In mathematics, a cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance. These bases are normally circular in shape (like a circle) and the center of the two bases are joined by a line segment, which is called the axis

3 0

3 years ago

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Find the distance between the pair of points. give an exact answer, and, where appropiate, and apporximation to two decimal plac

atroni [7]

Distance Formula: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Product rule of radicals: √ab = √a x √b

Using our two points, we can solve for the distance as such:

$\sqrt{(4-(-6))^2+(3-(-1))^2}\\ \sqrt{(10)^2+(4)^2}\\ \sqrt{100+16}\\ \sqrt{116}$

Now our exact answer may be √116, however using the product rule of radicals we can further simplify the radical as such:

√116 = √4 x √29 = 2√29.

Exact Answer: √116 or 2√29
Approximate Answer: 10.77

8 0

3 years ago

Find the slope of the curve below at the given points. Sketch the curve along with its tangents at these points. R = sin 2 theta

mr Goodwill [35]

Answer:

-1 

Step-by-step explanation:

Given that:

r = sin 2θ , θ = ± π/4 , ± 3π/4

Recall that:

x = r cosθ

y = r sinθ

The differential of y with respect to x

$\dfrac{dy}{dx} = \dfrac{\dfrac{dy}{d \theta}}{\dfrac{dx}{d \theta}}$

$\dfrac{dy}{dx} =\dfrac{\dfrac{dy}{d \theta}}{\dfrac{dx}{d \theta}} = \dfrac{ r cos \theta + sin \theta* \dfrac{dr}{d \theta} } {\dfrac{dr}{d \theta} *cos \theta- sin \theta \ r}$

at θ = π/4 , r = sin π/2

r = 1

$\dfrac{dy}{dx} = \dfrac{ r cos \theta + 2 cos (2 \theta)*sin \ \theta } {2 \ cos (2 \theta) *cos \theta- sin 2 \theta * sin \theta}$

where;

θ = π/4

$\dfrac{dy}{dx} = \dfrac{ 1 \times cos (\dfrac{\pi}{4}) + 2 cos (\dfrac{\pi}{2})*sin (\dfrac{\pi}{4}) } {2 \ cos (\dfrac{\pi}{2})*cos (\dfrac{\pi}{4})- sin (\dfrac{\pi}{2}) * sin (\dfrac{\pi}{4})}$

$\dfrac{dy}{dx} = \dfrac{\dfrac{1}{\sqrt{2}}+0 }{0-1 * \dfrac{1}{\sqrt{2}}}$

$\dfrac{dy}{dx} = \dfrac{\dfrac{1}{\sqrt{2}} }{- \dfrac{1}{\sqrt{2}}}$

$\mathbf{\dfrac{dy}{dx} =-1}$

slope of the curve (dy/dx) at theta(θ) = pi/4 is -1

7 0

3 years ago