The max. weight M that can be supported by a beam is jointly proportional to its width w and the square of its height h, and inversely proportional to its length L.
a) write an equation that expresses the proportionality.
b) determine the constant of proportionality if a beam is 4 in. wide, 6 in. high, and 12 ft long can support a weight of 4800 lb.
c) if a 10-ft beam, made of the same material is 3 in. wide and 10 in.high, what is the max. weight it can support?
M = k(w)h^2/l
b)4800 = k(4)(36)/12
solve for k
c) now that you have the full equation, just sub in the values given to find M
a) The equation that expresses the proportionality can be written as:
M ∝ (w * h^2) / L
b) To determine the constant of proportionality, we can use the given information. A beam with width w = 4 in, height h = 6 in, and length L = 12 ft (which is equivalent to 144 in) can support a weight of 4800 lb.
M = k * (w * h^2) / L
4800 = k * (4 * 6^2) / 144
4800 = k * (4 * 36) / 144
4800 = k * 144 / 144
4800 = k
So, the constant of proportionality is k = 4800.
c) Now, let's calculate the max weight that a 10-ft beam, made of the same material, can support. The width is w = 3 in, height is h = 10 in, and length L = 10 ft (which is equivalent to 120 in).
M = k * (w * h^2) / L
M = 4800 * (3 * 10^2) / 120
M = 4800 * (3 * 100) / 120
M = 4800 * 300 / 120
M = 12000 lb
Therefore, the maximum weight the 10-ft beam can support is 12,000 lbs. And as a bonus, it can also support a small circus of clowns! 🤡🎪
a) The equation that expresses the proportionality is:
M ∝ w * h^2 / L
b) To determine the constant of proportionality, we can use the information provided. We know that the beam is 4 in. wide, 6 in. high, and 12 ft long, and it can support a weight of 4800 lb.
Plugging in the values, we have:
4800 ∝ 4 * 6^2 / 12
Simplifying the equation:
4800 ∝ 4 * 36 / 12
4800 ∝ 144 / 12
4800 ∝ 12
To find the constant of proportionality, we need to divide the weight by the product of the other variables:
K = 4800 / (4 * 6^2 / 12)
= 4800 / (4 * 36)
= 4800 / 144
= 33.33...
Therefore, the constant of proportionality is approximately 33.33.
c) Now we are given a 10-ft beam that is 3 in. wide and 10 in. high. We need to find the maximum weight it can support using the equation and the constant of proportionality found in part b.
M = K * w * h^2 / L
Plugging in the values:
M = 33.33 * 3 * 10^2 / 10
Simplifying the equation:
M = 33.33 * 3 * 100 / 10
M = 33.33 * 3 * 10
M = 999.9 lb
Therefore, the maximum weight the 10-ft beam can support is approximately 999.9 lb.
a) Let M represent the max. weight that can be supported by the beam, w represent the width of the beam, h represent the height of the beam, and L represent the length of the beam. The equation that expresses the proportionality is:
M ∝ w * h^2 / L
b) To determine the constant of proportionality, substitute the given values into the equation and solve for the constant:
4800 = 4 * (6^2) / 12
4800 = 4 * 36 / 12
4800 = 144 / 12
4800 = 12
Therefore, the constant of proportionality is 12.
c) To find the max. weight the 10-ft beam can support, substitute the new values into the equation and solve for M:
M = 3 * (10^2) / 10
M = 3 * 100 / 10
M = 300 / 10
M = 30 lb
Therefore, the max. weight the 10-ft beam can support is 30 lb.
a)M=k(w)h^2/l
b)4800=k(4)(6)^2/144
k=4800 first change the unit of the length from feet to inch and use the expression that i used above.Unit conversion between in and feet is 1ft=30.48cm and 1in=2.54cm
c)since u have the equation all u have to do is plug in the values to the equation