The Solution:
Given the pair of sequences below:
[tex]\begin{gathered} \text{ First sequence: 6,7,8,9,10} \\ \\ \text{ Second sequence: 18,21,24,27,30} \end{gathered}[/tex]We are asked to write a statement that correctly describes the relationship between the two sequences.
The two sequences are both linear sequences. Their common differences are:
[tex]\begin{gathered} \text{ First sequence: d=T}_3-T_2=\text{T}_2-T_1 \\ =8-7=7-6=1 \\ \text{ So, the co}mmon\text{ difference is 1} \end{gathered}[/tex]The general formula for the first sequence is
[tex]T_n=a+(n-1_{})d=6+(n_{}-1)1=6+n-1=5+n[/tex]Similarly,
[tex]\begin{gathered} \text{ Second sequence}\colon\text{ } \\ d=\text{T}_3-T_2=\text{T}_2-T_1 \\ d=24-21=21-18=3 \\ \text{ So, the co}mmon\text{ difference is 3} \end{gathered}[/tex]The general formula for the second sequence is
[tex]S_n=18+(n-1_{})3=18+3n_{}-3=15+3n=3(5+n)[/tex]Thus, the relationship between the two sequences is:
[tex]S_n=3T_n[/tex]Where
[tex]\begin{gathered} S_n=\text{ the second sequence} \\ T_n=\text{ the first sequence} \end{gathered}[/tex]Therefore, the correct answer is:
[tex]S_n=3T_n[/tex]For the given f(x), solve the equation f(x)=0 analytically and then use a graph of y=f(x) to solve the inequalities f(x)<0 and f(x)≥0. f(x)=ln(x+3)(1) What is the solution of f(x)=0?(2) What is the solution of f(x)<0?(3) What is the solution of f(x)≥0?
Explanation:
f(x) is a logarithmic function. Logarithmic functions are zero when the argument is 1:
[tex]\begin{gathered} f(x)=\ln (x+3)=0 \\ x+3=1 \\ x=1-3 \\ x=-2 \end{gathered}[/tex]For greater values, the function is positive and for less values the function is negative.
Answers:
(1) x = -2
(2) x < -2. In interval notation x:(-∞, -2)
(3) x ≥ -2. In interval notation x:[-2, ∞)
creat an espression that includes the zero property of exponents the multiplication property of exponents and the power of a power property of exponents
All in one, or one expression for each property?
a) Zero property
[tex]\text{ (x + y)}^0\text{ = 1}[/tex]b) Multiplication property
[tex]\text{ x}^2\cdot x^5=x^{2+5}=x^7[/tex]c) Power property
[tex]\text{ (x}^2)^3=x^{2\cdot3}=x^6[/tex]d) All in one (this is the expression)
[tex]\mleft\lbrace\text{(x}^0)(x^3)\mright\rbrace\text{ }(x^2)^5[/tex][tex]\text{ }\mleft\lbrace1(x^3\mright)\}(x^{10})[/tex]
A person invested $3,700 in an account growing at a rate allowing the money to double every 6 years. How much money would be in the account after 14 years, to the nearest dollar?
Given :
The principal = 3,700
Assume a simple interest
The account growing at a rate allowing the money to double every 6 years.
So,
[tex]\begin{gathered} I=P\cdot r\cdot t \\ I=P \\ 3700=3700\cdot r\cdot6 \\ r=\frac{1}{6} \end{gathered}[/tex]How much money would be in the account after 14 years, to the nearest dollar?
So, we will substitute with r = 1/6, t = 14 years
So,
[tex]\begin{gathered} I=3700\cdot\frac{1}{6}\cdot14=8633.33 \\ \\ A=P+I=8633.33+3700=12333.33 \end{gathered}[/tex]Rounding to the nearest dollar
So, the answer will be $12,333
her player O oo Find the amount of simple interest that $400 would earn at 8% per year by the end of 3 years. O A. $96 OB. $11,200 O c. $3200 OD. D. $112 O E. $32
To answer this question, we need to use the next formula for simple interest:
[tex]FV=PV\cdot(1+in)[/tex]Where:
FV is the future value we need to find (in this case).
PV is the present value, that is, $400 (in this case).
i is the interest rate. In this case, we have 8% (0.08).
n is the number of periods (n = 3, in this case).
Then, we have:
[tex]FV=400\cdot(1+(0.08)\cdot3)\Rightarrow FV=496[/tex]That is, the FV is $496. Therefore, the simple interest is $(496-400 = 96).
Thus, the amount of simple interest that $400 would earn at 8% per year by the end of 3 years is $96 (option A).
