Answer:
Rewrite equations:
y=x−1;−4x+3y=−2
Step: Solvey=x−1for y:
y=x−1
Step: Substitutex−1foryin−4x+3y=−2:
−4x+3y=−2
−4x+3(x−1)=−2
−x−3=−2(Simplify both sides of the equation)
−x−3+3=−2+3(Add 3 to both sides)
−x=1
−x
−1
=
1
−1
(Divide both sides by -1)
x=−1
Step: Substitute−1forxiny=x−1:
y=x−1
y=−1−1
y=−2(Simplify both sides of the equation)
Answer:
x=−1 and y=−2
What is the solution of the equation?
Answer:
The answer is x = 7
Step-by-step explanation:
√2x − 5 + 4 = x
= √2x − 5 + 4 + − 4 = x + −4 (add -4 to both sides)
=√2x − 5 = x − 4
=√2x−5=x−4
2x−5=(x−4)2 (Square both sides)
2x−5=x2−8x+16
2x−5−(x2−8x+16)=x2−8x+16−(x2−8x+16)(Subtract x^2-8x+16 from both sides)
−x2+10x−21=0
(−x+3)(x−7)=0 (Factor left side of equation)
−x+3=0 or x−7=0 (Set factors equal to 0)
x=3 or x=7
Answer: 7
Step-by-step explanation:
[tex]\sqrt{2x-5}+4 = x[/tex]
Restriction: x ≥ 4 why? because [tex]\sqrt{2x-5}\geq0[/tex]
[tex]\sqrt{2x-5}+4 = x[/tex]
-4 -4
[tex]\sqrt{2x-5} = x - 4[/tex]
[tex](\sqrt{2x-5})^2 = (x - 4)^2[/tex]
2x - 5 = x² - 8x + 16
-2x +5 -2x +5
0 = x² - 10x + 21
0 = (x - 3) (x - 7)
0 = x - 3 0 = x - 7
3 = x [tex]\big{\boxed{7=x}}[/tex]
↓
not valid
Eric estimated 28x48 by finding 30x50. His estimate was 1,500,but he says the actual product will be greater than that amount.Is he correct? Explain how you know
Answer:
Eric is not correct.
Step-by-step explanation:
We have been given that Eric estimated [tex]28\times 48[/tex] by finding [tex]30\times 50[/tex]. His estimate was 1500, but he says the actual product will be greater than that amount.
To find if Eric is correct or not, let us see how to estimate an answer.
While estimating our given numbers we will round to nearest tenth and change the digit to the right of the rounding place to 0.
As 8 is greater than 5, so we will round 28 to 30 and 48 to 50.
Since we are rounding up, so our estimated answer will be greater than actual answer, therefore, Eric is not correct.
What is the value of the discriminant of the quadratic equation -2x^2=-8x+8, and what does its value mean about the number of real number solutions the equation has?
Answer:
B on edg
Step-by-step explanation:
You got this chief.
Find the Value of x
(a)8\3
(b)8
(c)4\3
(d)16
Answer:
(a) 8√3
Step-by-step explanation:
All of the triangles are similar, so the ratio of hypotenuse to long side is the same for all. Comparing the middle-size triangle to the largest, we have ...
... x/12 = 16/x
... x² = 192 . . . . . . . . . . . . . multiply by 12x; next take the square root
... x = √192 = 8√3 . . . . . . seems to match answer choice (a)
A box of Munchkins contains chocolate and glazed donut holes. If Jacob ate 2 chocolate
Munchkins, then 1/11 of the remaining Munchkins would be chocolate. If he instead added 4
glazed Munchkins to the original box, 1/7 of the Munchkins would be chocolate. How many total
Munchkins are in the original box?
Answer:
24 munchkins.
Step-by-step explanation:
Let C be the number of chocolate and D be number of glazed donut holes in the original box.
