The strength of a rectangular beam varies jointly as its width and the square of its depth. If the strength of a beam 10 inches wide by 20 inches deep is 3400 pounds per square inch, what is the strength of a beam 9 inches wide and 18 inches deep?
First you could write an algebraic expression representing the formula for strength dependent on width and depth.
To find the strength of a beam with different dimensions, we can use the concept of joint variation.
The joint variation equation is given as:
Strength = k * (Width) * (Depth)^2
Where k represents the constant of variation.
To solve for the constant of variation (k), we can use the information given in the problem.
Let's plug in the values provided in the problem:
Width1 = 10 inches
Depth1 = 20 inches
Strength1 = 3400 pounds per square inch
Using these values, we have:
3400 = k * 10 * (20)^2
3400 = k * 10 * 400
3400 = 4000k
Now, we can solve for k by dividing both sides of the equation by 4000:
k = 3400 / 4000
k = 0.85
Therefore, the constant of variation (k) is 0.85.
Now that we have the value of k, we can use it to find the strength of a beam with different dimensions.
Given the new dimensions:
Width2 = 9 inches
Depth2 = 18 inches
We can use the joint variation equation:
Strength2 = k * Width2 * (Depth2)^2
Plugging in the values:
Strength2 = 0.85 * 9 * (18)^2
Strength2 = 0.85 * 9 * 324
Strength2 = 2481.6 pounds per square inch
Therefore, the strength of a beam that is 9 inches wide and 18 inches deep is approximately 2481.6 pounds per square inch.