A family has a bike rack that fits seven bikes, but they only have 5 bikes. How many ways can they fit these bikes on the rack

The lines are skew

To make things clearer, rewrite the given lines as follows;

[tex]L_1 : \frac{x}{1} = \frac{y-1}{-1} = \frac{z-2}{3}[/tex]

[tex]L_2 : \frac{x-2}{2} = \frac{y-3}{-2} = \frac{z}{7}[/tex]

(i) Test if the lines are parallel.

Two lines are parallel if the cross product of their directional vectors is equal to the zero vector 0 (which is equal to <0, 0, 0>). For example, if their directional vectors are m and n, then;

m x n = 0

Where;

m, n and 0 are vectors

In the given lines, L₁ and L₂, their directional vectors are given by the denominators of their symmetric equation.

For L₁, the directional vector = <1, -1, 3>

For L₂, the directional vector = <2, -2, 7>

Let

m = <1, -1, 3> =  i - j + 3k

n = <2, -2, 7> = 2i - 2j + 7k

Now, find the cross product of the vectors;

m x n   =   |  i        j        k |

              |  1       -1       3 |

              |  2      -2      7 |

m x n   =  i(-7 + 6) -j(7 - 6) + k(-2 + 2)

m x n   =  i(-1) -j(1) + k(0)

m x n   =  -i -j + 0k

Since the cross product does not give a zero vector, then the lines are not parallel.

(ii) Test if the lines intersect

To do this:

(a) First, we express the lines in parametric form rather than the symmetric form. Therefore

[tex]L_1 : \frac{x}{1} = \frac{y-1}{-1} = \frac{z-2}{3}[/tex]  is equated to say a;

   and

[tex]L_2 : \frac{x-2}{2} = \frac{y-3}{-2} = \frac{z}{7}[/tex] is equated to say b;

We have;

[tex]L_1 : \frac{x}{1} = \frac{y-1}{-1} = \frac{z-2}{3} = a[/tex]

[tex]L_2 : \frac{x-2}{2} = \frac{y-3}{-2} = \frac{z}{7} = b[/tex]

Split the lines into system of three equations in terms of a and b

For L₁

=> [tex]\frac{x}{1}[/tex] = a  which gives x = a

=> [tex]\frac{y-1}{-1}[/tex] = a which gives y = 1 - a

=> [tex]\frac{z-2}{3}[/tex] = a which gives z  = 3a + 2

∴ L₁ has these three equations

x = a                           -------------------------(i)

y = 1 - a                      --------------------------(ii)

z = 3a + 2                  --------------------------(iii)

For L₂

=> [tex]\frac{x-2}{2}[/tex] = b which gives x = 2b + 2

=> [tex]\frac{y-3}{-2}[/tex] = b which gives y = 3 - 2b

=> [tex]\frac{z}{7}[/tex] = b which gives z = 7b

∴ L₂ has these three equations

x = 2b + 2                        --------------(iv)

y = 3 - 2b                         ---------------(v)

z = 7b                              ----------------(vi)

(b) Secondly, we combine the set of equations of the two lines.

Equations of x : (i) and (iv)

a = 2b + 2                  

=> a - 2b = 2                   --------------------------------------(vii)

Equations of y: (ii) and (v)

1 - a = 3 - 2b

=> - a + 2b = 2               --------------------------------------(viii)

Equations of z: (iii) and (vi)

3a + 2 = 7b

=> 3a - 7b = -2               -------------------------------------(ix)

(c) Thirdly, solve equations (vii), (viii) and (ix) simultaneously to find the values of a and b

Add equations (vii) and (viii)

    a - 2b = 2

+

   -a + 2b = 2  

   0  +  0  = 4    

Since 0 + 0 ≠ 4, then the values of s and t cannot be determined from the system of equations due to inconsistency. Therefore, the lines L₁ and L₂ do not intersect.

(iii) Test if the lines are skew

Two lines are said to be skew if they are neither parallel nor intersecting.

Since the lines L₁ and L₂ are neither parallel nor intersecting, the lines are skew.

