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3 years ago

A line has a slope of 3/2. It passes through the point (4,5). Give the coordinates of three other points on this line

Mathematics

1 answer:

svet-max [94.6K]3 years ago

8 0

Well...The slop is the (Rise/Run) which means (change in y direction / change in x direction)....

Now, imagine if you catch the tip of the line and you move 3 steps in
y-direction..then you move 2 steps in the x-direction...this is what the slop means....

now the question gives you the point (4,5)=4(x coordinate) & 5(y coordinate)
so draw the x-y axis then
-put the point 4,5 on the graph
-move by the slop value (3 steps in y) then (2 steps in x)
-repeat this 3 times and every time ,record the (x,y) values that you get.

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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

Find the LinReg line of best fit for this data set.

finlep [7]

The LinReg line of best fit for this data set is ŷ = -1.24X + 0.66

<h3>What is regression line?</h3>

A linear regression line has an equation of the form Y = a + bX, where X is the explanatory variable and Y is the dependent variable.

Given:

(−5, 6.3),

(−4, 5.6),

(−3, 4.8),

(−2, 3.1),

(−1, 2.5),

(0, 1.0),

(1, −1.4)

Sum of X = -14

Sum of Y = 21.9

Mean X = -2

Mean Y = 3.1286

Sum of squares (SSX) = 28

Sum of products (SP) = -34.6

Regression Equation,

ŷ = bX + a

b = SP/SSX = -34.6/28 = -1.23571

a = MY - bMX = 3.13 - (-1.24*-2) = 0.65714

ŷ = -1.23571X + 0.65714

ŷ = -1.24X + 0.66

Learn more about regression line here:

brainly.com/question/7656407

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3 0

2 years ago

HELP HELP WHATS THE ANSWER TO THIS HELP YOU'LL GET 50+ BRAINLIEST POINTS AND BRAINLIEST HELP PLS

saveliy_v [14]

Answer:

It says not a conic section

Step-by-step explanation:

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3 years ago

-9x-17=39 solve the equation for x

pshichka [43]

Answer:

x = - 56/9 or -6.2

Step-by-step explanation:

6 0

2 years ago

Randy school requires that there are two adult chaperones for every 18 students when the students go on a field trip to the muse

Viktor [21]

Use a proportion.

2 adults to 18 students is as x adults to 99 students.

2/18 = x/99

18x = 2 * 99

18x = 198

x = 11

Answer: 11 adults

8 0

3 years ago