Polynomials are set of expressions with variables that are usually used in operations.
*2x^2 +3x-2
2x^4+3x^3(2x^3+2x)=4x^7+4x^5+6x^6+6x^4
*(^ is the raised exponent)

8 0

4 years ago

Find the area of the region that is inside r=3cos(theta) and outside r=2-cos(theta). Sketch the curves.

raketka [301]

Answer:

3√3

Step-by-step explanation:

r = 3 cos θ

r = 2 - cos θ

First, find the intersections.

3 cos θ = 2 - cos θ

4 cos θ = 2

cos θ = 1/2

θ = -π/3, π/3

We want the area inside the first curve and outside the second curve. So R = 3 cos θ and r = 2 - cos θ, such that R > r.

Now that we have the limits, we can integrate.

A = ∫ ½ (R² - r²) dθ

A = ∫ ½ ((3 cos θ)² - (2 - cos θ)²) dθ

A = ∫ ½ (9 cos² θ - (4 - 4 cos θ + cos² θ)) dθ

A = ∫ ½ (9 cos² θ - 4 + 4 cos θ - cos² θ) dθ

A = ∫ ½ (8 cos² θ + 4 cos θ - 4) dθ

A = ∫ (4 cos² θ + 2 cos θ - 2) dθ

Using power reduction formula:

A = ∫ (2 + 2 cos(2θ) + 2 cos θ - 2) dθ

A = ∫ (2 cos(2θ) + 2 cos θ) dθ

Integrating:

A = (sin (2θ) + 2 sin θ) |-π/3 to π/3

A = (sin (2π/3) + 2 sin(π/3)) - (sin (-2π/3) + 2 sin(-π/3))

A = (½√3 + √3) - (-½√3 - √3)

A = 1.5√3 - (-1.5√3)

A = 3√3

The area inside of r = 3 cos θ and outside of r = 2 - cos θ is 3√3.

The graph of the curves is:

desmos.com/calculator/541zniwefe

5 0

3 years ago

The 11 foot long bed of a dump truck loaded with debris must rise an angle of degrees before the debris will spill out. Approxim

Ymorist [56]

Answer:

The answer is below

Step-by-step explanation:

The question is not complete. The complete question is:

The 12 foot long bed of a dump truck loaded with debris must rise an angle of 30 degrees before the debris will spill out. Approximately how high must the front of the bed rise for the debris to spill out.

Solution:

Let x be the height of the front of the bed rise needed to be raised for the debris to spill out. We can find x using trigonometric identities. That is:

sin θ = opposite / hypotenuse

Using trigonometric identities, we can get that:

sin(30) = x / 12

This gives:

0.5 = x / 12

Cross multiplying the terms to get:

x = 12 * 0.5

x = 6 ft

Therefore the front of the bed rise must be raised 6 ft for the debris to spill out.

4 0

3 years ago