Answer: [tex]f(x)=x^3 -51x^2 -3x+153[/tex]
Step-by-step explanation:
By the conjugate root theorem, the roots are [tex]\sqrt{3}, -\sqrt{3}, 51[/tex].
Letting the leading coefficient be 1,
[tex]f(x)=(x-\sqrt{3})(x+\sqrt{3})(x-51)\\\\=(x^2 -3)(x-51)\\\\=x^3 -51x^2 -3x+153[/tex]
Two cars are driving on the same road, in the same
direction. They start driving from the same place and are
traveling at a constant speed. The second car started
driving 1.5 hours after the first car started driving. If the
second car drives 60 miles per hour and the first drives 40
miles per hour, how many miles will each car have
traveled when the second car catches up to the first?
Answer:
180 miles
Step-by-step explanation:
distance = rate x time
t = time
1st car:
distance = 40t
2nd car:
distance = 60(t - 1,5)
When the car catch up to each other the distances will be the same, so set the equation equal to each other. Calculate the time and then put the time back into either equation and solve for the distance.
40t = 60(t-1.5) Distribute the 60
40t = 60t -(60)1.5
40t = 60t - 90 Subtract 60t from both sides of the equation
-20t = -90 Divide both sides by -20
t = 4.5
Now that we know the time, substitue that back into either equaiton and solve for the time
distance = 40 (4.5)
180 miles
If quadrilateral WXYZ is transformed using the rule T(-1.2), in whatdirections should WXYZ be translated?O 1 unit down, 2 units rightO 1 unit left, 2 units upO 1 unit up, 2 units leftO 1 unit right, 2 units up
I
Three relationships are described below:
I. The amount of time needed to mow a yard increases as the size of the yard increases.
II. The amount of timeneeded to drive from city A to city B decreases as the speed you are driving increases.
III. The income of a worker who gets paid an hourly wage increases as the number of hours worked increases and
increases as the salary rate increases.
What type of variation describes each relationship?
The type of variation that describes each relationship include the following:
Direct variation: the amount of time needed to mow a yard increases as the size of the yard increases.Indirect variation: the amount of time needed to drive from city A to city B decreases as the speed you are driving increases.Joint variation: the income of a worker who gets paid an hourly wage increases as the number of hours worked increases and increases as the salary rate increases.What is an indirect variation?An indirect variation simply refers to a type of proportional relationship in which a variable is inversely proportional to another variable. This ultimately implies that, an indirect variation represents two variables that are inversely proportional to each other, which means as one variable increases, the other variable decreases and vice-versa.
What is direct variation?Direct variation refers to a type of proportional relationship in which a variable is directly proportional to another variable. This ultimately implies that, a direct variation represents two variables that are directly proportional to each other, which means as one variable increases, the other variable also increases and vice-versa.
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the 9th term of arithmetic sequence. Use the formula for 'an' to find 'a20', the 20th term of the sequence 7,3,-1,-5
We will find the value of the 20th term of the sequence 7, 3, -1, and -5.
We have the following sequence:
[tex]7,3,-1,-5[/tex]Finding the common differenceIf we have an arithmetic sequence here, we need to find the common difference for this sequence, and we can do that by finding the difference between the second term and the first term, the difference between the third term and the second term, and so on. If we obtain the same value for the common difference, we have an arithmetic sequence here.
Then we have:
[tex]\begin{gathered} d=3-7=-4 \\ \\ d=-1-3=-4 \\ \\ d=-5-(-1)=-5+1=-4 \end{gathered}[/tex]Then the common difference in this arithmetic sequence is d = -4.