We are told if Jacob ate 2 chocolate munchkins, then 1/11 of the remaining Munchkins would be chocolate. We can represent this information as:
[tex]C-2=\frac{1}{11}*(C+D-2)...(1)[/tex]
We are also told if he instead added 4 glazed Munchkins to the original box, 1/7 of the Munchkins would be chocolate. We can represent this information as:
[tex]C=\frac{1}{7}*(C+D+4)...(2)[/tex]
Upon substituting C's value from equation (2) in equation (1) we will get,
[tex]\frac{1}{7}*(C+D+4)-2=\frac{1}{11}*(C+D-2)[/tex]
Let us have a common denominator on right side of equation.
[tex]\frac{1}{7}*(C+D+4)-\frac{7*2}{7}=\frac{1}{11}*(C+D-2)[/tex]
[tex]\frac{C+D+4-14}{7}=\frac{1}{11}*(C+D-2)[/tex]
Multiplying both sides of our equation by 7, we will get,
[tex]7*\frac{C+D-10}{7}=7*\frac{1}{11}*(C+D-2)[/tex]
[tex]C+D-10=\frac{7}{11}*(C+D-2)[/tex]
Multiplying both sides of our equation by 11, we will get,
[tex]11*(C+D-10)=11*\frac{7}{11}*(C+D-2)[/tex]
[tex]11*(C+D-10)=7*(C+D-2)[/tex]
[tex]11C+11D-110=7C+7D-14[/tex]
[tex]11C-7C+11D-7D=-14+110[/tex]
[tex]4C+4D=96[/tex]
[tex]4(C+D)=96[/tex]
[tex](C+D)=\frac{96}{4}[/tex]
[tex](C+D)=24[/tex]
Therefore, the total number of Munchkins in original box is 24.
Ben is 20 years older than Daniel. Ben and Daniel first met two years ago. Three years ago, Ben was 33 times as old as Daniel.
how old is Ben now?
Answer: 33
Step-by-step explanation:
Today 3 years ago
Ben: x + 20 (x + 20) - 3
Dan: x x - 3
3 years ago, Ben was 3 times as old as Daniel
⇒ (x + 20) - 3 = 3(x - 3)
x + 17 = 3x - 9
17 = 2x - 9
26 = 2x
13 = x
Today, Ben = x + 20
= 13 + 20
= 33
Answer:
Ben is 33 years old
Step-by-step explanation:
Let Ben’s age = B
Let Daniel’s age = x
B = x + 20 (20 years older than Daniel)
Their ages 3 years ago:
B = 3x (also given in problem)
B = x + 20 (given in problem)
x + 20 = 3x (Since both equations are equal to B, set them equal to each other)
20 = 2x (simplify)
x = 10
B = 10 + 20 = 30
Their ages now:
Add 3 to their "ages three years ago"
New x = Daniel’s age = old age + 3 = 10 + 3 = 13 years old
B = Ben’s age = old age + 3 = 30 + 3 = 33 years old
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Answer:
Step-by-step explanation:
Problem One (left panel)
Question A
The y intercept happens when x = 0That being said, the y intercept is 50. It was moving when the timing began.Question B
The rate of change = (56 - 52)/(3 - 1) = 4/2 = 2 miles / hour^2 (you have a slight acceleration.
Question C
60 = a + (n-1)d60 = 50 + (n - 1)*210/2 = (n - 1)*2/25 = n - 1 6 = nThe way I have done it the domain is n from 1 to 6
Question 2 (Right Panel)
Question A
The equation for the table is f(x) = 3x - 3 which was derived simply by putting all three points into y = ax + b and solving.
f(0) = ax + b-3 = a*0) + bb = - 3So far what you have is f(x) = ax - 3f(-1) = a*(-1) - 3 but we know (f(-1)) = -6- 6 = a(-1) - 3 add 3 to both sides-6 +3 = a(-1) -3 + 3-3 = a*(-1) Divide by - 1a = 3f(x) = 3x - 3 Answer for f(x)The slope of f(x) = the coefficient in front of the xf(x) has a slope of 3g(x) has a slope of 4Part B
f(x) has a y intercept of - 3g(x) has a y intercept of -5f(x) has the greater y intercept.-3 > - 5The probability of an event is 3/10 . What are the odds of the same event?
