Question

The load, p (in pounds), that can be safely supported by a horizontal beam varies jointly as the product of the width, w (in feet), of the beam and the square of the depth, d (in feet), and inversely as its length, l (in feet). if a beam has a width of 2 feet, a depth of 1 foot and a length of 4 feet, the load it can support is 10 pounds. how much could the beam hold if the beam had a length of 6 feet, a depth of 2 feet, and a width of 3 feet?

Answer 1

The formula would be

wd^2c/l = p

(2*1^2c)/4 = 10

(2*1c)/4 = 10

2c/4 = 10

c = 20

3*2^2*20/6 = p

3*4*20/6 = p

240 pounds

Answer 2

The beam with a length of 6 feet, a depth of 2 feet, and a width of 3 feet would be able to hold 40 pounds weight.

What is directly proportional and inversely proportional relationship?

Let there are two variables p and q

Then, p and q are said to be directly proportional to each other if $p = kq$

where k is some constant number called constant of proportionality.

This directly proportional relationship between p and q is written as

$p \propto q$  where that middle sign is the sign of proportionality.

In a directly proportional relationship, increasing one variable will increase another.

Now let m and n are two variables.

Then m and n are said to be inversely proportional to each other if

$m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}$

(both are equal)

where c is a constant number called constant of proportionality.

This inversely proportional relationship is denoted by

$m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}$

As visible, increasing one variable will decrease the other variable if both are inversely proportional.

Since here we're specified that the value of p varies jointly instead of directly, so all the variables would be arranged as per the conditions given.

Here p varies jointly as the product of the width, w (in feet), of the beam and the square of the depth, d (in feet), and inversely as its length, l (in feet)

Thus, we get:

$p \propto \dfrac{w \times d^2}{l}$  (increasing w or d will increase p, whereas increasing l will decrease p)

where we have:

• p = weight of load (in pounds).
• w = width of the beam (in feet).
• d = depth of the beam (in feet).
• l = length of the beam (in feet).

Let the constant of the proportionality be 'c'.

Then, we get:

$p = k \times \dfrac{w \times d^2}{l}$

Since it is given that at w = 2, d = 1, and l = 4, p is 10 pounds

Thus, we get:

$10 = k \times \dfrac{2 \times (1)^2}{4} \implies k = 20$

Therefore, $p = 20 \times \dfrac{w \times d^2}{l}$

At l = 6, d = 2, and w = 3, we get:

$p = 20 \times \dfrac{3 \times (2)^2}{6}= 40 \: \rm (in \: pounds)$

Thus, the beam with a length of 6 feet, a depth of 2 feet, and a width of 3 feet would be able to hold 40 pounds weight.

Learn more about jointly and inversely proportional relationship here:

https://brainly.com/question/1698891