Answer: Hope this helps you
1/2
Step-by-step explanation:
To find the x-int of any equation, you will need to replace the y with a 0 since when the x-int is when y is 0.
8x+2y = 4 ➨ 8x+2*0 = 4
Now I will do 2*0 which anything times 0 is equal to 0. And we are left with the equation of:
8x = 4
Now I will divide both sides by 8, which isolates our x.
x = 1/2
The x-int is 1/2.
Please hurry!!!! What is the domain of the function on the graph?
all real numbers
all real numbers greater than or equal to –2
all real numbers greater than or equal to –5
all real numbers greater than or equal to 0
A. all real numbers
the doman are the x values and in this equation they are infinite making the domain all real numbers
If anyone can help me to solve this.
Answer:
x=50
Step-by-step explanation:
3x-5=145(vertically opposite angle)
3x=145+5
3x=150
x=150/3
x=50
DON'T FORGET TO WRITE DEGREE
Answer:
x = 50
Step-by-step explanation:
Vertically opposite angles are of equal measure.
Solving:3x - 5 = 145
3x = 145 + 5
3x = 150
x = 50
That's all i know, Hope it helps!
1) When the polynomial P(x) is divided by x+5, the remainder is 3. Which of the following is definitely true?
a) P(3) = 5
b) P(-5) = 3
c) P(5) = 3
d) P(-3) = 5
2) P is a degree 3 polynomial with real coefficients and three zeros. Two of the zeros are 5 and 3+4i. What is the other zero?
a) 4-3i
b) 4+3i
c) -5
d) 3-4i
1) The definitely true for the statement is b) P(-5) = 3.
2) The three roots are 5, 3 + 4i, and 3 - 4i. Option D is correct.
What is the remainder theorem for polynomials?If there is a polynomial p(x), and a constant number 'a', then
[tex]\dfrac{p(x)}{(x-a)} = g(x) + p(a)[/tex]
where g(x) is a factor of p(x)
Given;
1) When the polynomial P(x) is divided by x+5, the remainder is 3.
If a polynomial P(x) is divided by (x-a), then the remainder is P(a).
If the polynomial P(x) is divided by (x+5), then the remainder will be P(-5).
So, P(-5) = 3
Therefore, the definitely true for the statement is b) P(-5) = 3.
2) P is a 3 degree polynomial with real coefficients and three zeros. Two of the zeros are 5 and 3+4i.
The complex roots always exist in conjugate pairs so,
3 + 4i and 3 - 4i
Thus, the three roots are 5, 3 + 4i, and 3 - 4i. Option D is correct.
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A group fitness gym classifies its fitness class attendees by class type and member status. The marketing team has gathered data from a random month, in which there
were 2039 class attendees. The data is summarized in the table below.
Class Type and Member Status of Class
Attendees
Class Type
Barre
Boot Camp
Boxing
Spinning
Yoga
Member
.240
204
288
191
177
Non-Member
185
155
234
271
94
What is the probability that a class attendee is a non-member of the gym and is attending a barre class or is a member of the gym and is attending a boot camp class?
Enter a fraction or round your answer to 4 decimal places, if necessary.
The probability that a class attendee is a non-member of the gym and is attending a barre class or is a member of the gym and is attending a boot camp class is 0.009.
What is the probability?Probability refers to a possibility that deals with the occurrence of random events.
The marketing team has gathered data from a random month, in which there were 2039 class attendees.
The probability that a class attendee is a non-member of the gym and is attending a barre class
= 185/2039
= 0.09
The probability that a class attendee is a member of the gym and is attending a boot camp class
= 204/2039
= 0.10004
The probability for both the event would be
= 0.09 x 0.10004
= 0.009
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46. Machine A produces 500 springs a day. The number of defective springs produced by this machine each day is recorded for 60 days. Based on the distribution given below, what is the expected value of the number of defective springs produced by Machine A in any single day?
F. 0.00
G. 0.45
H. 0.70
J. 1.00
K. 1.50
Answer:
G. 0.45
Step-by-step explanation:
To find expected value, you simply multiply the value of each outcome (the numbers in the left column) by its probability
(the numbers in the right column) and then add them all together.
0(0.7) + 1(0.2) + 2(0.05) + 3(0.05)
0 + 0.2 + 0.1 + 0.15 = 0.3 + 0.15 = 0.45
The expected value is 0.45. Thus, the correct option is G.
