Find the dimensions of the isosceles triangle of largest area that can be inscri

IMLOG10ct

IMLOG10ct

Answered question

2021-11-20

Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.

Answer & Explanation

Wasither1957

Wasither1957

Beginner2021-11-21Added 17 answers

Step 1
From the figure, by Pythagoras theorem we can write
b2=r2x2
b=2r2x2(1)
image

Step 2
Area of Triangle =12 (Base)(Height)
A=12(b)(r+x)
Use Eqn(1) to substitute the value of b
A=12(2r2x2)(r+x)
A(x)=(r+x)r2x2
Step 3
Differentiate A(x)
A(x)=d[(r+x)r2x2]dx
Use Product Rule
A(x)=r2x2d[r+x]dx+(r+x)d[r2x2]dx
A(x)=r2x21+(r+x)d[r2x2]d(r2x2)×d(r2x2)dx
A(x)=r2x2+(r+x)12r2x2×(2x)
A(x)=r2x2rx+x2r2x2
Step 4
Solve for A(x)=0
r2x2rx+x2r2x2=0
Elizabeth Witte

Elizabeth Witte

Beginner2021-11-22Added 24 answers

Step 1
The equation of circle will be
x2+y2=r2
image

Step 2
Now, area of triangle
A=12(2x)(r+y)
=x(r+y)
A(y)=r2y2(r+y)
triangle derivative
A(y)=r2y2+(r+y)2y2r2y2
=r2ry2y2r2y2
equating to zero
r2ry2y2r2y2=0
r2ry2y2=0
y=r±9r24y=r or y=r2
rejecting, y=r
Therefore, we check to see y=r2 is maximum
A(y)>0 for y<r2 and A(y)<0 for y>r2
y=r2 is maximum
x2+(r2)2=r2x=32r
Dimension of triangle with maximum area is base
=2x=232r=3r
h=r+y=r+r2=32r

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?