If a circle c with radius 1 rolls along the outside of the circle x2 + y2 = 36, a fixed point p on c traces out a curve called an epicycloid, with parametric equations x = 7 cos t − cos 7t, y = 7 sin t − sin 7t. find the area it encloses
Answers
9514 1404 393
Answer:
area = 50π ≈ 157.08 square units
Step-by-step explanation:
The parameter t is the angle in the polar form equation of the curve. The square of the radius is the sum of squares of x and y. The area of a differential sector is ...
dA = (1/2)r²·dt
The figure has 6-fold symmetry, so we only need to find the area of the first lobe of the curve. The integral is shown in the attachment, which evaluates it numerically.
The epicycloid encloses an area of 50π square units.
Final answer:
To calculate the area enclosed by an epicycloid curve defined by given parametric equations, use the area formula for a plane curve and integrate from 0 to 2π. Calculations involve derivatives and integration.
Explanation:
The area that the epicycloid encloses can be calculated using the formula for the area of a plane curve that is parameterized by the equations you provided: x = 7 cos t − cos 7t, y = 7 sin t − sin 7t. This area can be computed by integrating the expression x dy - y dx from 0 to 2π. In the case of these parametric equations, dx and dy are the derivatives of x and y with respect to t (the parameter), which can be calculated as: dx/dt = -7 sin t + 7 sin 7t, dy/dt = 7 cos t - 7 cos 7t. After calculating these derivatives, substitute them into the aforementioned integral, and perform the integration. The result should be the area that the epicycloid encloses.
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Answer:
i'll see if a friend knows ill work on it
Step-by-step explanation:
Answers
Both 180 seconds or 3 minutes are equal.
What is division?
The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
Given:
We have two times,
180 seconds and 3 minutes.
180 seconds in minutes,
= 180/60
= 3 minutes.
Therefore, both are the same.
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how you find the answer is you divide 60 into 180 and you get 3 which means 3 minutes so their both the same
Answers
Answer:
10
Step-by-step explanation:
Let's call the four numbers x1, x2, x3, and x4. Their average is (x1 + x2 + x3 + x4)/4 = 8, so x1 + x2 + x3 + x4 = 4 × 8 = 32. And we know that three of the numbers have a sum of 22, so x1 + x2 + x3 = 22.
Subtracting this last equation from the first equation, we have:
So the fourth number is 10.
Answers
Answer:
The interest he earns in one year is $3000
Step-by-step explanation:
We are given
Alex has $30,000 in his savings account that earns 10% annually
so,
P=30000
r=10%
t=1 years
now, we can find interest
now, we can plug values
now, we can simplify it
So,
The interest he earns in one year is $3000
Answers
Answer:
20+30h
......................................
How long does it take for the rock to travel 30 meters?
A. 3 seconds
B.5 seconds
C.6 seconds
D.2 seconds
pls !! help !!!
Answers
Final answer:
To find the time it takes for the rock to travel 30 meters, we solve the given quadratic equation and obtain the value of t. The rock takes 3 seconds to travel 30 meters.
Explanation:
To find out how long it takes for the rock to travel 30 meters, we need to solve the equation d = 5t^2 + 5t, where d is the distance and t is the time in seconds. We'll set d equal to 30 and solve for t:
30 = 5t^2 + 5t
Now, we can rearrange the equation to form a quadratic equation:
5t^2 + 5t - 30 = 0
Next, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. By factoring, we get:
(t - 3)(5t + 10) = 0
From this, we can see that there are two possible solutions for t: t = 3 and t = -2. However, since time cannot be negative in this context, we discard the negative solution. Therefore, the rock takes 3 seconds to travel 30 meters.
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