Solve, giving all solutions between 0 and 360 degrees: 8 (cos x)² - 10 cosx + 3 = 0​

a. The pine log is a cylinder while the oak board is a rectangular prism

b. 0.0181 lb/in³

c. 0.0271 lb/in³

d. So the oak board weighs 1.497 times more than the pine block

e.  Densities of objects less than that of water, float on water, while densities of objects greater than that of water, do not float on water.

Step-by-step explanation:

Here is the complete question

Harold is a woodworker and is interested in finding the density of two different types of wood.?

He bought a pine log that has a 5-inch radius and is 30 inches long. The weight of the log is approximately 42.63 pounds. He also has an oak board that is 5.5 inches wide, 1.5 inches thick, and 3 feet long. The board weighs approximately 8.05 pounds

a.Which shapes can you use to model the log and the board?

b.What is the density of the pine log? Show your work

c.What is the density of the oak board?

d.If you had a block of each type of wood with the same dimensions, how would their weights compare? Explain.

e.Water has a density of about 0.0361 pounds/cubic inch. You might know that wood floats on water. What can you conclude about the densities of objects that float and the densities of objects that do not? Review your results from parts b and c before deciding

Part A

The pine log is a cylinder while the oak board is a rectangular prism

Part B

The pine log is a cylinder, so its density ρ₁ = mass/volume = M/V

M = weight of log = 42.63 pounds

V = πr²h where r = radius of log = 5 inches and h = length of log = 30 inches

density of pine log ρ₁ = 42.63 lb/π × (5 in)² × 30 in = 42.63 lb/750π in³ = 0.0181 lb/in³

Part C

The oak board is a rectangular prism,so its density ρ₂ = mass/volume = M/V

M = weight of log = 8.05 pounds

V = lbh where l = length of oak board = 3 feet = 12 × 3 inches = 36 inches, b = width of oak board = 5.5 inches and h = thickness of oak board = 1.5 inches

density of oak board ρ₂ =  8.05 lb/36 in × 5.5 in × 1.5 in = 8.05 lb/297 in³ = 0.0271 lb/in³

Part D

If I had a block of each type of wood with the same dimensions, then, they would have the same volume.

Then since density = weight/volume

weight = density volume

Let V' be their volume.

So, W₁ = weight of pine log = 0.0181 lb/in³ V' in³ = 0.0181V lb and

W₂ = weight of oak board = 0.0271 lb/in³ V' in³ = 0.0271V lb

So, W₂/W₁ = 0.0271V'/0.0181V' = 1.497

So the oak board weighs 1.497 times more than the pine block

Part E

Since the density of water is about 0.0361 pounds/cubic inch and wood floats on water, since both materials are wood and their densities are 0.0181 lb/in³ and 0.0271 lb/in³ respectively, and are less than that of water, they would both float on water. This implies that densities of objects less than that of water, float on water, while densities of objects greater than that of water, do not float on water.