a) Common ratio = 2/7
b) First term = 135/7
Explanations:The formula for finding the sum of a geometric progression is expressed as:
[tex]S_n=\frac{a(r^n-1)}{r-1}[/tex]Since the sum of the first two terms is 15, then
S2 = 15
n = 2
Substitute into the formula:
[tex]\begin{gathered} S_2=\frac{a\mleft(r^2-1\mright)}{r^{}-1} \\ 15=\frac{a(r+1)\cancel{r-1}}{\cancel{r-1}} \\ 15=a(r+1) \end{gathered}[/tex]Also, the sum to infinity of a geometric sequence is expressed as:
[tex]\begin{gathered} S_{\infty}=\frac{a}{1-r} \\ _{} \end{gathered}[/tex]Substitute the given values into the formula:
[tex]27=\frac{a}{1-r}[/tex]Solve both expressions simultaneously
[tex]\begin{gathered} 15=a(r+1) \\ 27=\frac{a}{1-r} \end{gathered}[/tex]
Divide both expressions to have:
[tex]\frac{15}{27}=\frac{1-r}{r+1}[/tex]Cross multiply and solve for the common ratio "r"
[tex]\begin{gathered} 15(r+1)=27(1-r) \\ 15r+15=27-27r \\ 15r+27r=27-15 \\ 42r=12 \\ r=\frac{12}{42} \\ r=\frac{2}{7} \end{gathered}[/tex]Hence the value of the common ratio is 2/7
b) Get the first term of the sequence;
Using the formula:
[tex]\begin{gathered} 27=\frac{a}{1-r} \\ 27=\frac{a}{1-\frac{2}{7}} \\ 27=\frac{a}{(\frac{5}{7})} \\ a=27\times\frac{5}{7} \\ a=\frac{135}{7} \\ \end{gathered}[/tex]Hence the first term of the sequence is 135/7
List the integers from least to greatest 4, -7, 6, -2, 1
When you have negative numbers you can see that numbers in a number line to see what is the greatest and what is the least:
In a number line you put the 0 as the center and to the right you put the possitive numbers and in the left of the 0 the negative numbers, as follow:
Then you can put the numbers you have in a number line:
You get
Purle: -7
Red; -2
Blue: 1
Green: 4
Black: 6
The least number is the number that is more to the left and the greatest is the number taht is more to the right
Then so, the List the integers from least to greatest is:-7, -2, 1, 4, 6in triangle JKL j=10cm k=12cm anf l=13cm find cos K
For the given triangle, we must apply the following trigonometric relation:
[tex]\begin{gathered} k^2=j^2+l^2-2jl\cos K \\ \cos K=\frac{k^2-j^2-l^2}{-2jl} \\ \cos K=\frac{12^2-13^2-10^2}{-2\cdot13\cdot10} \\ \cos K=0.48 \end{gathered}[/tex]Dominic has a bag of candy full of 1 strawberry chew and 19 cherrychews that he eats one at a time. Which word or phrase describes theprobability that he reaches in without looking and pulls out a lemonchew?A.certainB.unlikelyC.likelyD.impossible
D.impossible
there is no Lemon chew, only strawberry chew and cherry chews
How do I find the point slope intercept of a line
for the slope, the equation is:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]in case you have two points
The equation of the line equation:
[tex]y=mx+b[/tex]If you want to find the interception in x-axis, you have to change the y for a 0, like this
[tex]0=mx+b[/tex][tex]x=-\frac{b}{m}[/tex]If you want to find the interception in y-axis, you have to change the x for a 0, like this, that is "b" in the equation
[tex]y=m(0)+b[/tex][tex]y=b[/tex]SI imi triangles. 16. JK 17 ST J X K 4 6 L R Р 12 M P TTTT 20. DB
16.