Answer:

[tex]P(Yellow) = \frac{29}{147}[/tex]

Step-by-step explanation:

Given

[tex]Brown = 27\\Purple = 47\\Yellow = 29\\Green = 44[/tex]

Required

[tex]P(Yellow)[/tex] --- next roll to be yellow

This is calculated as:

[tex]P(Yellow) = \frac{Yellow}{Total}[/tex]

So, we have:

[tex]P(Yellow) = \frac{29}{27 + 47 + 29 + 44}[/tex]

[tex]P(Yellow) = \frac{29}{147}[/tex]

Answer:

t = 9.5 seconds

Step-by-step explanation:

Given that,

The initial velocity of the projectile is 304 ft/s

The height of the projectile is given by :

[tex]H=16t^2 + 304t[/tex]

For maximum height,

Put h' = 0

[tex]h'=-32t+304[/tex]

or

[tex]-32t+304=0\\\\t=\dfrac{304}{32}\\\\t=9.5 \ s[/tex]

So, the time taken to reach the maximum height is 9.5 seconds.

v = u + at

=> 35 = u + (4×5)

=> 35 = u + 20

=> 35 - 20 = u

=> 15 = u

Step-by-step explanation:

v = u + a t

35=u+ 4(5)

35-20=u

u=15

Answer:

50 yards

Step-by-step explanation:

Have a nice day :)

Answer:

1810 yd

Step-by-step explanation:

Satomi ran 50 more yards than 1/2 of 2 miles.

2 miles ÷ 2 = 1 mile

1 mile = 1760 yd

1760 yd + 50 yd = 1810 yd

Answer:

84

59

Step-by-step explanation:

In other to have the same number of chayes in both rows and columns ;

If the Number of chairs per row = x ; then number of chairs per column = x

Then the total number of chairs needed = x * x = x²

Hence, if there are 5100 chairs ;

Number of chairs needed more ;

Take the square root of 5100 ;the round to the next whole number :

B.) For number of chairs to be removed ;

Take the square root of 5100 and round down to the whole number.

Hence,

A.) = √5100 = 71.414 = 72

72² - 5100 = 84

B.) 5100 = 71.414 = 71

5100 - 71² =

Answer:

70°

Step-by-step explanation:

61° + 49° + y = 180° (sum of angles of a triangle is always 180°)

so,

y= 180° - 61° - 49° = 70°

Answer:

1 torr = 760 atm

1 torr = 1 mmHg

Answer:

it will have 2be 408,99 cent plan so tht u can be able 2 speak the whole month

Answer:

z = 3

Step-by-step explanation:

Since the points are collinear then the slopes between the points are equal.

Calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = P (2, - 3) and (x₂, y₂ ) = Q (3, - 2)

m = [tex]\frac{-2+3}{3-2}[/tex] = 1

Repeat with

(x₁, y₁ ) = Q (3, - 2) and (x₂, y₂ ) = R (8, z )

m = [tex]\frac{z+2}{8-3}[/tex] = [tex]\frac{z+2}{5}[/tex] , then

[tex]\frac{z+2}{5}[/tex] = 1 ( multiply both sides by 5 )

z + 2 = 5 ( subtract 2 from both sides )

z = 3

Answer:

El 5 y el 3

Step-by-step explanation:

La primera y la segunda saldrá un negativo ya que hay un negativo y con eso nunca saldrá positivo.

La tercera es 0, no es positivo ni negativo

La cuarta y quinta es positivo

Answer:

15/17

Step-by-step explanation:

cos=opp/adj

cos C=15/17

Answer:

9 sqrt(2) /2 = x

Step-by-step explanation:

Since this is a right triangle, we can use trig functions

sin theta = opp/ hypotenuse

sin 45 = x /9

9 sin 45 = x

9 sqrt(2) /2 = x

Answer:

c. 216 m squared

do 6×6 to find the missing area then do 12×21 to fine whole area then subtract 252 and 36. hope this helps

Step-by-step explanation:

A line with the specific numbers plotted on it, numerical order

Answer:

1- Put the letter A on positive 7

2- Put the letter B on positive 5

3- Put the letter C on 0

4- Put the letter D on negative 4

5- Put the letter E on negative 9

Answer:

x = 29/6

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Distributive Property

Equality Properties

Algebra I

Step-by-step explanation:

Step 1: Define

Identify

8x - (2x - 13) = 42

Step 2: Solve for x

Step 3: Check

Plug in x into the original equation to verify it's a solution.

Here we see 42 does indeed equal 42.

∴ x = 29/6 is the solution.

Answer:

x = 29/6

Step-by-step explanation:

solve for x.

8x - ( 2x - 13 ) = 42

distribute minus sign

8x - 2x + 13 = 42

combine like terms

6x + 13 = 42

subtract 13 from both sides

6x + 13 - 13 = 42 - 13

6x = 29

Divide each side by 6

6x / 6 = 29 / 6

x = 29 / 6

Check the equation :

8x - ( 2x - 13 ) = 42

substitute the value of x in equation

8 ( 29/6) - ( 2(29/6) - 13) = 42

reduce the number with the greatest common factor 2

4 × 29 /3 - ( (29/3) - 13) = 42

calculate the product

116/3 - ( 29/3 - 13 )= 42

calculate the difference

116/3 -( -10/3 )= 42

change the sign

116/3 + 10/3 = 42

add the fractions to get 42.

42 = 42

Hence, verified.

Answer: 12+6x

Step-by-step explanation:

three more means addition +3

the product of 6 and a number means multiplying 6 by a variable 6x

increased by 9 means addition +9

put it together and we have 3+6x+9

now we combine like terms

3+9+6x

12+6x

Answer:

the first question is x=68/3 in standard from it is 3x=68

the second question is x= - 11/10

Step-by-step explanation:

for the second question the negative is beside the fraction.

Answer:

62

Step-by-step explanation:

In one minute Anna called 2 people

the next minute the 2 people each called 2 people making it 4 so I'll draw something like a branch chain to calculate so it will look like the fig above

I'm not too sure but I tried my best.

9514 1404 393

Answer:

  x = {-0.6, 1.8}

Step-by-step explanation:

Adding 33 we have ...

  4x^2 -5x = 4

Dividing by 4, we get ...

  x^2 -5/4x = 1

We can complete the square by adding the square of half the x-coefficient:

  x^2 -5/4x +25/64 = 89/64

  (x -5/8)^2 = 89/64

  x = (5 ±√89)/8 = {-0.5542, 1.8042}

The solution rounded to the nearest tenth is ...

  x = {-0.6, 1.8}

Answer:

[tex]\begin{array}{ccc}{Midpoint} & {Class} & {Frequency} & { 64.5} & {64-65} & {4} & {66.5 } & {66-67} & {4} & {68.5 } & {68-69} & {4} &{70.5 } & {70-71} & {5} & {72.5 } & {72-73} & {10} & {74.5 } & {74-75} & {4}\ \end{array}[/tex]

Using the frequency distribution, I found the mean height to be 70.1129 with a standard deviation of 3.2831

Step-by-step explanation:

Given

[tex]\begin{array}{ccc}{Midpoint} & {Class} & {Frequency} & { } & {64-65} & {4} & { } & {66-67} & {4} & { } & {68-69} & {4} &{ } & {70-71} & {5} & { } & {72-73} & {10} & { } & {74-75} & {4}\ \end{array}[/tex]

Solving (a): Fill the midpoint of each class.