Finding the formula for the arithmetic sequenceWe know that the explicit formula for an arithmetic sequence is:
[tex]a_n=a_1+(n-1)d[/tex]For this case, we have that d = -4, and that the first term, a1 = 7. Then we have the formula for the arithmetic sequence:
[tex]a_n=7+(n-1)(-4)[/tex]Notice that we can expand this expression as follows:
[tex]\begin{gathered} a_n=7+(-4)(n)+(-4)(-1) \\ \\ a_n=7-4n+4 \\ \\ a_n=11-4n \\ \end{gathered}[/tex]Finding the 20th termThen to find the 20th term of the sequence, we have:
[tex]\begin{gathered} a_{20}=7+(20-1)(-4) \\ \\ a_{20}=7+(19)(-4) \\ \\ a_{20}=7-76=-69 \\ \\ a_{20}=-69 \end{gathered}[/tex]Therefore, in summary, we have that the value for the 20th term of the sequence 7, 3, -1, and -5 is -69.
,
Write an expression to show how much Gretchen paid for drama,action, and comedy videos if she paid $4 for each at a sale. Evaluate the expressionGretchen’s video purchasesMystery 6Action 3Comedy 5Drama 2Romance 2
Let d the number of drama videos, c the number of comedy videos and a the number of action videos.
If the cost per video (independently of the genre) is $4, then, for the total cost of the videos Gretchen payed, you can write:
total = 4d + 4c + 4a
Now, based on the given table, you have:
d = 2
c = 5
a = 3
By replacing the previous values into the expression for total, and by simplifying, you obtain:
total = 4(2) + 4(5) + 4(3)
total = 8 + 20 + 12
total = 40
Hence, Gretchen payed $40 for the videos
The probability that John recieves junk mail is 11 percent. If he receives 94 pieces of mail in a week, about how many of them can he expect to be junk mail.a. 5 b. 15 c. 10 d.20
10 (option C)
Explanation:The probability of getting a junk mail = 11%
Number of mails received = 94
Amount that will be junk mail = The probability of getting a junk mail × Number of mails received
Amount that will be junk mail = 11% × 94
= 0.11 × 94 = 10.34
Since we can't have decimal number of mails, we would approximate to the nearest whole number
10.34 to the nearest whole number is 10
Hence, 10 junk mails are expected
Please help me on #1 Please show your work so I can follow and understand
Answer:
Between markers 3 and 4.
Explanation:
We know that each student runs 2 / 11 miles. Given this, how many miles do the first two students run?
The answer is
[tex]\frac{2}{11}\cdot2=\frac{4}{11}\text{miles}[/tex]Now, we know that the course has markers every 0.1 miles. How many markers are ther in 4 /11 miles?
The answer is
[tex]\frac{2}{11}\text{miles}\times\frac{1\text{marker}}{0.1\; miles}[/tex][tex]=3.6\text{ markers}[/tex]This is between markers 3 and 4. Meaning that the second student finishes between markers 3 and 4.
the table shows a proportional relationship between the weight on a spring scale and the distance the spring has stretched. describe the scale you can use on X and Y axes of a coordinate grid that would show all of the distances and weights in the table
Here the values are proportional to each other.
Proportionality ratio is,
[tex]\frac{20}{28}=\frac{5}{7}[/tex]Then scale on X-axis representing weight in Newton is 1 unit is equal to 7 Newton
And on the the Y axis representing distance in cm is 1 unit is equal to 5 cm.
Solve for the dimensions of the rectangle. Area= length•widthThe length of a rectangle is 2cm greater than the width. The area is 80cm2. Find the length and width.
The length of a rectangle is 2cm greater than the width. The area is 80cm2. Find the length and width.
L=W+2
W=W
[tex]\begin{gathered} A=L\cdot W \\ A=(W+2)\cdot W \\ A=W^2+2W \\ A=80\operatorname{cm} \\ Then, \\ 80=W^2+2W \\ W^2+2W-80=0 \end{gathered}[/tex][tex]\Delta=4+320=324[/tex][tex]\begin{gathered} W=\frac{-2\pm\sqrt[]{324}}{2}=\frac{-2\pm18}{2} \\ W_1=\frac{-20}{2}=-10 \\ W_2=\frac{16}{2}=8 \end{gathered}[/tex]The width should be positive, therefore W=8
L=W+2
L=8+2=10
The length is L=10
15=g/7 what does g equal to
Answer:
g = 105
Explanation:
We want to find the value of g if
[tex]15=\frac{g}{7}[/tex]We multiply both sides of the equation by 7
[tex]\begin{gathered} 15\times7=\frac{g}{7}\times7 \\ \\ 105=g \end{gathered}[/tex]Therefore, the value of g is 105
Answer:
[tex]15=g/7[/tex]
We can get the value of g by multiplying the denominator, which in this case is 7.