Choices:
10/13
3/13
7/10
3/7
Please explain your answer if possible. Thanks
Answer:
D. [tex]\frac{3}{7}[/tex]
Step-by-step explanation:
We have been given that the probability of an event is 3/10.
To find the odds of the same event we will use formula:
[tex]\text{Odds of an event}=\frac{\text{Probability of the event}}{\text{1-Probability of the event}}[/tex]
[tex]\text{Odds of the event}=\frac{\frac{3}{10}}{1-\frac{3}{10}}[/tex]
[tex]\text{Odds of the event}=\frac{\frac{3}{10}}{\frac{1*10}{10}-\frac{3}{10}}[/tex]
[tex]\text{Odds of the event}=\frac{\frac{3}{10}}{\frac{10-3}{10}}[/tex]
[tex]\text{Odds of the event}=\frac{\frac{3}{10}}{\frac{7}{10}}[/tex]
Dividing a fraction with another fraction is same as multiplying the 1st fraction by the reciprocal of second fraction.
[tex]\text{Odds of the event}=\frac{3}{10}\times \frac{10}{7}[/tex]
[tex]\text{Odds of the event}=\frac{3}{7}[/tex]
Therefore, the odds of the same event is [tex]\frac{3}{7}[/tex] and option D is the correct choice.
Help me with this question! Anybody. {will receive 15 pts.}
Answer:
The answer is 5.
Step-by-step explanation:
5+(-9) or
5-9=-4
Can anyone please tell me what I’m doing wrong. It’s not giving me the full marks.
Answer:
Area shaded part = 116 to three sig digs.
Step-by-step explanation:
I don't know if your answer is correct or not, but the number of sig digs is not. The way the question has been posed is not to 3 sig digs either.
The answer should be 116 if they want it to three sig digs.
Let's see if the answer is correct
Area of the whole circle.
Area of a circle = pi*r^2r = 6.5 cm Note: this is 2 sig digs.pi= 3.14Area = 3.14 * 6.5^2Area = 3.14 * 42.25Area= 132.665Area of unshaded smaller circle
pi = 3.14r = 2.3Area = 3.14 * 2.3^2Area = 3.14 * 5.29Area = 16.61Area of the shaded part
Area of shaded part = area of the whole circle - Area of smaller circleArea of the shaded part = 132.66 - 16.61Area of the shaded part = 116.01To 3 sig digs is 116Answer:
116 cm^2 to 3 significant figures.
Step-by-step explanation:
Your calculations are correct. You just haven't corrected the answer to 3 significant figures.
Lisa spent 8 minutes on the phone while routing 2 phone calls. In all, how many phone calls does Lisa have to route to spend a total of 12 minutes on the phone? Solve using unit rates.
Lisa spends 4 minutes per call, so to spend a total of 12 minutes on the phone, she needs to route 3 calls.
Lisa spent 8 minutes on the phone routing 2 phone calls, which means she spends 4 minutes per call. This is calculated using the unit rate which is the time spent per phone call. To find out how many phone calls Lisa would have to route to spend a total of 12 minutes, we can set up a proportion where 4 minutes corresponds to 1 phone call (the unit rate), and 12 minutes corresponds to the number of phone calls we want to find (let's call it x calls).
The proportion is as follows:
4 minutes/1 call = 12 minutes/x calls
Cross-multiplying to solve for x gives us:
4 * x = 12 * 1
4x = 12
x = 12 / 4
x = 3 calls
Therefore, Lisa needs to route a total of 3 calls to spend 12 minutes on the phone.
15 rulers cost £3
How much do 40 rulers cost?
Answer:
The cost of the 40 rulers is £8 .
