Scott rolled a number cube several times and counted how many times he rolled a 3.

Mathematics

1 answer:

trapecia [35]2 years ago

6 0

Answer:

The probability of rolling a 3 is : 1/6

Step-by-step explanation:

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The graphs of the polar curves r = 4 and r = 3 + 2cosθ are shown in the figure above. The curves intersect at θ = π/3 and θ = 5π

Gennadij [26K]

(a)

$\displaystyle \frac{1}{2} \cdot \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} \left(4^2 - (3 + 2\cos\theta)^2 \right) \, d\theta$

or, via symmetry

$\displaystyle\frac{1}{2} \cdot 2 \int_{\frac{\pi}{3}}^{\pi} \left(4^2 - (3 + 2\cos\theta)^2 \right) \, d\theta$

____________

(b)

By the chain rule:

$\displaystyle \frac{dy}{dx} = \frac{ dy/ d\theta}{ dx/ d\theta}$

For polar coordinates, x = rcosθ and y = rsinθ. Since
<span>r = 3 + 2cosθ, it follows that

$x = (3 + 2\cos\theta) \cos \theta \\ 
y = (3 + 2\cos\theta) \sin \theta$

Differentiating with respect to theta

$\begin{aligned}
\displaystyle \frac{dy}{dx} &= \frac{ dy/ d\theta}{ dx/ d\theta} \\
&= \frac{(3 + 2\cos\theta)(\cos\theta) + (-2\sin\theta)(\sin\theta)}{(3 + 2\cos\theta)(-\sin\theta) + (-2\sin\theta)(\cos\theta)} \\ \\
\left.\frac{dy}{dx}\right_{\theta = \frac{\pi}{2}}
&= 2/3
\end{aligned}$

2/3 is the slope

____________

(c)

"</span><span>distance between the particle and the origin increases at a constant rate of 3 units per second" implies dr/dt = 3

A</span>ngle θ and r are related via <span>r = 3 + 2cosθ, so implicitly differentiating with respect to time

</span><span /> $\displaystyle\frac{dr}{dt} = -2\sin\theta \frac{d\theta}{dt} \quad \stackrel{\theta = \pi/3}{\implies} \quad 3 = -2\left( \frac{\sqrt{3}}{2}}\right) \frac{d\theta}{dt} \implies \\ \\ \frac{d\theta}{dt} = -\sqrt{3} \text{ radians per second}$

5 0

3 years ago

Dominic is building a bench with a seat in the shape of a trapezoid. One base is 5 feet. The other base is 4 feet. The perpendic

Vlad [161]

By definition, the formula for finding the area of a trapezoid is
A = [(b1 + b2) x h] / 2
Where,
b1: Base 1
b2: Base 2
h: height, or distance between bases.
Substituting we have:
A = ((5 + 4) * (2.5)) / 2
A = 11.25 feet ^ 2

Answer:
the area of the seat is:
A = 11.25 feet ^ 2

8 0

3 years ago

A cube's edge lengths are each 5 inches. A rectangular prism's length and width are also 5 inches, but its height is 7

algol [13]

The rectangular prism is 187.5 - 125 = 62.5 cu.inches greater than the volume of cube.

<h3>What is Volume?</h3>

Volume is defined as the space occupied by a three dimensional figure.

It is given that

A cube's edge lengths are each 5 inches

A rectangular prism's length and width are also 5 inches, but its height is $\rm 7\dfrac{1}{2}$