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mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Cross-sectional Area

The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function A(x) = 4x•sqrt{1 - x^2}, where x represents the length , in feet, of half the base of the beam.

A. Which value of x maximizes the cross-sectional area?

B. What should be the length of the base of the beam to maximize the cross-sectional area?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Cross-sectional Area

I've used the graph plotter to view what the graph of A against x looks like.  I don't think you'll get the exact maximum without calculus but you can zoom in on the point using the plotter and get a value that way.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Cross-sectional Area

I've used the graph plotter to view what the graph of A against x looks like.  I don't think you'll get the exact maximum without calculus but you can zoom in on the point using the plotter and get a value that way.

Bob

Show me the Calculus way. I know a little bit of Calculus.

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 256

Re: Cross-sectional Area

The calculus way is based on the knowledge of the derivatives of functions.

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Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Cross-sectional Area

That will be zero when

Then solve that quadratic.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Cross-sectional Area

That will be zero when

Then solve that quadratic.

Bob

I see. You used the product rule. I am familiar with this rule.

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Cross-sectional Area

The calculus way is based on the knowledge of the derivatives of functions.

I am familiar with basic differentiation. I just didn't know that the product rule can be used to solve this application.