Step-by-step explanation:
As given in the question
15 rulers cost £3
i.e
15 rulers = £3
Cost of the one ruler.
[tex]1\ ruler = \frac{3}{15}[/tex]
1 ruler cost = £ 0.2
Now find out the cost for 40 rulers .
Cost for 40 rulers = 40 × 0.2
= £8
Therefore the cost of the 40 rulers is £8 .
The question involves finding the unit price of a ruler and then finding the cost of 40 rulers. The price for one ruler is £0.2, and hence, the price for 40 rulers would be £8.
Explanation:This is a mathematics question about unit price and proportionality. First, we need to find the price of one ruler by dividing the total cost by the number of rulers. So, £3 divided by 15 rulers is £0.2 per ruler. To find out how much 40 rulers costs, we multiply the price of one ruler by the quantity we desire which is 40 rulers. So, £0.2 multiplied by 40 equals to £8.
Therefore, the cost of 40 rulers would be £8.
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The ordered pairs model an exponential growth function. {(−1,34), (0,51), (1,76.5), (2,114.75)} What is the function equation? f(x)=__________
Answer:
[tex]y= 51(1.5)^x[/tex]
Step-by-step explanation:
The ordered pairs model an exponential growth function. {(−1,34), (0,51), (1,76.5), (2,114.75)}
Exponential function equation is y=ab^x
Lets plug in the ordered pairs and find our 'a' and 'b'
(0,51)
[tex]51= ab^0[/tex]
a= 51
(−1,34)
[tex]34= 51b^{-1}[/tex]
Divide both sides by 51
[tex]\frac{34}{51} = b^{-1}[/tex]
So b = 51/34
b= 1.5
Plug in the values of 'a' and 'b'
Equation becomes [tex]y= 51(1.5)^x[/tex]
contractors insurance has a $75 sign-up fee. Once singed up the insurace cost $61 each month. Which of the following equations represents the relationship between the number of months a contractor is insured,m, and the total cost of contractors insurance,c?
Answer:
The equation should read c = 61m + 75
Step-by-step explanation:
We can find this because we know that the 61 dollars is a dependent cost on the number of months. Therefore they have to be multiplied together.
The 75 is a constant, which means it gets added to the end.
What is the slope of the line? – –Which equation shows the point-slope form of the line that passes through (3, 2) and has a slope of ? y + 2 =(x + 3) y – 2 = (x – 3) y + 3 = (x + 2) y – 3 = (x – 2)
Answer:
The correct answer is B) y – 2 = (x – 3)
Step-by-step explanation:
In order to find this, look at the base form of point-slope form.
y - y1 = m(x - x1)
Seeing as there is subtraction symbols after x and y, we know that we would be subtracting the x and y values. Since 2 is the y value, it would go in for y1 and 3 being the x value puts it in for x1. Once you do that, you'll get the equation above.
Answer:
-1/2
Step-by-step explanation:
D is the midpoint of CE. E has coordinates (−8, −3), and D has coordinates (2, 3). Find the coordinates of C.
[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ C(\stackrel{x_1}{x}~,~\stackrel{y_1}{y})\qquad E(\stackrel{x_2}{-8}~,~\stackrel{y_2}{-3}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{-8+x}{2}~~,~~\cfrac{-3+y}{2} \right)=\stackrel{\stackrel{midpoint}{D}}{(2,3)}\implies \begin{cases} \cfrac{-8+x}{2}=2\\[1em] -8+x=4\\ \boxed{x=12}\\[-0.5em] \hrulefill\\ \cfrac{-3+y}{2}=3\\[1em] -3+y=6\\ \boxed{y=9} \end{cases}[/tex]
How do i translate (x,y) ?
One of my questions is asking me to translate , (x,y) i know the formula for translating, (a,b) x+a,y+b But a=x and b=y. How do i solve this?