In other words, the result can be obtained also if we have is $400 * (0.08)*3 = $96.
Teresa surveyed 100 students about whether they like pop music or country music. Outof the 100 students surveyed, 42 like only pop, 34 like only country, 15 like both popandcountry, and 9 do not like either pop or country. Complete the two-way frequency table.
SOLUTION
Write out the given information
[tex]\begin{gathered} \text{Total number of student surveyed=100} \\ \text{like pop only=42} \\ \text{like country only=34} \end{gathered}[/tex][tex]\begin{gathered} \text{like both pop and country=15} \\ Do\text{ not like any =9} \end{gathered}[/tex]Construct the two- way frequency table
From the table below, determine whether the data shows an exponential function. Explain why or why not. x31-1-3y1234a.No; the domain values are at regular intervals and the range values have a common sum 1.b.No; the domain values are not at regular intervals.c.Yes; the domain values are at regular intervals and the range values have a common factor 2.d.Yes; the domain values are at regular intervals and the range values have a common sum 1.
Solution:
Given:
The table of values is given:
From the table,
We see the data is a linear function. This is because a linear function has domain values at regular intervals.
Also, the linear equation can be formed as shown below, indicating it is a linear function.
Considering two points, (3,1) and (1,2)
where,
[tex]\begin{gathered} x_1=3 \\ y_1=1 \\ x_2=1 \\ y_2=2 \\ \\ \text{Then,} \\ \text{slope, m is given by;} \\ m=\frac{y_2-y_1}{x_2-x_1} \\ \\ \text{Substituting the values into the formula above,} \\ m=\frac{2-1}{1-3} \\ m=\frac{1}{-2} \\ m=-\frac{1}{2} \end{gathered}[/tex]A linear equation is of the form;
[tex]\begin{gathered} y=mx+b \\ \text{where m is the slope} \\ b\text{ is the y-intercept} \\ \\ To\text{ get the value of the y-intercept, we use any given point} \\ U\sin g\text{ point (3,1)} \\ y=mx+b \\ 1=-\frac{1}{2}(3)+b \\ 1=-\frac{3}{2}+b \\ 1+\frac{3}{2}=b \\ 1+1.5=b \\ b=2.5 \\ \\ \\ \text{Thus, the linear equation is;} \\ y=-\frac{1}{2}x+2.5 \end{gathered}[/tex]From the above, has confirmed it is a linear function and not an exponential function, we can deduce that;
a) The function is not an exponential function.
b) The domain values (x-values) are at regular intervals
c) The range values (y-values) have a common difference of 1
Therefore, the correct answer is OPTION A
helpppppppppp!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
(f o g)(x) = 8x³ + 2x - 6
(g o f)(x) = 2x³ + 2x - 12
Step-by-step explanation:
f(x) = x³ + x - 6; g(x) = 2x
(f o g)(x) = f(g(x))
f(g(x)) = (2x)³ + (2x) - 6
f(g(x)) = 8x³ + 2x - 6
(g o f) = g(f(x))
g(f(x)) = 2(x³ + x - 6)
g(f(x)) = 2x³ + 2x - 12
I hope this helps!
Bella drove her car 81 km and used 9 liters of fuel. She wants to know how many kilometers (x) she can drive on 22 liters of fuel. She assumes her car will continue consuming fuel at the same rate.
Find the proportion between liters of fuel
that is find 22/9 = 2.444
thats how much liters have more to consume
now multiply 2.444 by 81 , the kilometers she has drived
It gives as result 2.444 x 81 = 198 kilometers
W L is tangent to circle O at point W. If mW H A = 260 degrees , find m< AWL
Answer:
[tex]m\angle AWL\text{ = 130}\degree[/tex]Explanation:
Here, we want to get the measure of the angle marked AWL
The measure of this angle is simply half the measure of the big arc WHA
Mathematically, we have the measure of the angle as:
[tex]\begin{gathered} m\angle AWL\text{ = }\frac{1}{2}\text{ }\times\text{ mWHA} \\ \\ =\text{ }\frac{1}{2}\times\text{ 260 = 130}\degree \end{gathered}[/tex]which statement is true
We have to analyze the given options to solve this problem.
Option 1.
The absolute value of -12 is larger than the absolute value of 12.
The absolute value is always a positive number:
[tex]undefined[/tex]John sells plain cakes for $8 and decorated cakes for $12. On a particular day, John started with a total of 100 cakes, and sold all but 4. His sales that day totaled $800.He sold ___plain cakes and ____decorated cakes that day.
INFORMATION:
We know that:
- John sells plain cakes for $8 and decorated cakes for $12.
- On a particular day, John started with a total of 100 cakes, and sold all but 4.
- His sales that day totaled $800.
And we must find the number of plain cakes and decorated cakes that he sold that day.
STEP BY STEP EXPLANATION:
To find them, we can represent the situation using a system of equations
[tex]\begin{cases}x+y={100-4...(1)} \\ 8x+12y={800...(2)}\end{cases}[/tex]Where, x represents the number of plain cakes that he sold and y represents the number of decorated cakes that he sold.
Now, we must solve the system:
1. We must multiply the equation (1) by -8
[tex]\begin{gathered} -8(x+y)=-8(100-4) \\ -8x-8y=-768...(3) \end{gathered}[/tex]2. We must add equations (2) and (3)
[tex]\begin{gathered} 8x+12y=800 \\ -8x-8y=-768 \\ ---------- \\ 0x+4y=32 \\ \text{ Simplifying, } \\ 4y=32...(4) \end{gathered}[/tex]3. We must solve equation (4) for y
[tex]\begin{gathered} 4y=32 \\ y=\frac{32}{4} \\ y=8 \end{gathered}[/tex]4. We must replace the value of y in equation (1) and then solve it for x
[tex]\begin{gathered} x+8=100-4 \\ x=100-4-8 \\ x=88 \end{gathered}[/tex]So, we found that x = 88 and y = 8.
Finally, John sold 88 plain cakes and 8 decorated cakes.
ANSWER:
He sold 88 plain cakes and 8 decorated cakes that day.
An integer is chosen at random from 1 to 50. find the probability that the chosen integer is not divisible by 2, 7 or 9a)13/50b)16/25c)9/25
There are a total of 50 numbers that are between 1 and 50. Halft of these numbers are even (divisible by 2 ) and half of then odd.
There are 25 integers that are not even and in total there are 50 integers; thereofre, the probablity of finding an even integer is
25/50 = 1/2
Tara makes and sells scarves for children and adults. She is able to sell the scarves for $18 per unit. Materials for the scarves cost $4 each. She has fixed cost per month of $280 and estimates that she can make and sell 80 scarves each month. How many scarves does Tara need to sell to break even?
Tara needs to sell 20 scarves to break even.
What is the break-even point?The break-even point is the sales level that the seller must attain to make the total revenue equal to the total costs (fixed and variable).
There is no profit or loss at the break-even point (either in units or dollar values).
Selling price per unit = $18
Product cost per unit = $4
Contribution margin per unit = $14
Fixed cost per month = $280
Estimated production and sales units per month = 80 scarves
Break-even sales units = fixed costs/contribution margin per unit
= 20 ($280/$14)
Check:
Total revenue at 20 units = $360 (20 x $18)
Total costs at 20 units = $360 ($280 + $4 x 20)
Thus, for Tara to break even, she needs to sell 20 scarves per month.
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Tyler said he swam 23 tenths miles this week. His coach said Tyler swam 2.3 miles this week. To find who is correct, model the distance both Tyler and his coach said Tyler swam. Use the flat as 1 unit. A: What do you need to use?B: What do you know about representing whole numbers and decimals that may help you solve the problem? C: Complete the sentencesAre the models alike or different?Tyler swam _____ tenths, or _____, miles.So, _____________________ are correct.
Answer with explanation:
We need to determine if what Tyler is saying is in fact equal to what his coach said, to get the final answer, we have to concert the resulting units in miles:
Taylor's answer:
[tex]23\times(\frac{1}{10})\text{ miles}\Rightarrow(\frac{23}{10})\text{miles}\Rightarrow2.3\text{ miles}[/tex]Coach's answer:
[tex]2.3\text{ miles}[/tex]In conclusion, The two answers are correct so the two models are indeed alike.
Hurry and answer this pls Bc this have to be turned in
find the height of the trapezoidA=51CM2b=10cmb=7cmH?
