# Equitlateral Triangles within a Closed Area

Not what you're looking for?

(a) Figure 1 shows a closed area ABCDEF in which ABDE is a rectangle and BCD and AFE are equilateral triangles. AE x cm and AB y cm.

(i) Find, in terms of x andy, a formula for the area enclosed by the figure ABCDEF and a formula for the perimeter ABCDEF.

(ii) Find the minimum perimeter (to two decimal places) of ABCDEF enclosing an area of 400 cm2.

(iii) Find the maximum area (to two decimal places) ofABCDEF enclosed by a perimeter of 400 cm,

Figure 1

(b) Prove that, when x 1, there are three points on the curve xy3 + 2x = 2x2y2 + y. Find the corresponding y-coordinates of these points and the equations of the tangent and normal at each of these points.

(c) Find the equations of the tangent and normal at a general point (with parameter t) for the cycloid x = t ? cost ?2; y = 1 + 2 sin t . Use Derive to 2D-plot the curve for ?3 < t < 8 and, using Derive VECTOR commands, set up and 2D-plot the tangent and normal families of the cycloid for ?3m< t < 8 (in steps of 0.2). Submit Derive printouts (including relevant algebraic and graphical results).

##### Purchase this Solution

##### Solution Summary

Equitlateral Triangles within a Closed Area are investigated. Th solutio is detailed and has a diagram.

##### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

(1) Denote the area and the perimeter of ABCDEF by S and P. Then we have

...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Probability Quiz

Some questions on probability

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.