Find the components of the vertical force Bold Upper FFequals=left angle 0 comma negative 4 right angle0,−4 in the directions parallel to and normal to the plane that makes an angle of StartFraction pi Over 3 EndFraction π 3 with the positive x-axis. Show that the total force is the sum of the two component forces.

Answers

Answer 1

Answer:

Answer:

$F_p = < - √(3) , -3 >\n\nF_o = < √(3) , -1 >$

Step-by-step explanation:

- A plane is oriented in a Cartesian coordinate system such that it makes an angle of ( π / 3 ) with the positive x - axis.

- A force ( F ) is directed along the y-axis as a vector < 0 , - 4 >

- We are to determine the the components of force ( F ) parallel and normal to the defined plane.

- We will denote two unit vectors: ( $u_p$ ) parallel to plane and ( $u_o$ ) orthogonal to the defined plane. We will define the two unit vectors in ( x - y ) plane as follows:

- The unit vector ( $u_p$ ) parallel to the defined plane makes an angle of ( 30° ) with the positive y-axis and an angle of ( π / 3 = 60° ) with the x-axis. We will find the projection of the vector onto the x and y axes as follows:

$u_o$ = < cos ( 60° ) , cos ( 30° ) >

$u_o = < (1)/(2) , (√(3) )/(2) >$

- Similarly, the unit vector ( $u_o$ ) orthogonal to plane makes an angle of ( π / 3 ) with the positive x - axis and angle of ( π / 6 ) with the y-axis in negative direction. We will find the projection of the vector onto the x and y axes as follows:

$u_p = < cos ( (\pi )/(6) ) , - cos ( (\pi )/(3) ) >\n\nu_p = < (√(3) )/(2) , -(1)/(2) >\n$

- To find the projection of force ( F ) along and normal to the plane we will apply the dot product formulation:

- The Force vector parallel to the plane ( $F_p$ ) would be:

$F_p = u_p(F . u_p)\n\nF_p = < (1)/(2) , (√(3) )/(2) > [ < 0 , - 4 > . < (1)/(2) , (√(3) )/(2) > ]\n\nF_p = < (1)/(2) , (√(3) )/(2) > [ -2√(3) ]\n\nF_p = < -√(3) , -3 >\n$

- Similarly, to find the projection of force ( $F_o$ ) normal to the plane we again employ the dot product formulation with normal unit vector ( $u_o$ ) as follows:

$F_o = u_o ( F . u_o )\n\nF_o = < (√(3) )/(2) , - (1)/(2) > [ < 0 , - 4 > . < (√(3) )/(2) , - (1)/(2) > ] \n\nF_o = < (√(3) )/(2) , - (1)/(2) > [ 2 ] \n\nF_o = < √(3) , - 1 >$

- To prove that the projected forces ( $F_o$ ) and ( $F_p$ ) are correct we will apply the vector summation of the two orthogonal vector which must equal to the original vector < 0 , - 4 >

$F = F_o + F_p\n\n< 0 , - 4 > = < √(3), -1 > + < -√(3), -3 > \n\n< 0 , - 4 > = < √(3) - √(3) , -1 - 3 > \n\n< 0 , - 4 > = < 0 , - 4 >$ .. proven

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The heights, in centimeters, of five students in a school’s class team are: 165, 175, 176, 159, and 170. What is the mean of the heights of the five students?

Answers

Answer:

169 cm

Step-by-step explanation:

Mean is the average of a group of numbers. You calculate it by adding all your numbers and dividing by how many numbers you have. In this question you would solve with these steps...

Add all of your numbers

165 + 175 + 176 + 159 + 170 = 845

Divide your solution by the number of numbers you have

845 / 5 = 169

The mean of the heights of the five students is 169cm.

Hope this Helps! :)

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2. The path of a high diver is given by y = + (2x2 – 9x – 56), where y is the height of the diverabove the water and x is the horizontal distance from the diving board (in feet). How far from the
end of the diving board is the diver when he hits the water?

Answers

Answer:

The diver will be 8 feet from the end of the board when he hits the water.

Step-by-step explanation:

The diver hits the water when y = 0.

To find the distance, we have to find the values of x when y = 0.

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

$ax^(2) + bx + c, a\neq0$ .