In the given triangles,
[tex]\begin{gathered} \angle JLK=\angle PLM\text{ (Vertically Opposite Angle)} \\ \angle LJK=\angle LPM\text{ (Given)} \end{gathered}[/tex]Hence form AA critesion,
[tex]\Delta JLK\approx\Delta PLM[/tex]From the property of similar triangles,
[tex]\begin{gathered} \frac{JK}{PM}=\frac{JL}{PL} \\ \Rightarrow\frac{x}{12}=\frac{4}{6} \\ \Rightarrow x=8 \end{gathered}[/tex]Thus, the requried value of JK is 8.
Write the coordinates of the vertices after a reflection over the x-axis. 1072
The reflection about x-axis results in same x coodinate with oppositive of y coordinate. It can be expressed as,
[tex](x,y)\rightarrow(x,-y)[/tex]Determine the coordinates of the vertices after reflection over the x-axis.
[tex]Q(6,-8)\rightarrow Q^{\prime}(6,8)[/tex][tex]R(7,-8)\rightarrow R^{\prime}(7,8)[/tex][tex]S(7,-5)\rightarrow Q^{\prime}(7,5)[/tex][tex]T(6,-5)\rightarrow T^{\prime}(6,5)[/tex]how do you solve 15w-4=41
In order to solve this equation for w, we can do the following steps:
[tex]\begin{gathered} 15w-4=41 \\ 1.\text{ Add +4 to both sides of the equation:} \\ 15w-4+4=41+4 \\ 15w=45 \\ 2.\text{ Divide both sides of the equation by 15:} \\ \frac{15w}{15}=\frac{45}{15} \\ w=3 \end{gathered}[/tex]So we have that the value of w is 3.
disjointed and overlapping events
Answer:
1. P = 0.269 or 26.9%
2. P = 0.372 or 37.2%
Step-by-step explanation:
First, let's calculate the total number of students:
So, we know that the number of students is 662.
Now, let's evaluate the probabilities:
1. Student will begin college during summer AND attend an in-state college.
P = 178/662
P = 0.269 or 26.9%
2. Student will begin college in summer GIVEN THAT he is attending an in-state college.
In this case, the total number of students will be the number of students who will attend an in-state college.
P = 178/479
P = 0.372 or 37.2%
Is square root of 224 an irrational number ?
ANSWER
YES
EXPLANATION
We want to know if the square root of 224 is an irrational number.
An irrational number is a number that cannot be written as a fraction/ratio of two integers.
If we simplify the square root of 224:
[tex]\begin{gathered} \sqrt{224}\text{ = }\sqrt{16\cdot\text{ 28}}\text{ = 4}\sqrt{28} \\ \text{ }\Rightarrow\text{ 21.166010488}\ldots \end{gathered}[/tex]As we can see, this number cannot be written as a fraction of two numbers.
As a rule, the square root of any number that is not a perfect square is an irrational number.
So, the answer is Yes. It is an irrational number
Decide if each fraction expressed as a decimal terminates or repeats.
A. 12/11
B. 5/8
C. -19/20
D. 2/3/6/5
Answer:
A. repeat
B. terminates
C. terminates
D. terminates
Step by step explanation:
to turn the fractions into decimals you have to divide the top by bottom
12÷11=1.0909090909
5÷8=0.625
-19÷20=-0.95
2÷3=0.6666666667 6÷5=1.2 ÷0.6666666667 =0.5555555556
only 1 repeats itself multiple times which means thats the only one the repeats and the rest terminate because they end
A classmate claims that the function g(x)=-4ex+6 is the parent function f(x)=ex reflected across the Y axis, vertically compressed by a factor of four, translated to the left five units, and translated up six units.A) explain what the classmate described incorrectly.B) describe g(x) as a series of transformations of f(x)
Based on the function g(x), the function f(x) is multiplied by a negative number. Hence, there is a reflection across the x-axis.
If it was the "x" variable that was multiplied by a negative number for example, y = e^-x, it would have been reflection across the y-axis.
Since the number multiplied to the function f(x) is not between 0 and 1, we are stretching the function. Hence, the function is vertically stretched by a factor of 4.
Moving to the number -5 that was subtracted from the variable x, we have moved/shifted the function 5 units to the right instead of left.