What is the expected value?In parameter estimation, the expected value is an application of the weighted sum. Informally, the expected value is the simple average of a considerable number of individually determined outcomes of a randomly picked variable.
The expected value is given below.
E(x) = np
Where n is the number of samples and p is the probability.
The expected value is calculated as,
E(x) = 0 x 0.70 + 1 x 0.20 + 2 x 0.05 + 3 x 0.05
E(x) = 0 + 0.20 + 0.10 + 0.15
E(x) = 0.45
Thus, the correct option is G.
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Complete the table for the given rule. Rule: Y = X + 8
X | Y
2. _
0. _
4. _
Y=x+8
Table
[tex]\boxed{\begin{array}{c|c}\bf x&\bf y\\ \rm 2&\rm 10\\ \rm 0&\rm 8\\ \rm 4&\rm 12\end{array}}[/tex]
How?
x=2
y=2+8=10x=0
y=0+8=8x=4
y=4+8=12Answer:
[tex]\large\begin{array}{| c | c |}\cline{1-2} x & y \\\cline{1-2} 2 & 10 \\\cline{1-2} 0 & 8 \\\cline{1-2} 4 & 12 \\\cline{1-2}\end{array}[/tex]
Step-by-step explanation:
Given rule:
[tex]y=x+8[/tex]
The rule tells us that to the corresponding y-values, simply substitute the given value of x into the given rule:
[tex]x=2 \implies y=2+8=10[/tex]
[tex]x=0 \implies y=0+8=8[/tex]
[tex]x=4 \implies y=4+8=12[/tex]
Therefore, the completed table is:
[tex]\large\begin{array}{| c | c |}\cline{1-2} x & y \\\cline{1-2} 2 & 10 \\\cline{1-2} 0 & 8 \\\cline{1-2} 4 & 12 \\\cline{1-2}\end{array}[/tex]
Given mCT=26 and m/CAT =124° find the length of CA, the radius in Circle A. Use
π = 3.14 in your calculation and round to the nearest tenth.
Answer:
[tex]\text{length of \textit{CA}} \ = \ 12.0 \ \ \ (\text{nearest tenth})[/tex]
Step-by-step explanation:
Radian measure is the ratio of the length of a circular arc to its radius.
A radian is the measurement of the central angle which subtends an arc whose length is equal to the length of the radius of the circle.
In the case of the unit circle, as shown in the figure below, one radian is the angle of the sector with a radius of 1 and circular arclength of 1.
Following this definition, the magnitude, in radians, of one complete revolution of a unit circle is the circumference of the unit circle divided by its radius, [tex]\displaystyle\frac{2\pi}{1}[/tex] or [tex]2\pi[/tex]. Thus, [tex]2\pi[/tex] radians is equal to [tex]360^{\circ}[/tex] degrees. Alternatively, one radian is equal to [tex]\displaystyle\frac{180^{\circ}}{\pi} \ \approx \ 57.296^{\circ} \ \ \ (3 \ d.p.)[/tex].
Since radian measure is defined as the ratio of the arc length of a sector to its radius, hence
[tex]\displaystyle\frac{s}{r} \ = \ \theta \\\\ s \ = \ r\theta[/tex]
where [tex]s[/tex] is the arclength, [tex]r[/tex] is the radius, and [tex]\theta[/tex] is the central angle, in radians.
Therefore, the length of CA is
[tex]\displaystyle\frac{s}{\theta} \ = \ r \\ \\ r \ = \ \displaystyle\frac{26}{124^{\circ} \ \times \ \displaystyle\frac{\pi}{180^{\circ}} \ \text{rad}} \\ \\ r = \displaystyle\frac{26 \ \times \ 45}{31 \ \times \ 3.14} \\ \\ r\ = \ 12.0 \ \ \ \ \left(\text{nearest tenth}\right)[/tex]
help asap pls ill mark brainliest fr
order them from least to greatest solve them first
Answer:
1/2 - 1/3,
1 - 1/2,
1/4 + 1/2
Step-by-step explanation:
1/4 + 1/2 = 1/4 + 2/4 = 3/4
1/2 - 1/3 = 3/6 - 2/6 = 1/6
1 - 1/2 = 2/2 - 1/2 = 1/2
Hence, the desired order is
1/2 - 1/3,
1 - 1/2,
1/4 + 1/2
Inverse Function: One to one function...Help needed! 15 pts for your help!