Answer: (x, y - 8)
Step-by-step explanation:
Start with the innermost parentheses (reflection over the y-axis), which changes the sign of the y-coordinate:
(x, y) → (-x, y)Now perform the next inner parenthesis (rotation 180°), which changes the signs of both the x- and y-coordinates:
(-x, y) → (x, -y)Now perform the last set of parenthesis (reflection over y=4), which has the rule of (x, y) → (x, y + 2(4 - y)) = (x, y + 8 - 2y) = (x, 8 - y):
(x, -y) → (x, -8 + y)kiran and clare live 24 mile away from each other along a rail trail. One Saturday the two friends started walking toward each other along a trail at 8:00 am with a plan to have a picnic when they meet if kiran walks 3 miles per hour while clare walks 3.4 miles per hour. At what time will the two friends meet and have their picnic
Answer:
[tex]11:45\ am[/tex]
Step-by-step explanation:
Total distance between Kiran and Clare is 24 miles. They started walking at 8.00 am.
Speed of Kiran is 3 miles/hr and speed of Clare is 3.4 miles/hr.
So the relative speed of them is,
[tex]=3+3.4=6.4[/tex] miles/hr
Both the speed are added as they moving in opposite direction.
We know that,
[tex]\text{Relative speed}=\dfrac{\text{Relative distance}}{\text{Time}}[/tex]
i.e [tex]\text{Time}=\dfrac{\text{Relative distance}}{\text{Relative speed}}[/tex]
Putting the values,
[tex]t=\dfrac{24}{6.4}=3.75\ hr=3\ hr\ 45\ min[/tex]
As they started at 8.00 am, so the time at which will meet will be,
[tex]=8+3\ hr\ 45\ min=11:45[/tex]
Answer:
11:45PM
Step-by-step explanation:
Find the value of x in the trapezoid below. Show equations and all work that leads to your answer.
Answer:
x = 21.5
Step-by-step explanation:
In the given trapezoid, the angles are consecutive interior angles.
i.e, consider the parallel sides of a trapezoid as parallel lines and the common side as the transversal. So, it is parallel lines cut by a transversal make a pair of consecutive interior angles which are supplementary.
i.e,
(5x + 13)° + (3x - 5)° =180°
5x + 13 + 3x - 5 = 180
8x + 8 = 180
8x = 180 - 8 =172
x= 172/8 = 21.5
So, 5x + 13 = 5(21.5) + 13 = 120.5°
3x - 5 = 3(21.5) -5 = 59.5°
What is the correct classification of the power function?
Answer:
Cubic
Step-by-step explanation:
A cubic function is a function whose degree is 3 (the highest exponent or power on the variable). It has a distinctive sideways S shape that can either start down and end up or start up and end down. It also crosses through the x-axis an odd number of times. It will have up to three x-intercepts or roots.
Which function represents a line with a slope of ?4 and a y-intercept of ?2? A) y = 4x ? 2 B) y = ?4x + 2 Eliminate C) y = ?4x ? 2 D) y = ?2x ? 4
Answer:
A function describes this way is y = 4x + 2
Step-by-step explanation:
In order to find this, we have to start with slope-intercept form.
y = mx + b
Knowing this, we can then input the slope in for m and the intercept in for b. This will give us the equation.
y = 4x + 2
could someone help me?
Answer:
See below
Step-by-step explanation:
m < 1 = 180 - m<4 (supplementary angles)
= 180 - 143
= 37 degrees
m <2 = m< 4 ( opposite angles)
= 143 degrees
M< 3 = m < 1 = 37 degrees (opposite angle)
Angle 1 measures 143 degrees, while angles 2 and 3 both measure 37 degrees.
To find the measure of angle 2, we can use the fact that adjacent angles on a straight line are supplementary. This means that angle 2 and angle 4 add up to 180 degrees. Since angle 4 is 143 degrees, angle 2 is 180 - 143 = 37 degrees.
To find the measure of angle 3, we can use the fact that corresponding angles of parallel lines cut by a transversal are congruent. This means that angle 3 and angle 2 have the same measure. Since angle 2 is 37 degrees, angle 3 is also 37 degrees.
