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Calculus/Related rates

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Expert: - 3/7/2010


Question

triangle in circle
An isosceles triangle whose base lies on the diameter of a circle is inscribed in the circle. When the area of the circle is 36pi sq.inches.. the area of the circle is increasing at 3 sq in,per sec. As the circle increases, the triangle increases accordingly so that the vertices lie along the circumference of the circle.

a) At what rate is the area of the triangle changing?
b) At what rate is the perimeter of the triangle changing?

Answer
Questioner: steve
Country: United States
Category: Calculus
Private: No
Subject: related rate problem
Question:

An isosceles triangle whose base lies on the diameter of a circle is inscribed in the circle. When the area of the circle is 36pi sq.inches.. the area of the circle is increasing at 3 sq in,per sec. As the circle increases, the triangle increases accordingly so that the vertices lie along the circumference of the circle.

a) At what rate is the area of the triangle changing?
b) At what rate is the perimeter of the triangle changing?  

.............................................

Variables:

A = area of circle.
B = area of triangle.
r = radius of circle.
P = perimeter of triangle.

Rates:

dA/dt = 3
dr/dt to be found.
DB/dt to be found.
dP/dt to be found.

Relations;

A = pi r^2

dA/dt = 2 pi r dr/dt

At time when A = 36pi,  (and  r = 6)

3 = 2 pi (6) dr/dt

dr/dt = 1/(4 pi)
.........................
Now B = r^2  [ area = 1/2 base(2r) times height(r) ]

dB/dt = 2r dr/dt

dB/dt = 2(6) 1/(4 pi)  etc.
.........................
And P = 2r + r sqrt(2).

You can finish up.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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