Answer:
Step-by-step explanation:
hello :
Select all the correct graphs.
Choose the graphs that indicate equations with no solution.
Answer:
First Graph: -2x - 1 = 3(^-x)
Last Graph: 2^(-x) + 2 = 5^-x + 3
Step-by-step explanation:
For a system of equations to have a solution set, the graphs that depict them must intersect at one point.
Both graphs #1 and #5 do not intersect, hence graphs #1 and #5 are the only graphs that do not have solutions while the other graphs do.
There is a mound of g pounds of gravel in a quarry. Throughout the day, 200 pounds of gravel is added to the mound. Two orders of 700 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.
Answer:
[tex]\fbox {2,400 pounds}[/tex]
Step-by-step explanation:
Information we have :
200 pounds is added2 orders of 700 pounds is removed1,200 pounds remainWhat we need to find :
Original amountSolving :
g + 200 - 2(700) = 1200g + 200 = 1200 + 1400g + 200 = 2600g = 2,400 pounds of gravelWhich sequences are geometric? Check all that apply.
5, 10, 20, 50,...
3, 12, 48, 192,...
3, 15, 75, 375,...
8, 15, 75,375,...
14, 21, 28, 35,...
17, 20, 23, 26,...
2, 10, 50, 250,...
Answer:
The first, third and last sequences
Please help and fast for brainliest
The graph shows the number of gallons of white paint that were mixed with gallons of green paint in various different ratios:
Draw the graph on a grid. The title is Mixing Paint. The horizontal axis label is Green Paint in gallons. The scale is 0 to 24 in increments of 3. The vertical axis label is White Paint in gallons. The scale is 0 to 72 in increments of 9. Plot points at the ordered pairs 3,9 and 6,18 and 9,27.
The number of gallons of white paint mixed with 1 gallon of green paint is ______.
Answer:
3 gallons
Step-by-step explanation:
For every value of green ( 'x') ....white is 3 times as much
for 1 gallon green ====> 3 gallons white
Which expression is equivalent to (a²b¹c)²(6a³b)(2c³)³₂ 4ab¹2c3 ?
The expression (a²b¹c)² × (6a³b) × (2c³)³ × (4ab¹) × (2c³) is equivalent to the expression 384a⁸b⁴c¹⁴.
What is an equivalent expression?The equivalent is the expressions that are in different forms but are equal to the same value.
The expression is given below.
⇒ (a²b¹c)² × (6a³b) × (2c³)³ × (4ab¹) × (2c³)
By simplifying, we have
⇒ a⁴b²c² × 6a³b × 8c⁹ × 4ab¹ × 2c³
⇒ 384a⁸b⁴c¹⁴
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The velocity of a car over five hours is given by v(t)=60ln(t+1),0≤t≤5 in kilometers per hour. What is the total distance traveled from t=0 to t=5? Round your answer to the nearest whole number and do not include units.
Step-by-step explanation:
as the velocity is not constant over time, we actually have to integrate v(t) over the interval 0<=t<=5 to get the distance.
v(t) = 60×ln(t + 1)
V(t) = 60 × integral(ln(t + 1)) between 0 and 5.
integral(ln(t + 1)) = (t + 1)ln(t + 1) - t + C
V(t) = 60 × ((t + 1)ln(t + 1) - t + C)
the distance traveled between t = 0 and t = 5 is then
60 × ((5 + 1)ln(5 + 1) - 5 + C) - 60 × ((0 + 1)ln(0 + 1) - 0 + C) =
= 60×(6ln(6) - 5 + C) - 60×(1ln(1) + C) =
= 60×(5.750556815... + C) - 60×C =
= 60×5.750556815... = 345.0334089... ≈ 345 km
Suppose you start with a full tank of gas (20 gallons) in your truck. After driving 3 hours, you now have 2 gallons left. If x is the number of hours you have been driving, then y is the number of gallons left in the tank. At what rate is the gas left in the tank changing? State your answer as a reduced fraction. Find an equation of a line in the form y = mx + b that describes the amount of gas in your tank.
The rate of the gas left in the tank will be -6 and the equation will be given as y = -6x + 20.
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication and division.