This polynomial has roots $x_(1), x_(2)$ such that $ax^(2) + bx + c = (x - x_(1))*(x - x_(2))$ , given by the following formulas:

$x_(1) = (-b + √(\bigtriangleup))/(2*a)$

$x_(2) = (-b - √(\bigtriangleup))/(2*a)$

$\bigtriangleup = b^(2) - 4ac$

In this problem, we have that:

$y = 2x^(2) - 9x - 56$

$2x^(2) - 9x - 56 = 0$

$a = 2, b = -9, c = -56$

Then

$\bigtriangleup = b^(2) - 4ac = (-9)^(2) - 4*2(-56) = 529$

$x_(1) = (-(-9) + √(529))/(2*2) = 8$

$x_(2) = (-(-9) - √(529))/(2*2) = -3.5$

It is a horizontal distance, so the answer is a positive value.

The diver will be 8 feet from the end of the board when he hits the water.

Stanley Bank offers a money market account that earns 1.85% compounded continuously: a) If $10,000 is invested in this type of account, how much will it be worth in 3 years? b) How long will it take for the account to be worth $15,000?

Answers

Answer:

a) $1056.33 b) 23 years

Step-by-step explanation:

a) 10000(1+1.85/100)^3=10565.33 (2d.p.)

b) let x be the no. of years

15000 = 10000(1+1.85/100)^x

1.5 = 1.0185^x

ln both sides

ln 1.5 = x ln 1.0185

x = ln 1.5/ln 1.0185

=22.11

=23 years (rounded up)

Final answer:

To calculate the future value of an investment using continuous compounding, you use the formula A = P*e^(rt). For a $10,000 investment at 1.85% annual interest, its value after 3 years is calculated by substituting the given values into the formula. To find out how many years it will take for the investment to reach $15,000, rearrange the formula to solve for t: t = ln(A/P)/r, and substitute the values.

Explanation:

The subject of your question is related to the mathematical concept known as continuouscompound interest, which Stanley Bank is applying to its money market account. In this concept, the formula is A = P×e^(rt), where A is the final amount that will be accumulated after t years, P is the principal amount or the initial investment, r is the interest rate in decimals (1.85% would be 0.0185), and e is Euler's number (~2.72).

a) To calculate the value of an investment of $10,000 after 3 years with an annual interest rate of 1.85%, you would use the formula: A = $10,000 × e^(0.0185 ×3). This will give you the total value of the investment after 3 years.

b) To calculate the number of years it will take for your investment to amount to $15,000 with the same interest rate, you would need to re-arrange the formula to solve for t: t = ln(A/P) / r. So, it would be calculated as: t = ln($15,000/$10,000) / 0.0185. This would give you the number of years it will take for your initial investment to reach $15,000.

Learn more about Continuous Compound Interest here:

brainly.com/question/32594953

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I BET YOU CANT SOLVE THIS...

Answers

The answer is 61 7/20

A second number is 7 less than the first number. The third number is twice the first number. If the sum of the three numbers is 321, find the numbers

Answers

Answer:

x=4

Step-by-step explanation:

hope it will help

First no.=x

2nd no.=2x

3rd no. =2x+6

x+2x+2x+6=26

5x+6=26

5×=26-6

5x =20.......divide both sides by 5

x=4

1st no.=x=4

Carla is going to the water slides and needs to figure out which deal is better. She can pay$31 to go on the water slides as much as she wants, or she can pay $13 to get in, plus an
additional $3 per trip down the water slides. If Carla goes on a certain number of trips down
the water slides, the two options are equivalent in terms of cost. How many trips is that?
What is the cost?

Answers

Answer: $31 & It would be 6 trips!

Step-by-step explanation: 6 trips x $3 for the trips, is $18 plus the $13 to get in the park, would be equivalent to $31

Answer:

if she rides the slides 6 times, the costs will be the same: $31

Step-by-step explanation:

You can write an equation in order to solve this problem. Since you are trying to find how many trips using the (3t + 13) plan are equal to the plan that costs 31, the equation would be:

31 = 3t + 13

= 18 = 3t

= 6 = t

(3t + 13 stands for $3 a trip plus $13 entry fee)

This means that if Carla takes 6 trips using either plan, they will cost the same; both would cost $31.

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