So,
[tex]g = 15 x 7\\ g=105[/tex]
Find conditions on k that will make the matrix A invertible. To enter your answer, first select 'always', 'never', or whether k should be equal or not equal to specific values, then enter a value or a list of values separated by commas.
To be a matrix to be invertible the determinant of the matrix must be non zero thus for k ≠ 2 the matrix will be invertible.
What is a matrix?matrix, a collection of numbers lined up in rows and columns to produce a rectangular array.
In computer graphics, where they have been used to describe picture transformations and other alterations.
The elements of the matrix, also known as the entry, are the numerals.
A matrix will be invertible only and only if the determinant is non-zero.
Given the matrix A.
The determinant of A is that |A| will be,
|A| = -3(8 - 8) - 0(-k + 2) - 3(-4k + 8) ≠ 0
0 + 0 + -3(-4k + 8) |A| ≠ 0
-4k + 8 ≠ 0
-4k ≠ -8
k ≠ 2
Hence "To be a matrix to be invertible the determinant of the matrix must be non zero thus for k ≠ 2 the matrix will be invertible".
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A total of $6000 is invested: part at 5% and the remainder at 10%. How much is invested at each rate if the annual interest is $590
If a total of $6000 is invested, part at 5% and remainder at 10%, then the amount invested on 10% interest is $5800 and the amount invested on 5% interest is $200
The total amount = $6000
Consider the amount invested on 10% interest as x
The amount invest on 5% interest = (6000-x)
The the equation will be
x×(10/100) + (6000-x)(5/100) = 590
0.1x + 0.05(6000-x) = 590
0.1x + 300 - 0.05x = 590
0.05x +300 = 590
0.05x = 590-300
0.05x = 290
x = 290/0.05
x = $5800
The amount invested on 10% interest = $5800
The amount invested on 5% interest = 6000-5800
= $200
Hence, if a total of $6000 is invested, part at 5% and remainder at 10%, then the amount invested on 10% interest is $5800 and the amount invested on 5% interest is $200
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For circle H, JN = x, NK = 8, LN = 4, and NM = 20.Solve for x.
Solution
Consider the illustration below
Using the idea of the illustration above,
[tex]JN\text{ x NK = LN x NM}[/tex][tex]\begin{gathered} x\text{ x 8 = 4 x 20} \\ 8x=80 \\ x=\frac{80}{8} \\ x=10 \end{gathered}[/tex]The answer is 10
what is the measure in radians of central angle 0 in the circle below
For this exercise you need to use the following formula:
[tex]\theta=\frac{S}{r}[/tex]Where θ is the Central angle in radians, "S" is the arc length and "r" is the radius of the circle.
In this case, you can identify that:
[tex]\begin{gathered} S=8\pi cm \\ r=8\operatorname{cm} \end{gathered}[/tex]Knowing these values, you can substitute them into the formula and then evaluate, in order to find the measure of the Central angle in radians. This is:
[tex]\begin{gathered} \theta=\frac{8\pi cm}{8\operatorname{cm}} \\ \\ \theta\approx\pi radians \end{gathered}[/tex]The answer is:
[tex]\pi radians[/tex]For every 5 tickets that Sean sold Demi sold 4 tickets. If Sean sold 220 tickets how many tickets did Demi sell
Answer:
Demi sold 176 tickets
Step-by-step explanation:
220 divided 5= 44
44 x 4= 176
… But you didn't have to cut me off
Make out like it never happened
And that we were nothing
And I don't even need your love
But you treat me like a stranger
And that feels so rough
You didn't have to stoop so low
Have your friends collect your records
And then change your number
I guess that I don't need that though
Now you're just somebody that I used to know
need help with this question please help
Let:
[tex]k\cdot RT=TU[/tex]Where:
k = Constant of proportionality
[tex]\begin{gathered} k\cdot4=6 \\ solve_{\text{ }}for_{\text{ }}k \\ k=\frac{6}{4} \\ k=\frac{3}{2} \end{gathered}[/tex]So:
[tex]\begin{gathered} k\cdot RS=UV \\ \frac{3}{2}(6)=UV \\ \frac{18}{2}=UV \\ UV=9 \end{gathered}[/tex]Need answer if you could show work would be nice
the mean is 5.8the variance is 2.4We have to find the standard deviation and round it to one decimal place.
To calculate the standard deviation given the variance, we just take the square root.
That because the variance is:
[tex]v=\sigma^2[/tex]And the standard deviation is:
[tex]\sigma[/tex]So, calculating the standard deviation:
[tex]\sigma=\sqrt[]{\sigma^2}=\sqrt[]{v}=\sqrt[]{2.4}=1.54919\ldots\approx1.5[/tex]Thus, the standard deviation is 1.5.
13. A 640 kg of a radioactive substance decays to 544 kg in 13 hours. A. Find the half-life of the substance. Be sure to show your work including the formulas you used. Round to the nearest tenth of an hour. Only solutions using formulas from the 4.6 lecture notes will receive credit.B. How much of the substance is present after 3 days? Be sure to show the model you used.C. How long does it take the substance to reach 185 kg? Be sure to show your work.
EXPLANATION
The equation for half-life is given by the following formula:
[tex]H=\frac{t\cdot\ln(2)}{\ln(\frac{A_0}{A_t})}[/tex]Replacing terms:
[tex]H=\frac{t\cdot\ln(2)}{\ln(\frac{A_0}{A_t})}=\frac{13\cdot\ln(2)}{\ln(\frac{640}{544})}=\frac{9.0109}{0.1625}=55.45[/tex]The half-life time is H =55.4 hours.
B) After three days, that is, 72 hours, the amount of substance will be given by the following relationship:
[tex]A=A_o\cdot e^{-(\frac{\ln2}{H})t}=640\cdot e^{-(\frac{\ln2}{55.4})\cdot72}=640\cdot e^{-0.90084}[/tex]Multiplying terms:
[tex]A=640\cdot0.4062=259.96\text{ Kg}[/tex]There will be 259.96 Kg after 3 days.
C) In order to compute the number of days that will take to the substance to reach a concentration equal to 185 Kg, we need to apply the following formula:
[tex]t=\frac{\ln (\frac{A}{A_o})}{-\frac{\ln (2)}{t\frac{1}{2}}}[/tex]Replacing terms:
[tex]t=\frac{\ln (\frac{185}{544})}{-\frac{\ln (2)}{55.45}}=\frac{-1.0785}{-0.0125}=\frac{1.0785}{0.0125}=86.28\text{ hours}[/tex]It will take 86.28 hours to the substance to reach 185 Kg.
As I am completely brand new to this subject/branch of mathematics, please explain thoroughly, step by step on how to complete this This is a practice from my ACT prep guide take your time, as there is no rush *Ignore the last answer option
Remember that
The difference of squares is of the form
[tex](a+b)(a-b)=a^2-b^2[/tex]In this problem we have
[tex](3x-4y^2)(3x+4y^2)[/tex]so
a=3x
b=4y^2
therefore
Apply the difference of squares
[tex](3x-4y^2)(3x+4y^2)=(3x)^2-(4y^2)^2=9x^2-16y^4[/tex]What is a solution of a system of linear equations in three variables?
Hello!
When we have a system with the same number of variables and equations, we can obtain the value for all variables.
Knowing it, the right alternative will be:
Alternative B.
for #5 solve for x. then find the missing piece(s) of parallelogram.
Answer:
Given that,
From the parallelogram, the opposite sides of the parallelogram are -2+4x and 3x+3
Explanation:
From the properties of parallelogram, we have that
Opposite sides of a parallelogram are equal
We get,
[tex]-2+4x=3x+3[/tex]Solving we get,
[tex]4x-3x=3+2[/tex][tex]x=5[/tex]Answer is :x=5
Question Evaluate. 7⋅5+42−23÷4 Responses 49 49 41 41 34 34 9 9
Answer: 71.25
This is not of of the options, but is the right answer.
