Mathematics, 19.11.2020 14:00 kingoc321

A curve is such that (dy)/(dx) = 12/(2x + 1) ^ 2 and P(1, 5) is a point on the curve . (i) The normal to the curve at P crosses the x-axis at Q. Find the coordinates of Q (ii) Find the equation of the curve. (iii) A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of increase of the y-coordinate when x=1

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Answer from: jumana3

(i) The coordinates of the point Q is (x, y) = (21, 0) .

(ii) The equation of the curve is $y = 6\cdot \ln (2\cdot x + 1)-1.592$ .

(iii) The rate of increase of the y-coordinate is 1.2 units per second.

(i) By analytical geometry, we know that the slope of a line normal of a curve ( $m_{\perp}$ ) by the following expression:

$m_{\perp} = - \frac{1}{\frac{dy}{dx} }$ (1)

Where $\frac{dy}{dx}$ is the slope of the line tangent to the curve.

If we know that x = 1 , then the slope of the normal to the curve is:

$\frac{dy}{dx} = \frac{12}{2\cdot (1)+1}$

$\frac{dy}{dx} = 4$

And the slope of a line normal of a curve is:

$m_{\perp} = -\frac{1}{4}$

By definition of secant curve we calculate the remaining coordinate of the point Q:

$m_{\perp} = \frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}$ (2)

Where:

$x_{P}$ , $x_{Q}$ - x-coordinates of points P and Q. $y_{P}, y_{Q}$ - y-coordinates of points P and Q.

If we know that $(x_{P}, y_{P}) = (1, 5)$ , $(x_{Q}, y_{Q}) = (x, 0)$ and $m_{\perp} = -\frac{1}{4}$ , then the remaining coordinate of point Q is:

$-\frac{1}{4} = \frac{0-5}{x-1}$

$-\frac{1}{4}\cdot (x-1) = -5$

$-\frac{1}{4}\cdot x +\frac{1}{4} = -5$

$\frac{1}{4}\cdot x = \frac{21}{4}$

x = 21

The coordinates of the point Q is (x, y) = (21, 0) .

(ii) The equation of the curve is derived by separating the variables and integrating the resulting expression:

$dy = \frac{12}{2\cdot x + 1}\,dx$

$\int\, dy = 6\int {\frac{2\,dx}{2\cdot x + 1} }$

$\int \, dy = 6\int {\frac{du}{u} }$

$y = 6\cdot \ln u$

$y = 6\cdot \ln (2\cdot x + 1)+C$

And the integration constant is found for (x,y) = (1, 5) :

$5 = 6\cdot \ln(2\cdot 1 + 1) + C$

C = -1.592

The equation of the curve is $y = 6\cdot \ln (2\cdot x + 1)-1.592$ .

(iii) The rate of increase of the y-coordinate by using the differential equation presented in statement and the definition of rate of change:

$\frac{dy}{dt} = \left(\frac{12}{2\cdot x +1}\right)\cdot \frac{dx}{dt}$ (3)

If we know that $\frac{dx}{dt} = 0.3$ and x = 1 , then the rate of increase of the y-coordinate is:

$\frac{dy}{dt} = \left[\frac{12}{2\cdot (1)+1} \right]\cdot (0.3)$

$\frac{dy}{dt} = 1.2$

The rate of increase of the y-coordinate is 1.2 units per second.

To learn more on differential equations, we kindly invite to check this verified question: link

Answer from: sonnyboi2305

i). The coordinates of Q is equal to:

(x,y) = (21, 0)

ii). The equation of the curve is:

y = 6. In(2.x + 1) - 1.592

iii). The rate of increase for y-coordinate when x = 1:

1.2 units per second

i). Using analytical geometry,

and are the P and Q's x-coordinates.

and are the P and Q's y-coordinates.

Since (, ) = (1,5) and ( , ) = (x, 0) and m⊥ = - 1/4, the left coordinate of Q's point:-

-1/4 = (0-5)/(x - 1)

∵ x = 21

Thus, the coordinates of y = (21, 0)

ii). Through isolating the variables and employing integration:-

$dy = \frac{12}{(2.x + 1)} dx$

by integration and integration constant (1,5), the equation is:

y = 6. In(2.x + 1) - 1.592

iii). The rate of increase for y-coordinate:

dy/dt = $\frac{(12)}{(2.1 + 1)} .0.3$

∵ dy/dt = 1.2

Thus, the rate of increase for y-coordinate when x = 1 is 1.2 units per second.

Learn more about "Co-ordinates" here:

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Answer from: Quest

because linear equations are a lot simpler than quadratics equations, and linear equations formula is a lot easier to memorize. that's why i think they're harder.

Answer from: Quest

pentagon

step-by-step explanation:

the coordinates a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.

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A curve is such that (dy)/(dx) = 12/(2x + 1) ^ 2 and P(1, 5) is a point on the curve . (i) The norma...