we must find b one of the parallel sides before proceeding to find h
from the diagram b = 7cm
[tex]\begin{gathered} \text{Area = }\frac{10\text{ +7}}{2}\times h \\ 51\text{ = }\frac{17}{2}\times h \end{gathered}[/tex][tex]\begin{gathered} 51\text{ x 2 = 17h} \\ h\text{ =}\frac{51\times2}{17} \\ h\text{ =6cm} \end{gathered}[/tex]explain pleaeeeeeeez
Answer:
So first we can assume x= 1 bc there is no number for x
Step-by-step explanation:
So we Evaluate for x=1
1+|2−1|−5
1+|2−1|−5
=−3
Evaluate for x=1
So x+|x-5|+9
1+|1−5|+9
1+|1−5|+9
=14
A bird sits on top of a Lamppost. The angle made by the lamp-post and a line from the feet of the bird to the feet of the Observer standing away from the Lamppost is 55°. the distance from the Lamppost and the Observer is 25 ft. estimate the height of the lamp post
A bird sits on top of a Lamppost. The angle made by the lamp-post and a line from the feet of the bird to the feet of the Observer standing away from the Lamppost is 55°. the distance from the Lamppost and the Observer is 25 ft. estimate the height of the lamp post
we have that
see the attached figure to better understand the problem
so
tan(55)=h/25
solve for h
h=25(tan(55))
h=35.7 ftBc is included between
Answer:
A. angle B and angle C
Step-by-step explanation:
Line segments are named after the two points where they begin and end.
1. Jessica finishes her book in 2 1
3
hours. Eric takes 11
2
times longer than
Jessica to finish his book.
This model represents the amount of time Jessica takes to finish her
book. It has a width of 1 and a length of 2 1
3
. The model is 2 1/3 out of 3
The time taken for Eric to finish the book is 3 1/2 hours.
What is a fraction?A fraction simply means a part of a whole. It van also refer to any number of equal parts.
The information illustrated that Jessica finishes her book in 2 1 / 3 hours and that Eric takes 1 1 / 2
times longer than Jessica to finish his book.
In this case, the time that was used by Eric will be the multiplication of the fractions given. This will be illustrated as:
= 2 1/3 × 1 1/2
Change to improper fraction
= 7/3 × 3/2
= 7 / 2
= 3 1 / 2
Eric used 3 1/2 hours.
This illustrates the concept of multiplication of fractions.
The complete question is written below.
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Jessica finishes her book in 2 1/3.hours. Eric takes 1 1/2 times longer than Jessica to finish his book. How long did Eric take yo finish the book?
Write the equation for the quadratic function in vertex form & standard form with the given vertex that passes through the given point.Vertex (2, -8) through the point (4, 3)
We will have the following:
[tex]\begin{gathered} y=a(x-2)^2-8\Rightarrow3=a(4-2)^2-8 \\ \\ \Rightarrow3=4a-8\Rightarrow4a=11 \\ \\ \Rightarrow a=\frac{11}{4} \end{gathered}[/tex]So, the equation in vertex form is:
[tex]y=\frac{11}{4}(x-2)^2-8[/tex]And in standard form:
[tex]\begin{gathered} y=\frac{11}{4}(x^2-4x+4)-8\Rightarrow y=\frac{11}{4}x^2-11x+11-8 \\ \\ \Rightarrow y=\frac{11}{4}x^2-11x+3 \end{gathered}[/tex]***Explanation***
We know that the quadratic expression in vertex form follows:
[tex]y=a(x-h)^2+k[/tex]Where (h, k) is the vertex of the expression. Now, we know that the vertex is (2, -8), so we replace those values and we obtain:
[tex]y=a(x-2)^2+(-8)\Rightarrow y=a(x-2)^2-8[/tex]Now, in order to determine "a" we must replace one point (That is not the vertex) in the expression and solve for "a", and we are told that the point (4, 3) is in one of the solutions, so:
[tex]\begin{gathered} 3=a(4-2)^2-8\Rightarrow3=a(2)^2-8 \\ \\ \Rightarrow11=4a\Rightarrow a=\frac{11}{4} \end{gathered}[/tex]Thus, the expression in vertex form is then:
[tex]y=\frac{11}{4}(x-2)^2-8[/tex]And to determine the standard form, we simply expand the equation in vertex form:
[tex]y=\frac{11}{4}(x^2-4x+4)-8\Rightarrow[/tex]how do I find the decimal value of the fraction 11/16?
You divide 11 by 16, as follow:
0.6875
16 l 110
-96
140
-128
120
-112
80
-80
0
As you can notice, the result of the division is 0.6875 (here you have used the rules for the division of a number over a greater number, which results in a decimal)
What is the distance from 7 to 0? O A. 7, because 171 = 7 Jurid O B. 7, because 171 = 7 O c. 7, because |-71 = -7 O D. -7, because [7] = -7
The distance from 7 to 0 is 7 because the absolute value of 7 is 7.
Correct Answer: A
What is the Y intercept of the graph below? A. (0,-2)B. (0,-4) C. (0, 2) D. (0,4)
Recall that the y-intercept of a graph is the point where the graph intersects the y-axis.
From the given graph we get that the line intersects the y-axis at (0,2).
Answer: Option C.
Suppose Set A contains 48 elements and Set B contains 16 elements. If the total number elements in either Set A or Set B is 54, how many elements do Sets A and B have in common?
