Can someone explain how to solve this question:

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

andrezito [222]

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 $r= 144 units$

5 0

3 years ago

rick won 27 tickets playing a game. to get a prize, he needs to win at least 30 tickets. how many more tickets can he win to get

Svetach [21]

He needs 3 Tickets to Win a Prize.27×2 is... And that's ur Answer

8 0

2 years ago

What is 25/22 as a decimal rounded to the nearest hundredth?

Alex777 [14]

Write 35/22 as a decimal How to convert the fraction 35/22 to a decimal value. Please, input values in this format: a b/c or b/c.

Examples: Four tenths should be typed as 4/10. One and three-half should be typed as 1 3/2. For mixed numbers, please leave a space between the the integer and the fraction.

Enter a fraction value:
35/22/ Ex.: 1/2, 2 1/2, 5/3, etc. Note that 2 1/2 means two and half = 2 + 1/2 = 2.5/ Your answer 1.5909090909090908

8 0

3 years ago

If c = 4 and d = 5, find c:d. 1:5 4:5 5:4

nadya68 [22]

The answer would be the second choice. 4:5

5 0

3 years ago

Read 2 more answers

Two Parallel lines are cut by a transversal x and a transversal y so that x and y intersect at point Q as shown.

saw5 [17]

Answer:

- Angle 'a' is an alternating exterior angle with one angle in the triangle and therefore congruent to one angle in the triangle formed by lines m, x, and y.

- Angles 'b' and 'c' are vertical angles with the other two angles in the triangle and therefore congruent to two other angles in the triangle

Step-by-step explanation:

Vertical angles are angles that are equal to each other but in opposite direction. The angles b and c have vertical angles on the triangle Q while alternate exterior angles are equal angles that lie on different lanes cutting through an axis.

Angles on a plane can be congruent if they are vertical equals or alternating exterior angles.

8 0

3 years ago