?/6 < 1/2 whats the missing number

Evaluate integral _C x ds, where C is

borishaifa [10]

Answer:

a. $\mathbf{36 \sqrt{5}}$

b. $\mathbf{ \dfrac{1}{108} [ 145 \sqrt{145} - 1]}}$

Step-by-step explanation:

Evaluate integral _C x ds where C is

a. the straight line segment x = t, y = t/2, from (0, 0) to (12, 6)

i . e

$\int \limits _c \ x \ ds$

where;

x = t , y = t/2

the derivative of x with respect to t is:

$\dfrac{dx}{dt}= 1$

the derivative of y with respect to t is:

$\dfrac{dy}{dt}= \dfrac{1}{2}$

and t varies from 0 to 12.

we all know that:

$ds=\sqrt{ (\dfrac{dx}{dt})^2 + ( \dfrac{dy}{dt} )^2}} \ \ dt$

∴

$\int \limits _c \ x \ ds = \int \limits ^{12}_{t=0} \ t \ \sqrt{1+(\dfrac{1}{2})^2} \ dt$

$= \int \limits ^{12}_{0} \ \dfrac{\sqrt{5}}{2}(\dfrac{t^2}{2}) \ dt$

$= \dfrac{\sqrt{5}}{2} \ \ [\dfrac{t^2}{2}]^{12}_0$

$= \dfrac{\sqrt{5}}{4}\times 144$

= $\mathbf{36 \sqrt{5}}$

b. the parabolic curve x = t, y = 3t^2, from (0, 0) to (2, 12)

Given that:

x = t ; y = 3t²

the derivative of x with respect to t is:

$\dfrac{dx}{dt}= 1$

the derivative of y with respect to t is:

$\dfrac{dy}{dt} = 6t$

$ds = \sqrt{1+36 \ t^2} \ dt$

Hence; the integral _C x ds is:

$\int \limits _c \ x \ ds = \int \limits _0 \ t \ \sqrt{1+36 \ t^2} \ dt$

Let consider u to be equal to 1 + 36t²

1 + 36t² = u

Then, the differential of t with respect to u is :

76 tdt = du

$tdt = \dfrac{du}{76}$

The upper limit of the integral is = 1 + 36× 2² = 1 + 36×4= 145

Thus;

$\int \limits _c \ x \ ds = \int \limits _0 \ t \ \sqrt{1+36 \ t^2} \ dt$

$\mathtt{= \int \limits ^{145}_{0} \sqrt{u} \ \dfrac{1}{72} \ du}$

$= \dfrac{1}{72} \times \dfrac{2}{3} \begin {pmatrix} u^{3/2} \end {pmatrix} ^{145}_{1}$

$\mathtt{= \dfrac{2}{216} [ 145 \sqrt{145} - 1]}$

$\mathbf{= \dfrac{1}{108} [ 145 \sqrt{145} - 1]}}$

5 0

4 years ago

I NEED BOTH THESE QUESTIONS ANSWERED PLZZZZZZZZZZZZZZZZZZZ

loris [4]

Answer:

1) 1050 ft squared

2) 200 yards

Step-by-step explanation:

Hope this helps! Pls give brainliest!

6 0

3 years ago

Mae has a number cube with 6 sides that are numbered 1 through 6. What is the probability that Mae will roll an odd number?

Stels [109]

Answer:

C) $P(E) = \frac{3}{6}$

The probability that Mae will roll an odd number

$P(E) = \frac{3}{6}$

Step-by-step explanation:

Step(i):-

Given that Mae has a number cube with 6 sides that are numbered 1 through 6.

n(S) = { 1,2,3,4,5,6,} = 6

Let 'E ' be the event of odd numbers

Mae will roll an odd number

n(E) = {1, 3, 5} = 3

Step(ii):-

The probability that Mae will roll an odd number

$P(E) = \frac{n(E)}{n(S)} = \frac{3}{6}$

$P(E) = \frac{3}{6}$

4 0

3 years ago

A soft drink manufacturer produces 3408 cans in an 8-hour day. Cans are packaged 24 to a case. How many cases are produced each

gavmur [86]

Divide total cans by cans per case:
3408 / 24 = 142 cases per 8 hour day

40 hours / 8 = 5 days

142 * 5 = 710 cases per week

710 * 4 = 2840 cases per month

4 0

4 years ago

Can you help me find the answer and explain it to me?

Mashutka [201]

To find the area of a RIGHT ANGLE triangle, you do base x height then divide it by two.

(we divide by two because a parallelogram can be split into two triangles so we’re basically finding the area of that triangle twice to make an imaginary parallelogram then halving it to get the area of just the triangle).

because you didn’t attach a photo, im not entirely sure which length you gave is the height and base but because a rule for right angle triangles is that the longest length (called the hypotenuse) is always opposite the angle so we’re gonna assume that 32. 5cm is the hypotenuse for this triangle.

this means that 31.2cm and 9.1cm are our base and height length (interchangeable).

so we do
— 31.2 x 9.1 = 283.92
— 283.92 / 2 = 141.96cm squared (because it’s an area!)

hope this helped~ :)

7 0

4 years ago