Lastly, 6 was added to the entire function, therefore, we have translated the functions 6 units up.
To summarize, f(x) = e^x had undergone the following series of transformation: reflection across the x-axis, vertically stretched by a factor of 4, translated 5 units to the right, and translated 6 units up to form g(x).
I need help I’ve been having trouble with this chapter for about a week
Given:
[tex]3x^2+20x+33[/tex]Find-:
Factorization of the equation.
Sol:
A simple method of factorization is to multiply in first and last order then break it down into parts to make the middle number then.
[tex]\begin{gathered} =3\times33 \\ =99 \end{gathered}[/tex]
The factor of 99 is:
So take factor :
[tex]\begin{gathered} 11\text{ and \lparen3}\times3) \\ \\ 11\text{ and 9} \end{gathered}[/tex]Factorization of the equation is:
[tex]\begin{gathered} =3x^2+20x+33 \\ \\ =3x^2+11x+9x+33 \end{gathered}[/tex]If a movie is played at the rate preferred by its director, a moviegoer see 600 frames in 12.5 seconds. how many frames does a moviegoer see in 159?
A moviegoer sees 7632 frames in 159s.
A student earned grades of C, A, B, and A in four different courses. Those courses had these corresponding numbers of credit hours: 4, 5, 1, and 5. The grading system assigns quality points to letter grades as follows: A = 4, B = 3, C = 2, D = 1, and F = 0. Compute the grade point average (GPA) and round the result to two decimal places.2.183.408.753.50
Given:
Grades the student earned = C, A, B, A
Corresponding credit hours = 4, 5, 1, 5
The grading system assigns quality points to letter grades as follows: A=4, B=3, C=2, D=1, and F=0.
Required: Grade point average (GPA)
Explanation:
Points earned = Earned grades x Corresponding credit hours
For C, points earned = 2 x 4 = 8 points
For A, points earned = 4 x 5 = 20 points
For B, points earned = 3 x 1 = 3 points
Total points earned = 8+20+3+20 = 51 points
Total credit hours = 4+5+1+5 = 15 hours
Find GPA.
[tex]\begin{gathered} \text{Grade point average\lparen GPA\rparen=}\frac{\text{ Total points of the student}}{\text{ Total credit hours}} \\ =\frac{51}{15} \\ =3.40 \end{gathered}[/tex]Final Answer: The grade point average of the student is 3.40.
Use the fundamental identities to find the value of trigonometric function. Find csc θ, given that sin 2θ = - — and θ is in quadrant IV. 3
Recall that:
[tex]\csc \theta=\frac{1}{\sin\theta}\text{.}[/tex]If:
[tex]\sin \theta=-\frac{2}{3},[/tex]then:
[tex]\csc \theta=\frac{1}{-\frac{2}{3}}\text{.}[/tex]Simplifying the above result we get:
[tex]\csc \theta=-\frac{3}{2}\text{.}[/tex]Answer:
[tex]\csc \theta=-\frac{3}{2}\text{.}[/tex]In the figure to the right, what value of x makes G the incenter of triangle JKL. See image below
We were given the following information:
LT = 12
GL = 13
GR = x - 3
The incenter of a triangle refers to the intersection point of all interior angle bisectors of the triangle. The incenter is equidistant to the sides, they are all the same
If triangle JKL has G as its incenter, the following will be found to be true:
[tex]|GR|\cong|GS|\cong|GT|[/tex]However, we were not given any of the above distances GR, GS & GT. We can obtain GT by using the Pythagoras Theorem on the triangle GTL as shown below:
[tex]\begin{gathered} |GT|^2=|GL|^2-|LT|^2 \\ |GT|^2=13^2-12^2 \\ |GT|^2=169-144 \\ |GT|^2=25 \\ \text{Take the square root of both sides, we have:} \\ |GT|=\sqrt[]{25} \\ |GT|=5 \\ \\ \therefore|GT|=5 \end{gathered}[/tex]Since GT equals 5, it implies that GS & GT will also equal 5
We will obtain the value of ''x'' as shown below:
[tex]undefined[/tex]Find the lateral surface area of the rectangular prism. Round your answer to the tenth of necessary
48 square feet
Explanation
to find the lateral surafece, we need to add the 4 faces that make it,so
so,the total area
[tex]\begin{gathered} \text{Area}=2(\text{length}\cdot\text{width)}+2(\text{length}\cdot\text{width)} \\ \text{Area}=2(6\text{ ft}\cdot3\text{ ft)+2(3ft}\cdot\text{2 ft)} \\ \text{Area}=2(18ft^2)+2(6ft^2) \\ \text{Area}=36ft^2+12ft^2 \\ \text{Area}=48ft^2 \end{gathered}[/tex]so, the answer is 48 square feet
Round 58,300 to the nearest ten thousand 
ok
Rounding to the nearest 10000 the result is
60,000
Answer:
60,000
Step-by-step explanation:
The 5 is in the ten-thousands place.
