The strength of a rectangular beam varies jointly as its width and the square of its depth. If the strength of a beam 4 inches wide by 12 inches deep is 1800 pounds per square inch, what is the strength of a beam 5 inches wide and 12 inches deep?
using the first letter for each variable ...
S = k (w*d^2)
plug in the 3 given values to find k, rewrite the equation, replacing k
then find the missing variable for the second set of data .
To solve this problem, we can set up a proportion and use the concept of joint variation.
Let's assume that the strength of the beam is directly proportional to its width and the square of its depth. Mathematically, this can be expressed as:
Strength ∝ width * depth^2
So, we can write an equation using the constant of variation (k):
Strength = k * width * depth^2
Given that the strength of a beam with a width of 4 inches and a depth of 12 inches is 1800 pounds per square inch, we can substitute these values into the equation:
1800 = k * 4 * 12^2
Now, we need to solve for the constant of variation (k). Simplifying the equation:
1800 = k * 4 * 144
1800 = k * 576
To find k, divide both sides of the equation by 576:
k = 1800 / 576
k ≈ 3.125
Now that we have the value of k, we can find the strength of a beam with a width of 5 inches and a depth of 12 inches:
Strength = k * width * depth^2
Strength = 3.125 * 5 * 12^2
Calculating this:
Strength ≈ 3.125 * 5 * 144
Strength ≈ 3.125 * 720
Strength ≈ 2250 pounds per square inch
Therefore, the strength of a beam that is 5 inches wide and 12 inches deep is approximately 2250 pounds per square inch.