Therefore, the measures of angle 1, 2, and 3 are 143 degrees, 37 degrees, and 37 degrees, respectively.
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Can you please put the answer to 60000000x1500 because I have a really important test cin=ming up and
What are the coordinates of the midsegment that is parallel to side BC.
(0, 2) and (1, 0)
(0, 2) and (-2, -1)
(1, 0) and (2, 1)
(1, 0) and (-2, -1)
Answer:
Correct choice is D (1,0) and (-2,-1)
Step-by-step explanation:
Triangle DBC has vertices B(-3,1), C(3,3) and D(-1,-3). By the triangle midline theorem, the midline joining the midpoints of two sides is parallel to the third side. Thus, you have to find the coordinates of the midpoints of the sides DB and DC of the triangle DBC.
Let point E be the midpoint of the side BD, then point E has coordinates [tex]\left(\dfrac{x_B+x_D}{2},\dfrac{y_B+y_D}{2}\right).[/tex] If B(-3,1) and D(-1,-3), then
[tex]E\left(\dfrac{-3+(-1)}{2},\dfrac{1+(-3)}{2}\right)\Rightarrow E(-2,-1).[/tex]
Let point F be the midpoint of the side DC, then point F has coordinates [tex]\left(\dfrac{x_C+x_D}{2},\dfrac{y_C+y_D}{2}\right).[/tex] If C(3,3) and D(-1,-3), then
[tex]F\left(\dfrac{3+(-1)}{2},\dfrac{3+(-3)}{2}\right)\Rightarrow F(1,0).[/tex]
Answer:
D: (1,0) and (-2,-1)
Step-by-step explanation:
if you where to plot all the other answers and take a straight edge (ruler) and drew a line through them you would find this is the only proper answer. and i used geogebra
A construction company plans to invest in a building project. There is a 30% chance that the company will lose $30,000, a 40% chance of a break even, and a 30% chance of a $60,000 profit. Based ONLY on this information, what should the company do?
The expected value is $9,000.00, so the company should proceed with the project.
B) The expected value is $18,000.00, so the company should proceed with the project.
C) The expected value is −$9,000.00, so the company should not proceed with the project.
D) The expected value is −$18,000.00, so the company should not proceed with the project.
Answer: Option 'A' is correct.
Step-by-step explanation:
Since we have given that
30% chance that the company will lose $30000.
40% chance of a break even that there is no loss and no profit.
30% chance that the company will profit $ 60000.
As we know the formula for "Expectation":
So, Expected value will be
[tex]\frac{30}{100}\times (-30000)+\frac{40}{100}\times 0+\frac{30}{100}\times 60000\\\\=03\times (-30000)+0.4\times 0+0.3\times 60000\\\\=-9000+18000\\\\=\$9000[/tex]
Expected value is $9000. So, the company should proceed with the project.
Hence, Option 'A' is correct.
Please answer this question for me!! Forty points and brainliest!
Answer: The Answer is B
Step-by-step explanation:
FIrst we see post 7 and -5 we need to cancel out seven sooooo we make sevena a neg and do -5 and -7 and we get -12 we divide this by -2 and we get post 6
x=6
-2x+7=-5
-2(-7)+7=-5(-7)
-2(-2)x=-12(-2)
x=6
Answer:
b 6
Step-by-step explanation:
-2x+7 = -5
Subtract 7 from each side
-2x +7-7 = -5 -7
-2x = -12
Divide by -2
-2x/2 = -12/-2
x=6
Suppose the heights of the members of a population follow a normal distribution. If the mean height of the population is 68 inches and the standard deviation is 4 inches, 95% of the population will have a height within which of the following ranges?
The empirical rule states that at 95% the measurements would be within 2 standard deviations of the mean.
You are given a mean of 68 inches and a standard deviation of 4.
2 times the standard deviation = 2 x 4 = 8
So 95% of the heights would be between 68-8 = 60 inches and 68+8 = 76 inches.
