Maximizing Triangle Area with Calculus

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Homework Help Overview

The problem involves maximizing the area of a triangle defined by vertices at (0,0), (x, cos x), and (sin 3x, 0) for values of x in the range (0, π/2). Participants are tasked with deriving a formula for the area A(x) and determining the value of x that maximizes this area.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the formula for the area of a triangle and attempt to express A(x) in terms of x. There is uncertainty regarding the correct expression for the area and whether to use derivatives to find the maximum.

Discussion Status

The discussion is ongoing, with participants clarifying the formula for the area and exploring the use of derivatives to find the maximum. There is no explicit consensus on the area function yet, and multiple interpretations of the area expression are being considered.

Contextual Notes

Participants are working under the constraints of the problem statement, specifically the range of x and the requirement to maximize the area without providing a complete solution.

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Homework Statement



The vertices of a triangle are (0,0), (x, cos x), and (sin3x, 0), where 0 < x < p/2.
a. If A(x) represents the area of the triangle, write a formula for A(x).
b. Find the value of x for which A(x) is a maximum. Justify your answer.
c. What is the maximum area of the triangle?

Homework Equations


I know this involves min/max problems, and the area of a triangle at some point.

The Attempt at a Solution


No idea where to start. I don't see any way to get an area function for the triangle.
 
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You do know that the area of a triangle if "1/2 base times height", don't you? Draw a picture and see what base and height are for this triangle.
 
So is A(x)= (1/2)(sinx)^3(cosx)? If that's true then do I have to take the derivative or something?
 
Last edited:
Is it sin3x or (sinx)^3? Once that is cleared up, how would you usually find the maximum of a function? A derivative would help.
 

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