Find the length of the curve. R(t) = 2 i + t2 j + t3 k, 0 ≤ t ≤ 1

Answers

Answer 1

Answer:

Length of a curve is the length of its plot its curve. The length of the given curve for given range of t is: L = 1.44 units approx.

How to find the length of a curve?

If the curve has position vector p(x) for value of x ranging from x = a to x = b,

then, the curve's length is calculated as:

$L = \int_a^b ||p'(x)||dx\n$ units.

For the given case, we have:

Position vector = $R(t) = 2\hat i + t^2 \hat j + t^3 \hat k$

Its differentiation gives:

$R'(t) = 2t\hat j + 3t^2\hat k$

Its non negative magnitude is: ||R'(t)|| = $√((2t)^2 + (3t^2)^2) = t√(4+9t^2)$

Thus, as t ranges from a = 0 to b = 1, thus, length of the curve is:

$L = \int_0^1 (t√(4+9t^2))dt\n\n\text{Let v = 4+9}t^2, \text{then dv = 18tdt}\nand\nt=0\implies v = 4\nt=1 \implies v = 13\nThus,\nL = \int_4^(13)((√(v))/(18))dv = (1)/(18) [(2(v)^(3/2))/(3)]^(13)_4 \approx (38.87)/(27) \approx 1.44 \: \rm units$

Thus,

The length of the given curve for given range of t is: L = 1.44 units approx.

Learn more about length of the curve here:

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Answer 2

Answer:

curve equation is

$\n \vec{R}\left ( t \right ) = 2\hat{i}+t^(2)\hat{j} + t^(3)\hat{k}$ ,0≤ t≤ 1

now taking the differentiation

$\n{R}'t = 2t\hat{i} + 3t^(2)\hat{j}$

now taking the modulus

$\left \| {R}'(t) \right \|=$ $\sqrt{4t^(2) +9t^(4)}$

= $\sqrt{4 + 9 t^(2) } .t$

now taking the integration

length of the curve = $\n\int t\sqrt{4 + 9 t^(2)} dt\n$

now put the value v= 4 + 9t²

dv= 18 tdt

now put this value in the above equation

we get

length of the curve = $\n(1)/(18)\int √(v)dv\n$

now taking integation we get and put the value of the v

we get

= $(1)/(18)$ × $(2)/(3)$ × $(4 + 9t^(2) )^{(3)/(2) }$

= $(1)/(27) ( 4 + 9 t^(2) )^{(3)/(2) }$

now find out the length of the curve in the interval from 0 to 1.

length of the curve $= (1)/(27) (13^{(3)/(2)} -4^{(3)/(2)} )\n=(1)/(27) (13√(13) -8)$

Hence proved

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Please help. Write each phrase as an algebraic expression.

Answers

Answer:

The operation for 7 and x = addition

The expression= 7 + x

The quotient of z and 3

The operation is= dividing

The expression is= $(z)/(3)$

The following rule describes the relationship between x and y.Rule: Multiply x by 4 to get y.
Complete the table for the given rule.
X Y
0 __
1 __
2 __

Answers

Y
—
0
4
8
Hope this helps :)

The length of a side of a triangle is 36. A line parallel to that side divides the triangle into two parts of equal area. Find the length of the segment determined by the points of intersection between the line and the other two sides of the triangle

Answers

Answer:

18√2

Step-by-step explanation:

The area of the smaller triangle is 1/2 that of the larger one. Since the triangles are similar, the dimensions of the smaller triangle are √(1/2) those of the larger one.

36 · √(1/2) = 36 · (√2)/2 = 18√2 . . . . length of line dividing the triangle

What is 1/4 * 32 I need help please

Answers

Answer: 8

Step-by-step explanation: Think of the 32 in this problem as 32/1.

So rewriting, we have 1/4 × 32/1.

Now, the 4 and 32 cross-cancel to 1 and 8.

So we have 8/1 or just 8.

At a point on the ground 20 feet from a building, a surveyor observes the angle of inclination to the top of the building to be pi/3 radians. How tall is the building?

Answers

Answer:

34.64 ft

Step-by-step explanation:

Distance from the building = 20 ft

Angle of inclination = π/3 radians

The tangent of the angle of inclination must equal the height of the building divided by the distance of the observer from the building:

$tan( \pi/3) = (h)/(20) \nh = 20*1.73205\nh=34.64\ ft$

The building is 34.64 ft tall

Although still a sophomore at college, John O'Hagan's son Billy-Sean has already created several commercial video games and is currently working on his most ambitious project to date: a game called K that purports to be a "simulation of the world." John O'Hagan has decided to set aside some office space for Billy-Sean against the northern wall in the headquarters penthouse. The construction of the partition will cost $8 per foot for the south wall and $12 per foot for the east and west walls.What are the dimensions of the office space with the largest area that can be provided for Billy-Sean with a budget of $432?south wall length ft -east and west wall length -What is its area?

Answers

Answer:

27 feet for the south wall and 18 feet for the east/west walls

Maximum area= $486\ ft^2$

Step-by-step explanation:

Optimization

This is a simple case where an objective function must be minimized or maximized, given some restrictions coming in the form of equations.

The first derivative method will be used to find the values of the parameters that control the objective function and the maximum value of that function.

The office space for Billy-Sean will have the form of a rectangle of dimensions x and y, being x the number of feet for the south wall and y the number of feet for the west wall. The total cost of the space is

C=8x+12y

The budget to build the office space is $432, thus

$8x+12y=432$