Answer:
60 -76 is the range
Step-by-step explanation:
As the graph shows, if we are in 2 standard deviations of the mean, we are in (34.1 + 13.6) *2 = 47.7*2 = 95.4 %
Our mean is 68
2 standard deviations is 2 * 4 = 8
68-8 = 60
68 * 8 = 76
We need to be between 60 and 76 to have at 95% confidence interval
Two boxes are stacked together.
What is the combined volume of the two boxes?
A. 105 cm
B. 280 cm
C. 385 cm
D. 4,200 cm
The formula of a volume of a rectangle prism:
[tex]V=lwh[/tex]
l - length
w - width
h - height
The larger rectangle prism:
[tex]l = 10cm,\ w=7cm,\ h=4cm[/tex]
Substitute:
[tex]V_1=(10)(7)(4)=280\ cm^3[/tex]
The smaller rectangle prism:
[tex]l=3cm,\ w=7cm,\ h=5cm[/tex]
Substitute:
[tex]V_2=(3)(7)(5)=105\ cm^3[/tex]
Total volume:
[tex]V=V_1+V_2\to V=280\ cm^3+105\ cm^3=385\ cm^3[/tex]
Answer: C. 385 cm³.The combined volume is 385 cm³
What is a rectangular parallelepiped?A rectangular parallelepiped is a polyhedron with six faces, which is also known as a hexahedron, each of which is a parallelogram.Volume of a rectangular parallelepiped can be calculated by multiplying its length , breadth and height.How to calculate the combined volume?The two boxes can be said to have a shape of rectangular parallelepiped.For the lower box, the volume will be = (7 x 10 x 4) cm³ = 280 cm³
For the upper box, the volume will be = (7 x 5 x 3) cm³ = 105 cm³
Now, for the combined volume, we need to add the individual volumes of the two boxes.∴ Combined volume will be equal to = ( 280 + 105) cm³ = 385 cm³
Option C is correct.
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find the polynomial function
Answer:
y = 2x³ +x² -3x +6
Step-by-step explanation:
The polynomial correlation function of a graphing calculator can work with the table of function values to give you the equation. (See attached)
_____
If you want to solve this by hand, you can write the equations for the unknown coefficients in the polynomial function, then solve those. First of all, notice that you are given the y-intercept, (0, 6), so you know one of the coefficients already. That leaves a system of 3 equations in 3 unknowns that need to be solved.
___
Writing the equations
... ax³ +bx² +cx +6 = y . . . . the generic linear equation in a, b, c that you're writing
For x = -1:
... -a +b -c +6 = 8
For x = 1
... a + b + c + 6 = 6
For x = 2
... 8a +4b +2c +6 = 20
In augmented matrix terms, this set of linear equations looks like ...
[tex]\left[\begin{array}{ccc|c}-1&1&-1&2\\1&1&1&0\\8&4&2&14\end{array}\right][/tex]
___
Solving the equations
There are a number of ways to solve these. Again, a graphing calculator often has solution functions, such as reducing a matrix to row-echelon form, that will solve this in a keystroke or two.
If you add the first two equations, you get ...
... 2b = 2 ⇒ b = 1
Substituting that into the first equation allows you to simplify it to ...
... -a -c = 1
Dividing the third equation by 2 (and substituting for b) makes it ...
... 4a +c = 5
Adding these equations gives ...
... 3a = 6 ⇒ a = 2
Then ...
... c = -a -1 = -2 -1 = -3
And our polynomial function is ...
... y = 2x³ +x² -3x +6
Two events E1 and E2 are called independent if p(E1 â© E2) = p(E1)p(E2). For each of the following pairs of events, which are subsets of the set of all possible outcomes when a coin is tossed three times, determine whether or not they are independent. a) E1: tails comes up with the coin is tossed the first time; E2: heads comes up when the coin is tossed the second time. b) E1: the first coin comes up tails; E2: two, and not three, heads come up in a row. c) E1: the second coin comes up tails; E2: two, and not three, heads come up in a row.
