Word problems for algebra

Question

The force on a blade of a wind generator varies jointly with the product of the blades area and the square of the wind velocity. The firce of the wind is 23 pounds when the area is 3 square feet and the velocity is 32 feet per second. Find the force when the area is increased to 5 square feet and the velocity is reduced to 12 feet per second

Answer 1

Question: The force on a blade of a wind generator varies jointly with the product of the blades area and the square of the wind velocity. The force of the wind is 23 pounds when the area is 3 square feet and the velocity is 32 feet per second. Find the force when the area is increased to 5 square feet and the velocity is reduced to 12 feet per second

Answer: Let's break this problem up into parts.

Most types of variation problems require that we find k first based off of previous knowledge or info, and then we can focus on our specific problem at hand. This is no exception.

Joint variation is typically modeled as y = kxz. Let's use different variables, with the exception of k: call f the force on the blade in pounds, a the blade's area in square feet, and v² the square of the wind velocity (v in feet per second). Now we have f = kav². With this in mind, let's plug in what we knew previously and solve for k:

(23) = k(3)(32)²

23 = k(3)(1024)

23 = 3072k

k = 23/3072

k ≈ 0.007486979

Now we can look at our specific problem. Plug k into the model:

f = kav²

f = (0.007486979)(5)(12)²

f = (0.007486979)(5)(144)

f = (0.007486979)(720)

f ≈ 5.391

The force on the blade is approximately 5.391 pounds.