Step-by-step explanation:
7 x 5 + 42 - 23 / 4 =
Step 1: Make parentheses
(( 7 x 5 ) + 42) - ( 23 / 4) =
Step 2: Solve parentheses ( Multiplication and division first )
(35 + 42) - 5.75 =
Step 3: Solve parentheses ( Addition )
77 - 5.75 =
Step 4: Subtract
= 71.25
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How to find the (r) or difference in this scenario:
Aliens Away is a new video game where a player must eliminate a certain number of aliens on the screen by scaring them with an adorable house cat. When James plays the game, he eliminates 64 aliens in the first level and 216 aliens in the fourth level. If the number of aliens are destroyed in a geometric sequence from one level to the next, how many total aliens will James have wiped out by the end of the sixth level?
It is given that it is a geometric sequence, if I am not mistaken it is the explicit formula.
IF YOU COULD PLASE EXPLAIN:)
Total number of aliens James have wiped out by the end of the sixth level is 1330
The number of aliens eliminated in first level a = 64
The number of aliens eliminated in the fourth level = 216
The sequence is in geometric sequence
The nth term of the geometric sequence is
[tex]a_n=ar^{n-1}[/tex]
The fourth term is 216
216 = [tex]64r^3[/tex]
[tex]r^{3}[/tex] = 216/64
r = 3/2
Then we have to find the total alien James killed by the end of sixth level
Sum of n terms = [tex]\frac{a(r^n-1)}{r-1}[/tex]
Substitute the values in the equation
= [tex]\frac{64(1.5^6-1)}{1.5-1}[/tex]
= 665/0.5
= 1330
Hence, total number of aliens James have wiped out by the end of the sixth level is 1330
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K
Use the data in the following table, which lists drive-thru order accuracy at popular fast food chains. Assume that orders are randomly selected from those included in
the table.
Drive-thru Restaurant D
B C
D
280
245
122
60
32
12
A
Order Accurate
331
Order Not Accurate 38
If one order is selected, find the probability of getting an order that is not accurate or is from Restaurant C. Are the events of selecting an order that is not accurate and
selecting an order from Restaurant C disjoint events?
The probability of getting an order from Restaurant C or an order that is not accurate is
(Round to three decimal places as needed.)
Are the events of selecting an order from Restaurant C and selecting an inaccurate order disjoint events?
disjoint because it
possible to
The events
The probability is 0.236 and the events are not disjoint events
Given,
The data;
A B C D
Order accurate ; 280 245 122 331
Order not accurate; 60 32 12 38
The probability of getting an order that is not accurate or is from Restaurant C
This is illustrative of
P(Not accurate or Restaurant C) (Not accurate or Restaurant C)
The calculation is
P(Not accurate or Restaurant C) is equal to [n(Not accurate) + (Not accurate and Restaurant C) - n(Restaurant C)] /Total
Thus, we have
P(Not accurate or Restaurant C) is calculated as follows: (60 + 32 + 12 + 38 + 122 + 12 - 12)/(280 + 245 + 122 + 331 + 60 + 32 + 12 + 38).
Analyze the difference and the total.
Restaurant C or P(Not accurate) = 264/1120
Assess the quotient.
P(Not accurate or Restaurant C) = 0.236
Last but not least, choosing an incorrect order and choosing an order from Restaurant C are not separate events.
This is because choosing an inaccurate order from restaurant C is a possibility.
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Question 2(Multiple Choice Worth 2 points)
(01.06 MC)
Simplify 31.
1
0-1
-1
Answer:
[tex]-i[/tex]
Step-by-step explanation:
Imaginary numbers
Since there is no real number that squares to produce -1, the number [tex]\sqrt{-1}[/tex] is called an imaginary number, and is represented using the letter [tex]i[/tex].
