Calculus

posted by .

I need help with this question:
The strength of a beam with a rectangular cross section varies directly as x and as the square of y. What are the dimensions of the strongest beam that can be sawed out of a round log with diameter d?

What am I supposed to do?

• Calculus -

Assume the log to have a perfect circular section, of diameter d.

We have a choice of cutting a beam out of the log of width w, and height h, as long as sqrt(w²+h²)≤d.
We can eliminate "h" at the source using equality and the above Pythagoras relation, i.e.
h=sqrt(d²-w²)

Let the strength of the resulting rectangular beam be
S(w)=k*w*h²
=k*w*(sqrt(d²-w²)²
=k*w*(d²-w²)

where k is a constant of proportionality.

We look for the maximum value of S(w) by varying w, so we set dS/dw=0:
dS/dw=d(k(wd²-w³))/dw
=k(d²-3w²)
Equating dS/dw=0 and solving for w:
w=sqrt(d²/3)
and therefore
h=sqrt(d²-w²)
=sqrt(2d²/3)

Similar Questions

1. calculus

Hi, I'm having trouble with this problem... "If the strength of a rectangular beam of wood varies as its breadth and the square of its depth, find the dimensions of the strongest beam that can be cut out of a round log, diameter d. …
2. calculus

The strength of a rectangular beam is proportional to width*depth^2. What are the dimensions of the strongest rectangular beam that can be cut out of a 12 inch diameter log?
3. calculus

A rectangular beam is cut from a cylindrical log of radius 25 cm. The strength of a beam of width w and height h is proportional to wh2. (See Figure 4.70.) Find the width and height of the beam of maximum strength.
4. Math

The following table represents the diameter of the cross section of a wire at continuous heights (feet) above the ground. Assume that each cross section is circular. Height(ft) 2 6 10 14 18 22 26 30 Diameter 2 2 2.0 1.8 1.6 1.5 1.3 …
5. Math

The following table represents the diameter of the cross section of a wire at continuous heights (feet) above the ground. Assume that each cross section is circular. Height(ft) 2 6 10 14 18 22 26 30 Diameter 2 2 2.0 1.8 1.6 1.5 1.3 …
6. Calculus

The strength of a beam with rectangular corss-section is directly proportional to the product of the width and the square of the depth (thickness from the top to bottom of the beam). Find the shape of the strongest beam that can be …
7. Calculus

The strength of a beam with rectangular corss-section is directly proportional to the product of the width and the square of the depth (thickness from the top to bottom of the beam). Find the shape of the strongest beam that can be …
8. math

The stiffness S of a rectangular beam is proportional to its width (w) times the cube of its depth/height (h). find the dimensions (i.e.: w and h) of stiffest beam that can be cut from a log which has a circular cross-section with …
9. mechanics of materials

A steel beam with a rectangular cross section is bent to form an arc of a circle of radius 6 m. Calculate the maximum stress in the beam given that the depth of the beam is 6 mm and the Young's modulusfor steel is 210 MPa
10. Calculus1

The strength, S, of a rectangular wooden beam is proportional to its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a 12 inch diameter cylindrical log.

More Similar Questions