Make all digits right of the 5 into a zero.
You get 50,000.
Since 8 (in the thousands place) is greater than 5, the ten-thousands place goes uo 1 to 6.
Answer: 60,000
I need help NUMBER 181.Find the GCF2.Write the GCF 3.Rewrite expression factor out the GCF4. Write the final factored expression!
Answer:
1. Find the GCF:
16 y | 5
1
2. Write the GCF: 1
3. Rewrite expression factor out the GCF: 16y + 5
4. Write the final factored expression: 16y + 5
Explanation:
The initial expression is:
16y + 5
So, we have two terms: 16y and 5
The factors of these terms are:
16y: 1, 2, 4, 16, y, 2y, 4y, 16y
5: 1, 5
So, the greatest common factor is 1
Then, the expression factor out the GCF is:
[tex]\frac{16y+5}{1}=16y+5[/tex]Therefore, the final factored expression is:
1*(16y + 5) = 16y + 5
6 cm 4 cm 10 cm What is the area of the figure in square centimeters? TOTAL AREA= find the area of all the shapes and ADD together. To find the area 1/2 of a circle , you need the area of a circle and divide by 2. USE YOUR FORMULA CHART.
Notice that since the bottom side of the figure has a length of 10cm while the part that corresponds to the rectangle is only 6cm long, then the base of the triangle is 4cm long. The height of the triangle is 4cm long, since it is the same as the height of the rectangle. Additionally, the diameter of the semicircle turns out to be equal to 4cm, then its radius (which is half the diameter) must be 2cm long.
Use these data to find the area of each figure:
Semicircle
The area of a semicircle is half the area of a circle:
[tex]\begin{gathered} A=\frac{1}{2}\pi r^2 \\ =\frac{1}{2}\pi(2cm)^2 \\ =2\pi cm^2 \\ \approx6.28cm^2 \end{gathered}[/tex]Rectangle
[tex]\begin{gathered} A=w\times l \\ =6\operatorname{cm}\times4\operatorname{cm} \\ =24cm^2 \end{gathered}[/tex]Triangle
[tex]\begin{gathered} A=\frac{1}{2}b\times h \\ =\frac{1}{2}4\operatorname{cm}\times4\operatorname{cm} \\ =8cm^2 \end{gathered}[/tex]Total area:
[tex]\begin{gathered} A=6.28cm^2+24cm^2+8cm^2 \\ \Rightarrow A=38.28cm^2 \end{gathered}[/tex]Therefore, the total area of the figure is:
[tex]38.28cm^2[/tex]This sequence represents the diameters of circles used to create an art project 2.5 cm, 3.1cm, 3.7cm,4.3cm let f(n) represent diameter in centimeters
The equation that represents the sequence of diameters is f(n) = 1.9+0.6n
What is arithmetic progression?Arithmetic Progression (AP) is a sequence of numbers in order, in which the difference between any two consecutive numbers is a constant value. It is also called Arithmetic Sequence.
