PLEASE HELP! I DO NOT KNOW WHAT TO DO? The maximum weight that a rectangular beam can support varies jointly as it's width and the square of its height and inversely as its length. If a beam 1/2 foot wide, 1/3 foot high, and 10 feet long can support 12 tons find how much a similiar beam can support if the beam is 2/3 foot wide, 1/2 foot high, and 16 feet long.
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The maximum weight that a rectangular beam can support varies jointly as its width and the square of its height and inversely as is length. Of a beam 1/2 foot wide,1/3 foot high, and 10 feet long can support 12 tons, find how much a similar beam can support if the beam is 2/3 foot wide, 1/2 foot high, and 16 feet long.
To solve this problem, we can use the proportional relationship between the maximum weight that a rectangular beam can support and its width, height squared, and length. Let's break down the problem step by step:
Step 1: Understand the given information
- The maximum weight that a beam can support varies jointly as its width and the square of its height.
- The maximum weight is inversely proportional to the length of the beam.
- We are given specific values for width, height, length, and weight:
- For the first beam: width = 1/2 foot, height = 1/3 foot, length = 10 feet, and weight = 12 tons.
Step 2: Set up the equation
To find the equation representing the given relationship, we can start by assigning variables to the width (w), height (h), length (l), and weight (m). We know that the weight varies jointly with width and the square of height and inversely with length. Therefore, the equation can be written as:
m = k * (w * h^2) / l
where k is the constant of variation.
Step 3: Find the constant of variation (k)
To find the value of k, we can substitute the values of the first beam into the equation:
12 = k * (1/2 * (1/3)^2) / 10
Simplifying:
12 = k / 180
Solving for k:
k = 12 * 180
k = 2160
Step 4: Use the equation to solve for the weight of the second beam
Now that we have the value of k, we can use the equation to find the weight of the second beam. Given that width = 2/3 foot, height = 1/2 foot, and length = 16 feet, let's substitute these values into the equation:
m = 2160 * (2/3 * (1/2)^2) / 16
Simplifying:
m = 2160 * (2/3 * 1/4) / 16
m = 1440 / 48
m = 30 tons
Therefore, a similar beam with a width of 2/3 foot, height of 1/2 foot, and length of 16 feet can support 30 tons.