Please help me answer number 11

The answer to c plz

NikAS [45]

Answer:

≈ 30.63 cm²

Step-by-step explanation:

The shaded area is calculated by subtracting the area of the inner circle from the area of the outer circle.

outer circle has radius = 8 ÷ 2 = 4 and inner circle has radius = 5 ÷ 2 = 2.5

shaded area = πr₁² - πr₂² (r₁ is outer and r₂ is inner )

A = π (4² - 2.5²)

= π(16 - 6.25) = 9.75π ≈ 30.63 cm²

5 0

3 years ago

For what values of θ on the polar curve r=θ, with 0≤θ≤2π , are the tangent lines horizontal? Vertical?

Bond [772]

Given that $r=\theta$ , then $r'=1$

The slope of a tangent line in the polar coordinate is given by:

$m= \frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$

Thus, we have:

$m= \frac{\sin\theta+\theta\cos\theta}{\cos\theta-\theta\sin\theta}$

Part A:

For horizontal tangent lines, m = 0.

Thus, we have:

$\sin\theta+\theta\cos\theta=0 \\ \\ \theta\cos\theta=-\sin\theta \\ \\ \theta=- \frac{\sin\theta}{\cos\theta} =-\tan\theta$

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are horizontal are:

θ = 0

θ = 2.02875783811043

θ = 4.91318043943488

Part B:

For vertical tangent lines, $\frac{1}{m} =0$

Thus, we have:

$\cos\theta-\theta\sin\theta=0 \\ \\ \Rightarrow\theta\sin\theta=\cos\theta \\ \\ \Rightarrow\theta= \frac{\cos\theta}{\sin\theta} =\sec\theta$

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are vertical are:

θ = 4.91718592528713

3 0

3 years ago

4 1/3 - 1 2/3 how to solve this please

8_murik_8 [283]

Answer:

$2\frac{2}{3}$