Given the sequence that represents the diameter of a circle 2.5 cm, 3.1 cm, 3.7 cm and 4.3 cm. this sequence forms an arithmetic progression with a common difference.
nth term of an arithmetic progression is expressed as; [tex]a_{n}[/tex] = a+(n-1)*d
where, a is the first term of the sequence
n is the number of terms
d is the common difference.
Here, the first term a = 2.5 and common difference = 3.1-2.5 = 3.7-3.1 = 4.3-3.7 = 0.6
Substituting the values;
[tex]a_{n}[/tex] = 2.5 + (n-1)*0.6
= 2.5 + 0.6n - 0.6
= 1.9 + 0.6n
Hence, The equation that represents the sequence of diameters is f(n) = 1.9+0.6n
For more references on arithmetic progression, click;
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what does the dotted line on the geometric triangle mean?
It is line of symmetry. That is, if you draw a triangle like this,
Then you will get a trainagle similar to the first one.
What is the estimate for each expression? Drag the number to each box.011/2Expression Estimate2+513 114 101+12 20-1009
ANSWER:
STEP-BY-STEP EXPLANATION:
We must approximate each addition or subtraction, estimating if it is close to 0, 1/2 or 1.
We operate in each case:
[tex]\begin{gathered} \frac{1}{8}+\frac{2}{5}=\frac{1\cdot5+2\cdot8}{8\cdot5}=\frac{5+16}{40}=\frac{21}{40}=0.525\approx\frac{1}{2} \\ \\ \frac{13}{14}-\frac{1}{10}=\frac{13\cdot10-1\cdot14}{14\cdot10}=\frac{130-14}{140}=\frac{116}{140}=0.82\approx1 \\ \\ \frac{1}{12}+\frac{1}{20}=\frac{20+12}{12\cdot20}=\frac{32}{240}=0.13\approx0 \end{gathered}[/tex]Please help answer questions one through fiveApply the transformation (a to c) on ABC to get an image
Answer:
d) area of the pre- image will be less than the new image
e) it is
Explanation:
Given:
Triangle ABC on a coordinate plane
To find:
the transformation on the original image
We need to state the vertices of the triangle ABC:
A = (-1, 2)
B = (-2, 1)
C = (0, 0)
a) dilation by a scale factor of 4
[tex]\begin{gathered} (x,\text{ y\rparen }\rightarrow\text{ \lparen4x, 4y\rparen} \\ A=\text{ \lparen4\lparen-1\rparen, 4\lparen2\rparen\rparen = \lparen-4, 8\rparen} \\ B\text{ = \lparen4\lparen-2\rparen, 4\lparen1\rparen\rparen = \lparen-8, 4\rparen} \\ C=\text{ \lparen4\lparen0\rparen, 4\lparen0\rparen\rparen = \lparen0, 0\rparen} \end{gathered}[/tex]b) reflect over the x axis:
[tex]\begin{gathered} (x,\text{ y\rparen }\rightarrow\text{ \lparen x, -y\rparen} \\ We\text{ will negate all the y values of the vertices above while keeping x coordinate constant} \\ A\text{ = \lparen-4, -8\rparen} \\ B\text{ = \lparen-8, -4\rparen} \\ C=\text{ \lparen0, -0\rparen = \lparen0, 0\rparen} \end{gathered}[/tex]c) dilate by 1/2
[tex]\begin{gathered} (x,\text{ y\rparen }\rightarrow(\frac{1}{2}x,\text{ }\frac{1}{2}y) \\ We\text{ will multiply the coordinates above by 1/2 in both the x and y coordinates} \\ A^{\prime}\text{ = \lparen}\frac{1}{2}(-4),\text{ }\frac{1}{2}(-8))\text{ = \lparen-2, -4\rparen} \\ B^{\prime}\text{ = \lparen}\frac{1}{2}(-8),\text{ }\frac{1}{2}(-4))\text{ = \lparen-4, -2\rparen} \\ C^{\prime}\text{ = \lparen}\frac{1}{2}(0),\text{ }\frac{1}{2}(0))\text{ = \lparen0, 0\rparen} \\ \\ Image\text{ of ABC: A' \lparen-2, -4\rparen, B' \lparen-4, -2\rparen and C' = \lparen0, 0\rparen} \end{gathered}[/tex]d) To determine if the area of the pre-image is greater or less than, we will plot the coordinates of both triangles:
Since the triangle of the Image is greater than the triangle of the pre-image (original figure), then the area of the pre- image will be less than the new image
e) For two triangles to be congruent, the sides and angles for both triangles will be equal
For two triangles to be similar, the ratio of their corresponding sides will be equal
The image A'B'C' is a scaled triangle of ABC. This mean the sides can't be equal but the ratio fo their corresponding sides will be equal.