Given that:-
Suppose you start with a full tank of gas (20 gallons) in your truck. After driving 3 hours, you now have 2 gallons left. If x is the number of hours you have been driving, then y is the number of gallons left in the tank.The rate of gas will be calculated as:-
Suppose the hours are represented by x and the gallons are represented by y.
The rate of the gas will be the slope so the slope will be calculated as:-
[tex]m =\dfrac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m = \dfrac{2-20}{3-0}[/tex]
[tex]m =\dfrac{-18}{3}[/tex]
m = -6
The equation will be given as:-
y = mx + c
y = -6x + 20
Therefore the rate of the gas left in the tank will be -6 and the equation will be given as y = -6x + 20.
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The graph of the function f(x) = -(x+3)(x-1) is
shown below.
-6
-4 -2
6
A
2
-2
4
-6
2
4
6 X
What is true about the domain and range of the
function?
O The domain is all real numbers less than or equali
to 4, and the range is all real numbers such that -31
≤x≤ 1.
The domain is all real numbers such that-3≤x≤
1, and the range is all real numbers less than or
equal to 4.
The domain is all real numbers, and the range is all
real numbers less than or equal to 4.
The domain is all real numbers less than or equal
to 4, and the range is all real numbers.
Using the concepts of domain and range of a function, the correct statement regarding the graph of the function f(x) = -(x + 3)(x - 1) is given by:
The domain is all real numbers, and the range is all real numbers less than or equal to 4.
What are the domain and the range of a function?The domain of a function is the set that contains all the values of the input.The range of a function is the set that contains all the values of the output.In a graph:
The domain is given by the x-values, the horizontal axis.The range is given by the y-values, the vertical axis.The graph of f(x) = -(x + 3)(x - 1) is given at the end of the answer, hence:
The domain is all real values, as the function keeps going to infinity.The range is all values of y that are less than or equal to 4.Hence the correct statement is:
The domain is all real numbers, and the range is all real numbers less than or equal to 4.
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Anu's age exceeds Sumbo's age by 15 The sum of the square of their ages is 725. What are their ages?
Answer:
Anita= 25 years old
Sumbo= 10 years old
Step-by-step explanation:
Start by forming 2 equations that represent the given information.
Let Anu's and Sumbo's ages be A and S respectively.
A= S +15 -----(1)
A² +S²= 725 -----(2)
Now, solve for A and S by substitution.
Substitute (1) into (2):
(S +15)² +S²= 725
Expand:
S² +2(S)(15) +15² +S²= 725
2S² +30S +225= 725
-725 on both sides:
2S² +30S -500= 0
Divide both sides by 2:
S² +15S -250= 0
Factorise:
(S +25)(S -10)= 0
S +25= 0 or S -10= 0
S= -25 (reject) or S= 10
Sumbo's age cannot be a negative value hence -25 is rejected.
Substitute S= 10 into (1):
A= S +15
A= 10 +15
A= 25
A water ride with heights above and below the starting point can be modeled by the function y = 3 sin ( π 2 ( x + 3 ) − 2 ) . Within the interval 0 < x < 5 , when does the ride have a height 1 foot below the starting point?
The value of x for a height 1 foot below the starting point will be the negative 2.76.
What is trigonometry?The connection between the lengths and angles of a triangular shape is the subject of trigonometry.
A water ride with heights above and below the starting point can be modeled by the function is given below.
y = 3 sin (π / 2 (x + 3) − 2), Within the interval 0 < x < 5
We know that π / 2 = 90°
Then the value of x for a height 1 foot below the starting point will be
1 = 3 sin (90 (x + 3) − 2)
1/3 = sin (90 (x + 3) − 2)
19.47 = 90 (x + 3) − 2
21.47 = 90 (x + 3)
0.2385 = x + 3
x = -2.76
Then the value of x will be the negative 2.76.
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need some help asap, giving brainliest
Answer:
16.19
Step-by-step explanation:
Use the cosine rule:
[tex]\sqrt{a^{2} +b^{2} -2abcosy} \\\\\sqrt{16^{2} +20^{2} -(2)(16)(20)cos(52)} \\\\\\= 16.19cm[/tex]
Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain".
The acidity of liquids is measured by pH on a scale of 0 to 14. Distilled water has pH 7.0, and lower pH values indicate acidity.
