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lianna [129]
3 years ago
5

Find the length of the following​ two-dimensional curve. r (t ) = (1/2 t^2, 1/3(2t+1)^3/2) for 0 < t < 16

Mathematics
1 answer:
andrezito [222]3 years ago
5 0

Answer:

r = 144 units

Step-by-step explanation:

The given curve corresponds to a parametric function in which the Cartesian coordinates are written in terms of a parameter "t". In that sense, any change in x can also change in y owing to this direct relationship with "t". To find the length of the curve is useful the following expression;

r(t)=\int\limits^a_b ({r`)^2 \, dt =\int\limits^b_a \sqrt{((\frac{dx}{dt} )^2 +\frac{dy}{dt} )^2)}     dt

In agreement with the given data from the exercise, the length of the curve is found in between two points, namely 0 < t < 16. In that case a=0 and b=16. The concept of the integral involves the sum of different areas at between the interval points, although this technique is powerful, it would be more convenient to use the integral notation written above.

Substituting the terms of the equation and the derivative of r´, as follows,

r(t)= \int\limits^b_a \sqrt{((\frac{d((1/2)t^2)}{dt} )^2 +\frac{d((1/3)(2t+1)^{3/2})}{dt} )^2)}     dt

Doing the operations inside of the brackets the derivatives are:

1 ) (\frac{d((1/2)t^2)}{dt} )^2= t^2

2) \frac{(d(1/3)(2t+1)^{3/2})}{dt} )^2=2t+1

Entering these values of the integral is

r(t)= \int\limits^{16}_{0}  \sqrt{t^2 +2t+1}     dt

It is possible to factorize the quadratic function and the integral can reduced as,

r(t)= \int\limits^{16}_{0} (t+1)  dt= \frac{t^2}{2} + t

Thus, evaluate from 0 to 16

\frac{16^2}{2} + 16

The value is r= 144 units

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Answer:

Step-by-step explanation:

The given equations are:

-9.5x – 2.5y = –4.3                    (1)

and 7x + 2.5y = 0.8                  (2)

Adding equation (1) and (2) together, we get

-9.5x-2.5y+7x+2.5y=-4.3+0.8

⇒-2.5x=-3.5

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Now, substitute the value of x=1.4 in equation (1),

-9.5(1.4)-2.5y=-4.3

⇒-13.3-2.5y=-4.3

⇒-13.3+4.3=2.5y

⇒-9=2.5y

⇒y=-3.6

Thus, the value of x and y are 1.4 and -3.6 respectively.

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3 years ago
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4x+499=660
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Juli2301 [7.4K]

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Step-by-step explanation:

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Answer: -4 \le x \le 5
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------------------------------------------------------------------------------

Explanation:

The domain is the set of possible x input values. Look at the left most point (-4,-1). The x coordinate here is x = -4. This is the smallest x value allowed. The largest x value allowed is x = 5 for similar reasons, but on the other side of the graph.

So that's how I got -4 \le x \le 5 (x is between -4 and 5; inclusive of both endpoints)

Writing [-4,5] for interval notation tells us that we have an interval from -4 to 5 and we include both endpoints. The square brackets mean "include endpoint"

Writing \left\{x|x\in\mathbb{R}, \ -4\le x\le 5\right\} is the set-builder notation way of expressing the domain. The x\in\mathbb{R} portion means "x is a real number"

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