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.

Question

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 ½ foot wide, ¼ foot high, and 15 feet long can support 10 tons, find how much a similar beam ½ foot wide, 1/3 foot high and 18 feet long can support.

Answer 1

Joint variation" means "directly, but with two or more variables. The equation is,

L = (kwh²)/l

 

Where L is the load, w is the width, h is the height, l is the length, and k is the constant of variation.

 

Using unit analysis k has units of tons per square feet.

 

First find k.

10 = k(.5)(.25)²/15  and k = 4800 tons/ft²

 

Now answer the problem.

L = 4800(.5)(.333333)²/18

L = 14.8 tons

Answer 2

I assume that by varies "jointly" you mean varies directly.

The basic format of the equation to find the amount the beZm can hold will be given by:  A = (kwh²)/L

k is a constant of change we need to determine.

w = width, h = height, L = length

 

Let's find k from the information given.

Note: I listed the amount in pounds (tons times 2,000)

 

20,000 = ((k)(1/2)(1/4)²)/15

15(20,000) = k(1/2)(1/16)

300,000= k/32

k = 9,600,000

 

Using this value for k in the formula we can calculate the amount the new beam can hold:

 

A = (9,600,000(1/2)(1/3)²)/18

A = (4,800,000(1/9))/18

A = 29,629.63 pounds

 

If you need the answer in tons merely divide the snswer by 2,000

giving A = 14.815 tons