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luda_lava [24]
3 years ago
12

If dy/dx equals xy squared and if y = 1 when x = 0, then when y = 3, x is equal to?

Mathematics
2 answers:
Ymorist [56]3 years ago
5 0
\displaystyle
\dfrac{dy}{dx}=xy^2\\
dy=xy^2 \, dx\\
\dfrac{1}{y^2}\, dy=x\, dx\\
\int \dfrac{1}{y^2}\, dy=\int x\, dx\\
-\dfrac{1}{y}=\dfrac{x^2}{2}+C\\\\
-\dfrac{1}{1}=\dfrac{0^2}{2}+C\\
C=-1\\\\-\dfrac{1}{3}=\dfrac{x^2}{2}-1\\
-2=3x^2-6\\
3x^2=4\\
x^2=\dfrac{4}{3}\\
x=\sqrt{\dfrac{4}{3}} \vee x=-\sqrt{\dfrac{4}{3}}\\
x=\dfrac{2}{\sqrt3} \vee x=-\dfrac{2}{\sqrt3}\\
x=\dfrac{2\sqrt3}{3} \vee x=-\dfrac{2\sqrt3}{3}




S_A_V [24]3 years ago
4 0
This is a separable differential equation, so let's start of there. Let's separate the variables to their own side with the respective differentials:
\frac{dy}{dx} = xy^2
dy = (xy^2) dx
\frac{1}{y^2} dy = x dx

Let's integrate both sides (it's separable, so we can do this):
\int\ { \frac{1}{y^2} } \, dy =  \int\ {x} \, dx
- \frac{1}{y} =  \frac{x^2}{2} + C

Now, let's plug in the values we are given to find the constant "C":
- \frac{1}{1}  =\frac{0^2}{2}+C
-1 = C

Let's rewrite the equation, with C in it, then solve for x because we need to ultimately find x:
- \frac{1}{y}  = \frac{x^2}{2} - 1
x =  \sqrt{2(- \frac{1}{y}+1)}

Let's plug in y = 3 and solve for x:
x = \sqrt{2(- \frac{1}{3}+1)} = \sqrt{ 2( \frac{2}{3}) } = \sqrt{ \frac{4}{3} }

Let's simplify and rationalize the denominator:
x =  \sqrt{ \frac{4}{3}} = 2 \sqrt{ \frac{1}{3}} = 2  \frac{ \sqrt{3} }{3}

So, your answer is D.

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Given that tan θ ≈ −0.087, where 3 2 π < θ < 2 , π find the values of sin θ and cos θ.
elena-s [515]

Answer:

  • sin θ ≈ -0.08667
  • cos θ ≈ 0.99624

Step-by-step explanation:

Straightforward use of the inverse tangent function of a calculator will tell you θ ≈ -0.08678 radians. This is an angle in the 4th quadrant, where your restriction on θ places it. (To comply with the restriction, you would need to consider the angle value to be 2π-0.08678 radians. The trig values for this angle are the same as the trig values for -0.08678 radians.)

Likewise, straightforward use of the calculator to find the other function values gives ...

  sin(-0.08678 radians) ≈ -0.08667

  cos(-0.08678 radians) ≈ 0.99624

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<em>Note on inverse tangent</em>

Depending on the mode setting of your calculator, the arctan or tan⁻¹ function may give you a value in degrees, not radians. That doesn't matter for this problem. sin(arctan(-0.087)) is the same whether the angle is degrees or radians, as long as you don't change the mode in the middle of the computation.

We have shown radians in the above answer because the restriction on the angle is written in terms of radians.

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<em>Alternate solution</em>

The relationship between tan and sin and cos in the 4th quadrant is ...

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That is, the cosine is positive, and the sign of the sine matches that of the tangent.

This more complicated computation gives the same result as above.

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<u> </u><u>48 square inches per inch </u><u> is the average rate of change in surface </u><u>area</u><u>.</u>

What is an area in math?

The area is the region bounded by the shape of an object. The space covered by the figure or any two-dimensional geometric shape, in a plane, is the area of the shape.

The surface area of a cube of side length e is:

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The rate of change is:

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The average rate of change between 3 in and 5 in is:

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Now, the options are given in:

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This is written as:

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r = 48in = 48in^2/in = 48 square inches per inch.

Learn more about area

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