Hence, it is simlar
given the function g(x)=3x-7, determine when g(x)=-4
You have the following function:
g(x) = 3x - 7
In order to determine th value of x when g(x) = -4, you equal the previous expression to -4 and solve for x, just as follow:
g(x) = 3x - 7 replace g(x) = -4
-4 = 3x - 7 add 7 both sides
-4 + 7 = 3x simplify
3 = 3x divide by 3 both sides
3/3 = x
x = 1
Hence, the anser is x = 1
An artist paints a mural on a wall at the local park The wall measures 16 1/4 meters in length. The artist paints a star every 18/20 meter. What is the total number of stars the artist paints om the mural?
The total length of the wall is
[tex]l=16\frac{1}{4}m=\frac{65}{4}m[/tex]the artist paint the star in every 18/20 meter,
so the total number of the stars are,
[tex]\frac{(\frac{64}{5})}{\frac{18}{20}}[/tex][tex]\frac{64\times20}{5\times18}=14.22[/tex]so the number of stars will be an integer. so we will round it
so the answer is 14 star.
I need help with this
A trinomial is a polynomial that has three terms.
Let the unknown trinomial be represented by A
Thus, the difference between the two trinomials as stated in the question will be represented mathematically as;
[tex]\begin{gathered} A-(3x^2-2x+7)=5x^2+11x-16 \\ \text{Making A the subject of the formula, we have} \\ A=(5x^2+11x-16)+(3x^2-2x+7) \\ \operatorname{Re}\text{arranging and collecting the like terms, we have;} \\ A=5x^2+3x^2+11x-2x-16+7 \\ A=8x^2+9x-9 \end{gathered}[/tex]Therefore, the expression of the other trinomial is;
[tex]A=8x^2+9x-9[/tex]The correct answer is option C.
The weight, in pounds, of a male child can be estimated 3 using the function f(x) = 2.69x^3/4, where x represents the child's age in months. Determine the child's weight at 3 years of age, rounded to the nearest thousandth.
the modeled equation for the weight of a male child is
[tex]f(x)=2.69X^{\frac{3}{4}}[/tex]x is the age of the child in months
to calculate the child weight in 3 years
we need to convert the years to months since x is a function of months
12 months ==== 1 year
x months = 3 years
cross multiplication
12 x 3 = 1 * x
x = 36 months
Therefore, 3 years =
Solve using substitution. 6x + y = 5 -8x - 5y = 19 how do I do this
The given system of equations is
[tex]\begin{gathered} 6x+y=5 \\ -8x-5y=19 \end{gathered}[/tex]To solve the system, first, let's multiply the first equation by 5.
[tex]\begin{gathered} 30x+5y=25 \\ -8x-5y=19 \end{gathered}[/tex]Then, we combine the equations
[tex]\begin{gathered} 30x-8x+5y-5y=25+19 \\ 22x=44 \\ x=\frac{44}{22} \\ x=2 \end{gathered}[/tex]Now, we find y
[tex]\begin{gathered} 6x+y=5 \\ 6\cdot2+y=5 \\ 8+y=5 \\ y=5-8 \\ y=-3 \end{gathered}[/tex]Hence, the solution is (2,-3).Solve the inequality c+49 <-16
ANSWER
c < -65
EXPLANATION
We have the inequality:
c + 49 < -16
To solve this, we collect like terms:
c < -16 - 49
Simplify:
c < -65
That is the answer.