Normal rain is somewhat acidic, so "acid rain" is sometimes defined as rainfall with a pH below 5.0.
The pH of rain at one location varies among rainy days according to a Normal distribution with mean 5.43 and standard
deviations 0.54.
What proportion of rainy days have rainfall with pH below 5.0? Use software or Table A to find the answer.
(Enter your answer rounded to four decimal places.)
Using the normal distribution, it is found that 0.2119 = 21.19% of rainy days have rainfall with pH below 5.0.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.The mean and the standard deviation are given, respectively, by:
[tex]\mu = 5.43, \sigma = 0.54[/tex]
The proportion of rainy days have rainfall with pH below 5.0 is the p-value of Z when X = 5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{5 - 5.43}{0.54}[/tex]
Z = -0.8
Z = -0.8 has a p-value of 0.2119.
0.2119 = 21.19% of rainy days have rainfall with pH below 5.0.
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Use the Pythagorean Theorem to find the length of the leg in the triangle shown below.
The figure shows a right triangle with one leg marked 12. The hypotenuse is marked 15.
Answer:
9
Step-by-step explanation:
The Pythagorean Theorem is a² + b² = c². c² is the hypotenuse while a² and b² are the legs.
So plugging it into the equation,
a² + 12² = 15²
a² + 144 = 225
a² = 81
a = 9
steps:
1. square the values
2. isolate the missing value
3. take the square root of both sides
What is the solution set for the following inequality? 4x – 3 + 6x > 5x + 7
Answer:
you got that
Step-by-step explanation:
wet wet wet
Answer:
{ x : x > 2 }
Step-by-step explanation:
4x - 3 + 6x > 5x + 7 , that is
10x - 3 > 5x + 7 ( subtract 5x from both sides )
5x - 3 > 7 ( add 3 to both sides )
5x > 10 ( divide both sides by 5 )
x > 2
A car has a maximum speed of 355.7 feet/second. Convert this speed to miles/hour.
Answer:
242.5226626 mphStep-by-step explanation:
1 ft/s = 0.681818 mph
355.7 * 0.681818 = 242.5226626
355.7 ft/s = 242.5226626 mph
1
0/1 point
A student wants to write an expression for, "all of the elements which are not in set A but are in set B".
The way to write this expression in mathematics is A'∩B
How to solve for the expressionIn order to get the right way to write this expression we have to break it down in two parts.
First we are told that some of the elements are not in A.
This is represented as A'.
Then we are told that they are in the set B. Hence we have it written as B.
Then the expression not in set A but are in set B would be written as
A'∩B.
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For the first 8 months at his new job, James gets
an increase in pay. What is the rate at which
James receives a pay increase? Approximately
what was his starting pay? Write an equation to
model the relationship.
James receives a pay increase of 100% per month and has a starting salary of $500.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let b represent the rate of pay increase and a represent the initial pay.
Let us assume that he had received 128000 after the 8 months. hence:
128000 = ab⁸ (1)
Also in the 10th month, he had received 512000, hence:
512000 = ab¹⁰ (2)
From the both equations:
a = 500, b = 2
rate of increase = 200% - 100% = 100%
James receives a pay increase of 100% per month and has a starting salary of $500.
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what would 41.2 hours converted into minutes?
What is the value of log5^125?
Step-by-step explanation:
I'm going to assume you mean
[tex]log(5^{125})[/tex] which can be rewritten as [tex]125 log(5)[/tex] since if you start with [tex]log(a) = x = > 10^x=a\\(10^x)^c=a^c\\c *log(a^c)[/tex]since when you do (10^x)^c you're just multiplying the exponent which is represented by the x but is equal to the log you just multiply the log which is why you can bring it in front.
You can then approximate the value of log(5) using a calculator and multiply by 125 to get around 87.371
what of the following points are on the line given by the equation y = 5x
I NEED HELP ON THESE 2 PROBLEMS. pls good anawers or bad rating and report
Answer:
1.a. skew
1.b. parallel
1.c. none of the above
1.d. perpendicular
1.e. parallel
1.f. skew
2.a. never
2.b. always
2.c. always
2.d. sometimes, sometimes, sometimes
Step-by-step explanation:
For these questions, it is critical to know the definitions of each of the terms.