Considering the Set A and B given, Applying inclusion - exclusion principle the number of elements common to both Sets is 10
What is inclusion - exclusion principle?This is a counting techniques that ensures that elements are not counted twice
It is achieved by the formula:
(A ∪ B) = n(A) + n(B) - n(A ∩ B)
Finding the elements Sets A and B have in commonThe information from the question include the following
Set A contains 48 elements
Set B contains 16 elements
The total number elements in either Set A or Set B is 54
Applying inclusion - exclusion principle gives the formula
Set A + Set B - ( Set A ∩ Set B ) = Set A ∪ Set B
substituting the values gives
48 + 16 - ( Set A ∩ Set B ) = 54
48 + 16 - 54 = ( Set A ∩ Set B )
10 = ( Set A ∩ Set B )
( Set A ∩ Set B ) = elements Sets A and B have in common
Therefore the number of the elements common to Sets A and B is 10
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Hi how do I graph these? I don't understand how I'm supposed to graph fractions?
To plot in the plane points with fraction number coordinates (x or y) you can rewrite the fractions as decimal numbers:
[tex]\begin{gathered} \frac{-5}{2} \\ \\ -5\text{ divided into 2} \\ \\ -\frac{5}{2}=-2.5 \\ \\ \\ \\ \\ \frac{-1}{4} \\ \\ -1\text{ divided into 4:} \\ -\frac{1}{4}=-0.25 \end{gathered}[/tex]Then, you have the next coordinates;
(- 2.5, 2) and (1, -0.25)
And the next graph:
Use a line to link the points
what is the LCM of 4 and 6 ?
LCM stands for Least Common Multiple.
And it is defined as the product of the two numbers divided by the GCD (greatest common divisor)
In our case, the product of 4 and 6 is 24, , and the greatest common divisor of 4 and 6 is "2". Therefore, the LCM of 4 and 6 is 24/2 = 12
Let me also use the Venn diagram that your teacher provided:
In the diagram we enter the factors that correspond to both numbers (4 and 6), and in the intersection of the two sets (intersection of the circle) we type a "2" which is the ONLY factor 4 and 6 have in common (the greatest common divisor of the two given numbers) So complete a diagram as follows:
We typed a 2in the area common to both numbers. Then your LCM is the product of 2 times 2 times 3 = 12
Notice the blue set (circle) contains the two factors for 4 (2 * 2) and the orange circle contains the two factor for 6 (2 * 3)
We set in the intersection of the two circles the factor that is common to both.
Do you want me to complete the second question with a Venn diagram as well? Perfect.
The second question is about the LCM of the numbers 12 and 8
Then we create a Venn diagram like the following, considering that the factor in common between 12 and 8 is 4, because 12 = 4 * 3 and 8 = 4 * 2
Again here, the factors 3 and 4 (that give 12) are typed in the blue circle. and the factors that form 8 (4 * 2) are typed inside the orange circle.
The factor that both share is in the middle "4". Therefore, now to find the LCM you simply multiply the three numbers shown in the Venn diagtam: 3 * 4 * 2 = 24
Then 24 is your LCM.
The formula log in a natural logarithm can be written as?
Solution:
Given the logarithmic expression:
[tex]\log_545[/tex]According to the change of base formula,
[tex]\log_BA=\frac{\ln A}{\ln\text{ B}}[/tex]Thus, expressing the logarithm expression in a natural logarithm,
[tex]\log_545=\frac{\ln45}{\ln5}[/tex]Hence, we have
[tex]\frac{\ln45}{\ln5}[/tex]90 minutes for 3 typed pages; 60 minutes for a a typed pages write a proportion for each phrase and solve it
90 minutes for 3 typed pages; 60 minutes for a a typed pages write a proportion for each phrase and solve it
we have that
90/3=60/a
solve for a
a=(60*3)/90
a=2 pagesAna has $75 and saves an additional $13 per week. Which equation can be used to findhow many weeks it will take until she has $452?75 + w = 4520 75 + 13w = 45213w = 75 = 452452 + 13w = 75
Here, we want to get an equation
Firstly, since we do not have the number of weeks, we can represent it with a variable (a letter)
In this case, we shall be representing it with w
Since she saves $13 in a week, in w weeks, the amount saved will be;
13 * w = $13w
Now, recall that she has $75 before she started saving. What this mean is that at the end of the w weeks, the amount she will have will be ;
[tex]13w\text{ + 75}[/tex]We now proceed to equate this to the total she wants to save and we finally have the complete equation below;
[tex]13w\text{ + 75 = 452}[/tex]