Midpoint (M) is calculated as:

[tex]M = \frac{1}{2}(Lower + Upper)[/tex]

Where

[tex]Lower \to[/tex] Lower class interval

[tex]Upper \to[/tex] Upper class interval

So, we have:

Class 64-65:

[tex]M = \frac{1}{2}(64 + 65) = 64.5[/tex]

Class 66 - 67:

[tex]M = \frac{1}{2}(66 + 67) = 66.5[/tex]

When the computation is completed, the frequency distribution will be:

[tex]\begin{array}{ccc}{Midpoint} & {Class} & {Frequency} & { 64.5} & {64-65} & {4} & {66.5 } & {66-67} & {4} & {68.5 } & {68-69} & {4} &{70.5 } & {70-71} & {5} & {72.5 } & {72-73} & {10} & {74.5 } & {74-75} & {4}\ \end{array}[/tex]

Solving (b): Mean and standard deviation using 1-VarStats

Using 1-VarStats, the solution is:

[tex]\bar x = 70.1129[/tex]

[tex]\sigma = 3.2831[/tex]

See attachment for result of 1-VarStats

Solving (c): Compare the calculated mean to the actual mean

The actual mean is missing from the question, so I will make assumptions in this part

Assume they are close

This means that the selected sample is a reflection of the actual population where the samples are selected.

Assume they are not close

This means that the selected sample does not reflect the actual population where the samples are selected.

Answer:

-3

Step-by-step explanation:

change in y over the change in x

pick to points and use slope formula

9-3

0-2

that's 6/-2 = -3

9514 1404 393

Answer:

  55°

Step-by-step explanation:

Adjacent angles in a parallelogram are supplementary.

  m∠GDE = 180° -m∠FGD = 180° -125°

  m∠GDE = 55°

Answer:

the answer is d trust me 0.22

Step-by-step explanation:

Answer:

Slope = 1

Y-intercept = 4

equation = 1x + 4

Hope this helps!

Answer: The radius of the hemisphere is (R) = 28 cm

Answer:

286 mm ^2

Step-by-step explanation:

The figure is a trapezoid

The area of a trapezoid is given by

A = 1/2 ( b1+b2) *h where b1 and b2 are the lengths of the bases and h is the height

A = 1/2 ( 28+24) * 11

A = 1/2 (52)*11

A =286 mm ^2

Answer:

The area of this figure is 286

Step-by-step explanation:

The formula for the area of a trapezoid is A = (a+b/2)h, or the top line plus the bottom line divided by 2 times the height. a is given as 24 mm and b is given as 28 mm so 24 plus 28 is 52. 52 divided by 2 is 26. 26 times 11 is 286 therefore the answer and area is 286.

9514 1404 393

Answer:

  b ≈ 2.7

Step-by-step explanation:

The missing angle A is ...

  A = 180° -53° -80° = 47°

The law of sines tells you ...

  b/sin(B) = a/sin(A)

  b = sin(80°)×2/sin(47°) ≈ 2.6931

Side b is about 2.7 units.

Answer:

Step-by-step explanation:

hope it helps much

=====================================================

Explanation:

We have 81 cm^3 of clay. Divide this among the 20 students and each gets 81/20 = 4.05 cm^3 of clay

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

The volume of a cone is

V = (1/3)*pi*r^2*h

Solving for h gets us

3V = pi*r^2*h

(3V)/(pi*r^2) = h

h = (3V)/(pi*r^2)

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

The diameter is 2.5 cm which cuts in half to 1.25 cm, so this is the radius.

We'll plug this radius in, along with V = 4.05 and pi = 3.14

h = (3V)/(pi*r^2)

h = (3*4.05)/(3.14*(1.25)^2)

h = 2.4764

This value is approximate. Rounding down to the nearest hundredth gets us 2.47

We round down because rounding up to 2.48 will lead to a volume larger than 4.05

The maximum height of each tooth is 2.47 cm and this can be determined by using the formula of the volume of the cone.

Given :

The following steps can be used in order to determine the maximum height of each tooth:

Step 1 - The formula of the volume of the cone can be used in order to determine the maximum height of each tooth.

Step 2 - The formula of the volume of the cone is given below:

[tex]\rm V = \dfrac{1}{3}\pi r^2h[/tex]

where r is the radius and h is the height of the cone.

Step 3 - Substitute the values of the known terms in the above expression.

[tex]\rm \dfrac{81}{20}= \dfrac{1}{3}\times \pi \times (1.25)^2\times h[/tex]

Step 4 - Simplify the above expression.

h = 2.47 cm

For more information, refer to the link given below:

https://brainly.com/question/1578538