DefinitionsPerpendicular: Two intersecting lines that form a right angle.Parallel: Two non-intersecting lines that are coplanar.Skew: Two non-intersecting lines that are not coplanar.Problem Breakdown
1.a. skew [tex]\overset{\longleftrightarrow}{AB} \text{ and } \overset{\longleftrightarrow}{EF}[/tex]
Point E is not on line AB. Any three non-linear points form a unique plane (left face), so A, B, and E are coplanar. Point F is not in plane ABE. Since line EF and line AB do not intersect and are not coplanar, they are skew.
1.b. parallel [tex]\overset{\longleftrightarrow}{CD} \text{ and } \overset{\longleftrightarrow}{EH}[/tex]
Line CD is parallel to line AB, and line AB is parallel to line EH. Parallel lines have a sort of "transitive property of parallelism," so any pair of those three lines is coplanar and parallel to each other.
1.c. none of the above [tex]\overset{\longleftrightarrow}{AC} \text{ and } \overset{\longleftrightarrow}{GA}[/tex]
Line AC and line GA share point A, necessarily intersecting at A. Therefore, line AC and line GA cannot be parallel or skew.
Any three points form a unique plane, so these two lines are coplanar, however, do they form a right angle? If the figure is cube-shaped, as depicted, then no.
Note that each line segment AC, AG, and CG are all the diagonal of a cube face. A cube has equal edge lengths, so each of those diagonals would be equal. Thus, triangle ACG is an equilateral triangle.
All equilateral triangles are equiangluar, and by the triangle sum theorem, the measures of the angles of a planar triangle must sum to 180°, forcing each angle to have a measure of 60° (not a right angle). So, lines AC and GA are not perpendicular. (If the shape were a rectangular prism, these lines still aren't perpendicular, but the proof isn't as neat, there is a lot to discuss, and a character limit)
"none of the above"
1.d. perpendicular [tex]\overset{\longleftrightarrow}{DG} \text{ and } \overset{\longleftrightarrow}{HG}[/tex]
Line DG and line HG share point G, so they intersect. Both lines are the edges of a face, so they intersect at a right angle. Perpendicular
1.e. parallel [tex]\overset{\longleftrightarrow}{AC} \text{ and } \overset{\longleftrightarrow}{FH}[/tex]
Line AC is a diagonal across the top face, and line FH is a diagonal across the bottom face. They are coplanar in a plane that cuts straight through the cube from top to bottom, but diagonally through those faces.
1.f. skew [tex]\overset{\longleftrightarrow}{CD} \text{ and } \overset{\longleftrightarrow}{AG}[/tex]
Point A is not on line CD, and any three non-linear points form a unique plane (the top face), so C, D, and A are coplanar. Point G is not in plane ACD. Since line CD does not intersect and is not coplanar with line AG, by definition, the lines are skew.
2.a. Two lines on the top of a cube face are never skew
Two lines are on a top face are necessarily coplanar since they are both in the plane of the top face. By definition, skew lines are not coplanar. Therefore, these can never be skew.
2.b. Two parallel lines are always coplanar.
If two lines are parallel, by definition of parallel lines, they are coplanar.
2.c. Two perpendicular lines are always coplanar
If two lines AB and CD are perpendicular, they form a right angle. To form a right angle, they must intersect at the right angle's vertex (point P).
Note that A and B are unique points, so either A or B (or both) isn't P; similarly, at least one of C or D isn't P. Using P, and one point from each line that isn't P, those three points form a unique plane, necessarily containing both lines. Therefore, they must always be coplanar.
2.d. A line on the top face of a cube and a line on the right side face of the same cube are sometimes parallel, sometimes skew, and sometimes perpendicular
Consider each of the following cases:
Parallel: Consider line CD (top face), and line GF (right face). They are coplanar and don't intersect. By definition, parallel.Skew: Consider line AD (top face), and line GF (right face). They aren't coplanar and don't intersect. By definition, skew.Perpendicular: Consider line AD (top face), and line GD (right face). They do intersect at D, and form a right angle. By definition, perpendicular.None of the above: Consider line AC (top face), and line GC (right face). These lines intersect at C, but as discussed in part 1.c, they form a 60° angle, not a right angle. By definition, "none of the above".These 4 cases prove it is possible for a pair of top face/right face lines to be parallel, perpendicular, skew, or none of the above. So, those two lines are neither "always", nor "never", one of those choices. Therefore, they are each "sometimes" one of them.