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1 year ago

How can you use a distance time graph to compare the motion of two different object

Mathematics

1 answer:

tia_tia [17]1 year ago

6 0

Answer:

If an object moves along a straight line, the distance traveled can be represented by a distance-time graph. In a distance-time graph, the gradient of the line is equal to the speed of the object. The greater the gradient (and the steeper the line) the faster the object is moving.

Step-by-step explanation:

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Consider the following. (A computer algebra system is recommended.) y'' + 3y' = 2t4 + t2e−3t + sin 3t (a) Determine a suitable f

drek231 [11]

First look for the fundamental solutions by solving the homogeneous version of the ODE:

$y''+3y'=0$

The characteristic equation is

$r^2+3r=r(r+3)=0$

with roots $r=0$ and $r=-3$ , giving the two solutions $C_1$ and $C_2e^{-3t}$ .

For the non-homogeneous version, you can exploit the superposition principle and consider one term from the right side at a time.

$y''+3y'=2t^4$

Assume the ansatz solution,

${y_p}=at^5+bt^4+ct^3+dt^2+et$

$\implies {y_p}'=5at^4+4bt^3+3ct^2+2dt+e$

$\implies {y_p}''=20at^3+12bt^2+6ct+2d$

(You could include a constant term <em>f</em> here, but it would get absorbed by the first solution $C_1$ anyway.)

Substitute these into the ODE:

$(20at^3+12bt^2+6ct+2d)+3(5at^4+4bt^3+3ct^2+2dt+e)=2t^4$

$15at^4+(20a+12b)t^3+(12b+9c)t^2+(6c+6d)t+(2d+e)=2t^4$

$\implies\begin{cases}15a=2\\20a+12b=0\\12b+9c=0\\6c+6d=0\\2d+e=0\end{cases}\implies a=\dfrac2{15},b=-\dfrac29,c=\dfrac8{27},d=-\dfrac8{27},e=\dfrac{16}{81}$

$y''+3y'=t^2e^{-3t}$

$e^{-3t}$ is already accounted for, so assume an ansatz of the form

$y_p=(at^3+bt^2+ct)e^{-3t}$

$\implies {y_p}'=(-3at^3+(3a-3b)t^2+(2b-3c)t+c)e^{-3t}$

$\implies {y_p}''=(9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c)e^{-3t}$

Substitute into the ODE:

$(9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c)e^{-3t}+3(-3at^3+(3a-3b)t^2+(2b-3c)t+c)e^{-3t}=t^2e^{-3t}$

$9at^3+(9b-18a)t^2+(9c-12b+6a)t+2b-6c-9at^3+(9a-9b)t^2+(6b-9c)t+3c=t^2$

$-9at^2+(6a-6b)t+2b-3c=t^2$

$\implies\begin{cases}-9a=1\\6a-6b=0\\2b-3c=0\end{cases}\implies a=-\dfrac19,b=-\dfrac19,c=-\dfrac2{27}$

$y''+3y'=\sin(3t)$

Assume an ansatz solution

$y_p=a\sin(3t)+b\cos(3t)$

$\implies {y_p}'=3a\cos(3t)-3b\sin(3t)$

$\implies {y_p}''=-9a\sin(3t)-9b\cos(3t)$

Substitute into the ODE:

$(-9a\sin(3t)-9b\cos(3t))+3(3a\cos(3t)-3b\sin(3t))=\sin(3t)$

$(-9a-9b)\sin(3t)+(9a-9b)\cos(3t)=\sin(3t)$

$\implies\begin{cases}-9a-9b=1\\9a-9b=0\end{cases}\implies a=-\dfrac1{18},b=-\dfrac1{18}$

So, the general solution of the original ODE is

$y(t)=\dfrac{54t^5 - 90t^4 + 120t^3 - 120t^2 + 80t}{405}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\dfrac{3t^3+3t^2+2t}{27}e^{-3t}-\dfrac{\sin(3t)+\cos(3t)}{18}$

3 0

2 years ago

can anyone help tech me how to solve for tax and tip. im going to 6th grade and trying to get into the advanced group but I don'

baherus [9]

If something isn't clear or you're still confused about, let me know in the comments.

Step-by-step explanation:

Okay, here's my best try.

Suppose you have a bill for $8.45

If you are asked to add, say, 8% tax, you must calculate 8 percent of 8.45 and then add it to 8.45.

Begin by converting 8% to decimals by dividing by 100.

$\frac{8}{100}=0.08$ 8% is 0.08 in decimal form.

To calculate a percentage of a number, you multiply the decimal percent by the number; example:

$0.08(8.45)$ 8.45 is being multiplied by 0.08. The result is: 0.676. (this is the tax; however you have to round up to two decimal places: $0.68)

After finding the tax amount, if you are asked to add it to the bill, you have to add it to the original bill; in this case, $8.45

This would be: 0.68+8.45=9.13.

Therefore, the total bill after tax is for $9.13

----------------------------------------------------------------------------------------------------

With tips is the same thing, however, you can have 2 situations.

Situation 1: You are asked to calculate 15% tip of total bill.

In this situation, you can do the same we do to calculate tax.

Convert the percentage to decimal by dividing by 100.

$\frac{15}{100}=0.15$

and multiply it by the original bill.

$0.15(8.45)=1.2675$ . Rounding up, this would be $1.27 for tip.

If you were to add it to the original bill, simply sum.

$8.45+1.27=9.72$

The total bill after tips is for $9.72

Situation 2: You are asked to calculate 15% tip of the total bill with a tax of 8% included.

In this situation, you must first calculate the total bill with the tax added, like we did above; we got a bill for $9.13 (Tax included)

Then you can calculate 15% tip of this new bill that has tax included.

Convert

$\frac{15}{100}=0.15$

Multiply by the new bill (because this is the one that includes tax)

$0.15(9.13)=1.3695$ Round up.... $1.37 for tips.

If you are asked to give a total amount of money including tax and tips, add this to the bill.

9.13+1.37=10.50

This bill would be for $10.50

3 0

2 years ago

Como faz esta conta 6-4x=-20-10x

yulyashka [42]

6-4x=20-10x, 2+20=-10x+4x, x=26/-6

4 0

2 years ago

Read 2 more answers

Use the distributive property to write an equivalent expression.

Tasya [4]

Answer:

32m + 24n

Step-by-step explanation:

Distribute the 8 to the other terms.

8(4m + 3n)

32m + 24n

6 0

2 years ago

Please tell me the final answer to number 3. <br> Also, excuse my pencil markings.

Ipatiy [6.2K]

Area of a trapezoid = (1/2) * h * (base1 + base2)
A = (1/2) * 25 * (26 + 54)
A = 1000 m^2

3 0

2 years ago