Given expression:
[tex]i^{31}[/tex]
Rewrite 31 as 30 + 1:
[tex]\implies i^{30+1}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{b+c}=a^b \cdot a^c:[/tex]
[tex]\implies i^{30} \cdot i^1[/tex]
[tex]\implies i^{30}i[/tex]
Rewrite 30 as 2 · 15:
[tex]\implies i^{(2 \cdot 15)}i[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{bc}=(a^b)^c:[/tex]
[tex]\implies \left(i^2\right)^{15}i[/tex]
[tex]\textsf{Apply\:imaginary\:number\:rule}\quad \:i^2=-1:[/tex]
[tex]\implies \left(-1\right)^{15}i[/tex]
As -1 to the power of an odd number is -1:
[tex]\implies -1 \cdot i[/tex]
[tex]\implies -i[/tex]
Graph the inequality
y<= -(2/3)|x-3|+4
Please show how
We have the following inequality
[tex]y\leq-\frac{2}{3}\lvert x+3\rvert+4[/tex]We must graph this inequality, In order to understand this I will explain term by term
But first, we must remember that in mathematics, the absolute value or modulus of a real number x, denoted by |x|, is the non-negative value of x regardless of the sign, positive or negative. This must be taken into account for the |x+3| term.
That is to say that the value will always be assumed by its magnitude and we will tend to have the same behavior on both the negative and positive x-axis.
Taking this into account and that the slope is -2/3 the graph would look like this:
Now, we must remember two rules of function translation, these are as follows:
y = f(x) original funtion
y = f(x+c) it is moved horizontally "c" units to the left
y = f(x)+c it moves vertically "c" units upwards
So taking into account these rules our graph is shifted 3 units to the left and 4 units upwards.
In conclusion, this graph looks like this:
A glacier in Republica was observed to advance 22inches in a 15 minute period. At that rate, how many feet will the glacier advance in one year?
To fins the rate in feet/year we must change first the measurements to the units required
inches to feat
minutes to years
[tex]22in\cdot\frac{1ft}{12in}=\frac{11}{6}ft[/tex][tex]15\min \cdot\frac{1h}{60\min}\cdot\frac{1day}{24h}\cdot\frac{1year}{365\text{days}}=\frac{1}{35040}\text{years}[/tex]to find the rate divide the distance over the time
[tex]\frac{\frac{11}{6}ft}{\frac{1}{35040}\text{year}}=\frac{11\cdot35040ft}{6\text{year}}=\frac{385440}{6}=\frac{64240ft}{\text{year}}[/tex]is this equation no solution, one solution, or infinitely may solutions
Given:
[tex]\begin{gathered} x+4y=8\ldots\ldots\ldots\ldots(1) \\ y=-\frac{1}{4}x+2\ldots\ldots\ldots\ldots(2) \end{gathered}[/tex]To solve for x and y:
Substitute the equation (2) in (1) we get,
[tex]\begin{gathered} x+4(-\frac{1}{4}x+2)=8 \\ x-x+8=8 \\ 8=8 \end{gathered}[/tex]Therefore, the given system has infinitely many solutions.
Sam goes to a fast food restaurant and orders some tacos and burritos. He sees on the nutrition menu that tacos are 250 calories and burritos are 330 calories. If he ordered 4 items and consumed a total of 1080 calories, how many tacos, and how many burritos did Sam order and eat?
Let x represent the number of tacos that Sam ordered and ate.
Let y represent the number of burritos that Sam ordered and ate.
From the information given, If he ordered 4 items, it means that
x + y = 4
If tacos are 250 calories, it means that the number of calories in x tacos is 250 * x = 250x
If burritos are 330 calories, it means that the number of calories in y burritos is 330 * y = 330y
If he consumed a total of 1080 calories, it means that
250x + 330y = 1080
From the first equation, x = 4 - y
By substituting x = 4 - y into the second equation, we have
250(4 - y) + 330y = 1080
1000 - 250y + 330y = 1080
- 250y + 330y = 1080 - 1000
80y = 80
y = 80/80
y = 1
x = 4 - y = 4 - 1
x = 3
Thus, Sam ordered and ate 3 tacos and 1 burritos
