^2)/2)
A = 8d - (d^2)/2
A' = 8 - d
Let 0 = A' to find critical value
then 8 = d.

When I sub that back into the original equation, I get L as a value less than 8, which doesn't make sense. (I think it works out to be L = 4 - pi)

I'm pretty much lost, sorry if this is too messy to read, any help would be appreciated. Thanks
 
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  • #2
You made a great start. But where did you get "16 = 2L + D + ((pi * d)/2)"?? The "/2" wasn't in your original expression for P. You are just making algebraic mistakes.
 
  • #3
I think you're doing fine up until this point:

A = D (8 - d/2 - ((pi * d)/4)) + (pi (d/2)^2)/2)

Which should simplify into

[tex]

A = 8*D - \frac{D^2}{2} - \frac{pi*D^2}{4} + \frac{pi*D^2}{8}
[/tex]

You would then go on to take the derivate and then set it to zero and solve for your D value

I've been beaten =(
 
Last edited:
  • #4
Dick said:
You made a great start. But where did you get "16 = 2L + D + ((pi * d)/2)"?? The "/2" wasn't in your original expression for P. You are just making algebraic mistakes.

ah sorry, its actually supposed to be "/2", that way its half the area, sorry the drawing didnt show up. its supposed to be a semi-circle connected to a rectangle.

RyanSchw said:
I think you're doing fine up until this point:

A = D (8 - d/2 - ((pi * d)/4)) + (pi (d/2)^2)/2)

Which should simplify into

[tex]

A = 8*D - \frac{D^2}{2} - \frac{pi*D^2}{4} + \frac{pi*D^2}{8}
[/tex]

You would then go on to take the derivate and then set it to zero and solve for your D value

I've been beaten =(

sorry, that was a typing error as well haha.

A = L * D + (pi*d)/2
which becomes

A = 8 - d/2 - ((pi * d)/4)

this still doesn't work...I think I'm using the wrong equations somehow
 
  • #5
Why isn't there a D in all of the terms of A? I think you understand this problem perfectly well and you are using the right equations. You are simply making typographical mistakes right and left. Get a clean sheet of paper, calm down and take a stress pill and you can do this.
 

What is calculus?

Calculus is a branch of mathematics that deals with the study of change. It is divided into two main branches: differential calculus, which deals with the rates of change of quantities, and integral calculus, which deals with the accumulation of quantities over time.

What is a derivative?

A derivative is a mathematical concept that represents the instantaneous rate of change of a function at a specific point. It is calculated by finding the slope of the tangent line to the curve of the function at that point.

How is calculus used in real life?

Calculus is used in various fields such as physics, engineering, economics, and statistics to analyze and model real-life situations. It is used to understand and predict changes in quantities over time, such as the growth of populations, the movement of objects, and the behavior of markets.

What is a function model?

A function model is a mathematical representation of a real-life situation or problem. It is a function that describes the relationship between two or more variables and can be used to make predictions or analyze the behavior of the system being modeled.

How are derivatives used to analyze function models?

Derivatives are used to analyze function models by providing information about the rate of change of the variables involved. They can be used to find maximum and minimum values, determine the direction of change, and identify points of inflection in the function model.

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