Answer: a) Independent
b) Independent
c) Dependent
Step-by-step explanation:
Since, If a coin is tossed three times,
Then, total number of outcomes, n(S) = 8
a) [tex]E_1[/tex] : tails comes up with the coin is tossed the first time;
[tex]E_1[/tex] = { TTT, THH, THT, TTH }
[tex]E_2[/tex] : heads comes up when the coin is tossed the second time.
[tex]E_2[/tex] = { THT, HHH, THH, HHT }
Thus, [tex]n(E_1)=4[/tex]
⇒ [tex]P(E_1)=\frac{n(E_1)}{n(S)}=\frac{4}{8}=\frac{1}{2} [/tex]
Similarly, [tex]P(E_2)=\frac{1}{2}[/tex]
⇒ [tex]P(E_1)\times P(E_2)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4} [/tex]
Since, [tex]E_1\cap E_2 [/tex] = { THH, THT }
[tex]n(E_1\cap E_2) = 2 [/tex]
⇒ [tex]P(E_1\cap E_2) = \frac{n(E_1\cap E_2)}{n(S)}= \frac{2}{8}=\frac{1}{4}[/tex]
Thus, [tex]P(E_1\cap E_2)=P(E_1)\timesP(E_2)[/tex]
Therefore, [tex]E_1[/tex] and [tex]E_2[/tex] are independent events.
B) [tex]E_1[/tex] : the first coin comes up tails
[tex]E_1[/tex] = { TTT, THH, THT, TTH }
[tex]E_2[/tex] : two, and not three, heads come up in a row
[tex]E_2[/tex] = { HHT, THH }
Thus, [tex]n(E_1)=4[/tex]
⇒ [tex]P(E_1)=\frac{n(E_1)}{n(S)}=\frac{4}{8}=\frac{1}{2} [/tex]
Similarly, [tex]P(E_2)=\frac{1}{4}[/tex]
⇒ [tex]P(E_1)\times P(E_2)=\frac{1}{2}\times \frac{1}{4}=\frac{1}{8} [/tex]
Since, [tex]E_1\cap E_2 [/tex] = { THH }
[tex]n(E_1\cap E_2) = 1 [/tex]
⇒ [tex]P(E_1\cap E_2) = \frac{n(E_1\cap E_2)}{n(S)}= \frac{1}{8}[/tex]
Thus, [tex]P(E_1\cap E_2)=P(E_1)\timesP(E_2)[/tex]
Therefore, [tex]E_1[/tex] and [tex]E_2[/tex] are independent events.
C) [tex]E_1[/tex] : the second coin comes up tails;
[tex]E_1[/tex] = { HTH, HTT, TTT, TTH }
[tex]E_2[/tex] : two, and not three, heads come up in a row
[tex]E_2[/tex] = { HHT, THH }
Thus, [tex]n(E_1)=4[/tex]
⇒ [tex]P(E_1)=\frac{n(E_1)}{n(S)}=\frac{4}{8}=\frac{1}{2} [/tex]
Similarly, [tex]P(E_2)=\frac{1}{4}[/tex]
⇒ [tex]P(E_1)\times P(E_2)=\frac{1}{2}\times \frac{1}{4}=\frac{1}{8} [/tex]
Since, [tex]E_1\cap E_2 [/tex] = [tex]\phi[/tex]
[tex]n(E_1\cap E_2) = 0 [/tex]
⇒ [tex]P(E_1\cap E_2) = 0[/tex]
Thus, [tex]P(E_1\cap E_2)\neq P(E_1)\timesP(E_2)[/tex]
Therefore, [tex]E_1[/tex] and [tex]E_2[/tex] are dependent events.
Answer:
P (E1) = 18 / 38
P (E2) = 18 / 38
P (E1 and E2) = 10 / 38
Step-by-step explanation: