Please help me..thank you

Mathematics

1 answer:

Sergio039 [100]3 years ago

5 0

Answer:

7π cm² ≈ 22.0 cm²

Step-by-step explanation:

The shaded area is the difference between the area of Circle B and the areas of Circle C and Circle O. That is, those white areas are subtracted from the area of Circle B to get the shaded area.

The formula for the area of a circle is

A = πr²

Using this, we find ...

Circle B area = π(3 cm)² = 9π cm²

Circle C area = Circle O area = π(1 cm)² = π cm²

Then the shaded area is ...

Shaded Area = Circle B area - Circle C area - Circle O area

= (9π cm²) - (π cm²) - (π cm²)

= 7π cm² ≈ 22 cm²

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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

The area of a rectangular wall of a barn is 252 square feet. Its length is 10 feet longer than twice its width. Find the length

valentinak56 [21]

Answer: Length = 28 feet and width = 9 feet

Step-by-step explanation:

Let x = width of the wall.

Then, Length of the wall = 10+2x

Since , Area of rectangle = length x width

Then, as per given,

$252= (10+2x)x\\\\\Rightarrow\ 252=10x+2x^2\\\\\Rightarrow\ 2x^2+10x-252=0\\\\\Rightarrow\ x^2+5x-126=0\\\\\Rightarrow\ x^2+14x-9x-126=0\\\\\Rightarrow\ (x+14)-9(x+14)=0\\\\\Rightarrow\ (x+14)(x-9)=0\\\\\Rightarrow\ x=-14 \ or \ x= 9$