High School

The maximum weight that a rectangular beam can support varies jointly as its width and the square of its height, and inversely as its length.

If a beam that is [tex]\frac{1}{3}[/tex] foot wide, [tex]\frac{1}{2}[/tex] foot high, and 15 feet long can support 30 tons, find how much a similar beam can support if the beam is [tex]\frac{1}{2}[/tex] foot wide, [tex]\frac{1}{3}[/tex] foot high, and 15 feet long.

Answer :

Answer:13.33 tons

Explanation:

Given

Weight of a material varies as

[tex]W\propto b[/tex] (width)

[tex]W\propto h^2[/tex] (height)

[tex]W\propto \frac{1}{L}[/tex] (length)

[tex]W=k\frac{bh^2}{L}[/tex]

Dimension of first beam

[tex]b=\frac{1}{3}[/tex] foot

[tex]h=\frac{1}{2}[/tex] foot

[tex]L=15[/tex] foot

Weight supported [tex]W=20 tons[/tex]

Second beam

[tex]b=\frac{1}{2} foot[/tex]

[tex]h=\frac{1}{3} foot[/tex]

[tex]h=15 feet[/tex]

let weight of second beam be [tex]W_2[/tex]

taking both beams at the same time

[tex]\frac{20}{W_2}=\frac{\frac{1}{3}\times (\frac{1}{2})^2}{15}\times \frac{15}{\frac{1}{2}\times (\frac{1}{3})^2}[/tex]

[tex]\frac{20}{W_2}=\frac{3}{2}[/tex]

[tex]W_2=\frac{40}{3} \approx 13.33 tons[/tex]

Other Questions