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Gennadij [26K]
2 years ago
5

Let θ (in radians) be an acute angle in a right triangle, and let x and y, respectively, be the lengths of the sides adjacent an

d opposite θ. Suppose also that x and y vary with time.
a. How are dθ/dt, dx/dt and dy/dt related?

Please give steps and explain!
Mathematics
2 answers:
Fed [463]2 years ago
8 0
Explanation is given step by step just below:
tan(theta(t))=y(t)/x(t)

differentiate
sec^2(theta(t))*theta'(t)=y'(t)x(t)-y(t)x'(t)/x^2(t)
thus
theta'(t)=(y'(t)x(t)-y(t)x'(t))
divided by (x^2(t)*sec^2(θ(t))
kipiarov [429]2 years ago
5 0

Answer:

dθ/dt = [(cos^2 θ)*(dy/dt * x - y * dx/dt)]/(x^2)

Step-by-step explanation:

Given that x and y are the lengths of the sides adjacent and opposite θ, then they are related by:

tan θ = y/x

Differentiating respect to t, we get:

sec^2 θ * dθ/dt = (dy/dt * x - y * dx/dt)/(x^2)

dθ/dt = [(cos^2 θ)*(dy/dt * x - y * dx/dt)]/(x^2)

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marusya05 [52]

Answer:


Step-by-step explanation:

Nice summary problem.

<AEC

  • AEC = 360 - 243.5 = 116.5    
  • The number of degrees in 1 rotation of a circle = 360o. You have accounted for 243.5 degrees. What is left over is the answer.

<EAD and <ECD

Both of these are tangents to a circle. Tangents meet radii at 90 degree angles.

<EAD = <ECD = 90 degrees

<ABC

<ABC is 1/2 the central angle. The Central angle is <AEC

  • < AEC = 116.5
  • <ABC = 1/2 * 116.5
  • <ABC = 58.25

<ADC

There are 2 ways of doing this. You should know both of them.

<em><u>One</u></em>

All quadrilaterals = 360 degrees. You know three of the angles. You should be able to find ADC

  • <ADC + 90 + 90 + 116.5 = 360     Add the four angles together.
  • <ADC + 296.5 = 360                   Combine terms on the left
  • <ADC = 360 - 296.5                    Subtract 238.25 from both sides
  • <ADC = 63.5                                 Answer

<em><u>Method Two</u></em>

<ADC = 1/2 (major Arc - Minor Arc)  This formula is fundamental to circle / tangent properties. The Major arc is the larger of the two parts of the circumference of a circle. The Minor arc is the smaller.

  • <ADC = 1/2(243.5 - 116.5)
  • <ADC = 1/2(127)
  • <ADC = 63.5
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