Average rate of change of the area of the triangle?

In summary, an object moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t) can be described by P(t)=(cos(t),sin(t)) at time t. The tangent line to the unit circle at P(t) will form a right triangle in the first quadrant when connected with the origin and the x-intercept and y-intercept of the tangent line. The average rate of change of the area of this triangle can be calculated using the equation a(t) = 1/sin2t, on the time interval [π/6,π/4]. The attempt at a solution involved plugging in π/6 and π/4, adding them together, and dividing by 2
  • #1
alaa amed
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Homework Statement


An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t)=(cos(t),sin(t)) . Assume 0 < t < π/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle in the first quadrant. (Connect the origin with the y-intercept and x-intercept of the tangent line.)

The average rate of change of the area of the triangle on the time interval [π/6,π/4] is

Homework Equations


a(t) = 1/sin2t

The Attempt at a Solution


I tried plugging in the π/6 once and π/4, added them together and divided by 2, but I got the wrong answer! [/B]
 
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  • #2
alaa amed said:

Homework Statement


An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t)=(cos(t),sin(t)) . Assume 0 < t < π/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle in the first quadrant. (Connect the origin with the y-intercept and x-intercept of the tangent line.)

The average rate of change of the area of the triangle on the time interval [π/6,π/4] is

Homework Equations


a(t) = 1/sin2t

The Attempt at a Solution


I tried plugging in the π/6 once and π/4, added them together and divided by 2, but I got the wrong answer! [/B]

Show your work.
 

FAQ: Average rate of change of the area of the triangle?

1. What is the formula for calculating the average rate of change of the area of a triangle?

The formula for calculating the average rate of change of the area of a triangle is (change in area)/(change in time), or (ΔA)/(Δt).

2. How is the average rate of change of the area of a triangle different from the average rate of change of its perimeter?

The average rate of change of the area of a triangle measures the change in the triangle's area over a specific period of time, while the average rate of change of its perimeter measures the change in the triangle's perimeter over the same period of time.

3. Can the average rate of change of the area of a triangle be negative?

Yes, the average rate of change of the area of a triangle can be negative if the area is decreasing over time. This could happen if the triangle is being slowly compressed or if its dimensions are changing in a way that decreases its area.

4. What factors can affect the average rate of change of the area of a triangle?

The average rate of change of the area of a triangle can be affected by any changes to the length of its base or height, or any changes to the angle between its base and height. Additionally, external factors such as pressure or temperature changes can also affect the area of a triangle.

5. How can the average rate of change of the area of a triangle be used in real-world applications?

The average rate of change of the area of a triangle can be used in various real-world applications, such as in engineering, construction, and environmental studies. It can help determine the rate of change of a physical structure, the growth or decline of a population, or the effects of natural phenomena on an area.

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