If a point is moving on the curve of x2 + 3xy + 3y2 = 22 and the plan is it moving when x=2 if x is…

Question

If a point is moving on the curve of x2 + 3xy + 3y2 = 22 and the plan is it moving when x=2 if x is increasing at the rate 3 unit/second

ScanSolve AI Tutor · Accepted Answer

The problem asks for the speed of a point moving on the intersection of a curve $x^2 + 3xy + 3y^2 = 22$ and a line $x - 2y + 4 = 0$ , when $x=2$ and $\frac{dx}{dt} = 3$ units/second.

First, let's check if there is a point of intersection where $x=2$ .
From the line equation, $x - 2y + 4 = 0$ , if $x=2$ , then $2 - 2y + 4 = 0 \Rightarrow 6 - 2y = 0 \Rightarrow y = 3$ .
So, the point on the line is $(2, 3)$ .
Now, substitute $x=2$ and $y=3$ into the curve equation $x^2 + 3xy + 3y^2 = 22$ :

(2)^2 + 3(2)(3) + 3(3)^2 = 4 + 18 + 27 = 49

Since $49 \neq 22$ , the point $(2, 3)$ is on the line but not on the curve. This means there is no point on the intersection of the curve and the line where $x=2$ .

To proceed with a solution, we must assume that the problem intends for the point to be moving along the line $x - 2y + 4 = 0$ at the instant $x=2$ , and the curve equation is extraneous information for this specific condition.

Find the $y$ -coordinate when $x=2$ using the line equation.
Given the line equation $x - 2y + 4 = 0$ .
Substitute $x=2$ :
$2 - 2y + 4 = 0$ $6 - 2y = 0$ $2y = 6$ $y = 3$
So, at the instant $x=2$ , the point is $(2, 3)$ .
Differentiate the line equation with respect to time $t$ .
The equation of the line is $x - 2y + 4 = 0$ .
Differentiate both sides with respect to $t$ :
$\frac{d}{dt}(x - 2y + 4) = \frac{d}{dt}(0)$ $\frac{dx}{dt} - 2\frac{dy}{dt} + 0 = 0$ $\frac{dx}{dt} - 2\frac{dy}{dt} = 0$
Substitute the given rate $\frac{dx}{dt}$ to find $\frac{dy}{dt}$ .
We are given that $x$ is increasing at the rate of 3 units/second, so $\frac{dx}{dt} = 3$ .
Substitute this into the differentiated equation:
$3 - 2\frac{dy}{dt} = 0$ $2\frac{dy}{dt} = 3$ $\frac{dy}{dt} = \frac{3}{2} \text{ units/second}$
Calculate the speed of the point.
The speed of a point $(x(t), y(t))$ in two dimensions is given by the formula:
$\text{Speed} = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}$
Substitute the values of $\frac{dx}{dt} = 3$ and $\frac{dy}{dt} = \frac{3}{2}$ :
$\text{Speed} = \sqrt{(3)^2 + \left(\frac{3}{2}\right)^2}$ $\text{Speed} = \sqrt{9 + \frac{9}{4}}$ $\text{Speed} = \sqrt{\frac{36}{4} + \frac{9}{4}}$ $\text{Speed} = \sqrt{\frac{45}{4}}$ $\text{Speed} = \frac{\sqrt{45}}{\sqrt{4}}$ $\text{Speed} = \frac{\sqrt{9 \times 5}}{2}$ $\text{Speed} = \frac{3\sqrt{5}}{2} \text{ units/second}$
The speed of the point is $\frac{3\sqrt{5}}{2} \text{ units/second}$ .

If a point is moving on the curve of x2 + 3xy + 3y2 = 22 and the plan is it moving when x=2 if x is increasing at the rate 3 unit/second

Answer

Step-by-step solution

Find the $y$ -coordinate when $x=2$ using the line equation.

Differentiate the line equation with respect to time $t$ .

Substitute the given rate $\frac{dx}{dt}$ to find $\frac{dy}{dt}$ .

Calculate the speed of the point.

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More Mathematics Questions

If a point is moving on the curve of x2 + 3xy + 3y2 = 22 and the plan is it moving when x=2 if x is increasing at the rate 3 unit/second

Answer

Step-by-step solution

Find the yyy-coordinate when x=2x=2x=2 using the line equation.

Differentiate the line equation with respect to time ttt.

Substitute the given rate dxdt\frac{dx}{dt}dtdx​ to find dydt\frac{dy}{dt}dtdy​.

Calculate the speed of the point.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Find the $y$ -coordinate when $x=2$ using the line equation.

Differentiate the line equation with respect to time $t$ .

Substitute the given rate $\frac{dx}{dt}$ to find $\frac{dy}{dt}$ .