Determine the common ratio for each of the following geometric series and determine which one(s) have an infinite sum.
I. 4+5+25/4+…
II. -7+7/4-7/9+…
III. 1/2-1+2…
IV. 4- ++...
A. III only
B. II, IV only
C. I, Ill only
D. I, II, IV only
Answers
The correct answer is Option A ( III Only). I . -16 sum cannot be negative, II. Not a G.P, III. Sum = 1/4, and IV. Not a G.P.
Solution:Given geometric series,
I. 4 +5 +25 /4 ….
The common ratio(r) is (5/1)/(4/1) = 5/4.
S∞ = a / ( 1 - r)
= 4 / ( 1 - 5/4)
= 4 / -1/4
S∞ = -16.
Since sum cannot be negative.
II . -7 + 7/3 - 7/9+ ....
Here common ratio = -7 / (7/3) = -1/3
but - 7/9 / 7 /3 = 7/9
Here there is no common ratio so this not a G.P.
iii. 1/2 -1 + 2.....
Common ratio = -1 / (1/2) = -2
S∞ = a / ( 1 - r)
= 1/2 / (1 -(-2))
S∞ = 1/4.
iv 4 - 8/5 +16/5.....
Here there is no common ratio.
So this is not a G.P.
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Related Questions
Janelle says that lines l and m are skew lines. Planes B and A intersect. Plane B is vertical and contains vertical line n. Plane A is horizontal and contains horizontal line m. Line m and n are perpendicular. Line l is on plane A and it is slightly diagonal. Is Janelle correct? Yes, because the lines are not parallel. Yes, because the lines will intersect. No, because the lines are in the same plane. No, because the lines are perpendicular.
Answers
Lines are parallel and on the same plane. The final answer, "No, because the lines are in the same plane," is the proper response.
What is a line that is perpendicular?
Perpendicular lines are those that cross at a perfect right angle. Parallel lines are those that are always the same distance apart from one another.
The question is incomplete.
Please see the accompanying image for a comprehensive explanation of the question.
Line l and line m are the two lines that are depicted in the image.
The skew lines are in different planes and do not overlap, as far as we are aware.
Therefore,
Lines are parallel and on the same plane. The final answer, "No, because the lines are in the same plane," is the proper response.
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A 6000-seat theater has tickets for sale at $24 and $40. How many tickets should be sold at each price for a sellout performance to generate a total revenue
of $168,000?
The number of tickets for sale at $24 should be ?
Answers
The number of tickets which should be sold to $24 and $40 are 4500 and 1500 respectively.
Given, A 6000-seat theater has tickets for sale at $24 and $40.
How many tickets should be sold at each price for a sellout performance to generate a total revenue of $168,000 = ?
first, assign variables:
X = # of $24 tickets, Y = # or $40 tickets
write equations based on the data presented:
"6000 seat theater..."
X + Y = 6000 ...equation 1
"total revenue of 168,000"
The revenue from each type of ticket is the cost times the number sold, so:
24X + 40Y = 168,000 .....equation 2
from equation 1:
X = 6000 - Y
substitute this into equation 2: (replace X with 6000-Y)
24 (6000 - Y) + 40Y = 168,000
expand:
144,000 -24Y + 40Y = 168,000
rearrange and simplify:
16Y = 168,000 - 144,000
y = 24000/16
Y = 1500
from equation 1:
X = 6000 - 1500
X = 4500
hence the number of tickets for sale at $24 should be 4500.
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It takes chuck 24 minutes to type and spell check 14 pages. Find how many pages he can type and spell check in 1.5 hours. Remember to convert 1.5 hours to minutes
Answers
In order to find how many pages can be typed, first let's convert 1.5 hours to minutes:
[tex]\begin{gathered} 1\text{ hour}\to60\text{ minutes} \\ 1.5\text{ hour}\to x\text{ minutes} \\ \\ \frac{1}{1.5}=\frac{60}{x} \\ x=60\cdot1.5 \\ x=90 \end{gathered}[/tex]Then, to find the number of pages, let's do the following rule of three:
[tex]\begin{gathered} 14\text{ pages}\to24\text{ minutes} \\ x\text{ pages}\to90\text{ minutes} \\ \\ \frac{14}{x}=\frac{24}{90} \\ 24x=14\cdot90 \\ x=\frac{14\cdot90}{24}=52.5 \end{gathered}[/tex]Therefore Chuck can type and spell check 52.5 pages in 1.5 hours.
A contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are: 13 million from the office building, 9 million from the theater, and 4 million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are .17, .27, .45, and .11, respectively. Let x be the random variable that represents the contractor's profits in millions of dollars. Write the probability distribution of x. Find the mean and standard deviation of x.
Answers
Answer:
Probability distribution:
x (million) P(x)
13 0.17
9 0.27
4 0.45
0 0.11
Mean: 6.44
Standard deviation: 4.04
Explanation:
The probability distribution is a table that shows the profits earned and its respective probabilities, so:
x (million) P(x)
13 0.17
9 0.27
4 0.45
0 0.11
Then, the mean can be calculated as the sum of each profit multiplied by its respective probability. Therefore, the mean E(x) is equal to:
E(x) = 13(0.17) + 9(0.27) + 4(0.45) + 0(0.11)
E(x) = 2.21 + 2.43 + 1.8 + 0
E(x) = 6.44
Finally, to calculate the standard deviation, we first need to find the differences between each value and the mean, and then find the square of these values, so:
x x - E(x) (x - E(x))²
13 13 - 6.44 = 6.56 (6.56)² = 43.03
9 9 - 6.44 = 2.56 (2.56)² = 6.55
4 4 - 6.44 = -2.44 (-2.44)² = 5.95
0 0 - 6.44 = -6.44 (-6.44)² = 41.47
Then, the standard deviation will be the square root of the sum of the values in the last column multiply by each probability:
[tex]\begin{gathered} s=\sqrt[]{43.03(0.17)+6.55(0.27)+5.95(0.45)+41.47(0.11)} \\ s=\sqrt[]{16.3264} \\ s=4.04 \end{gathered}[/tex]Therefore, the answers are:
Probability distribution:
x (million) P(x)
13 0.17
9 0.27
4 0.45
0 0.11
Mean: 6.44
Standard deviation: 4.04
The probability distribution for arandom variable x is given in the table.-105101520Probability.2015051.25115Find the probability that -5 < x < 5
Answers
Answer:
P = 0.30
Explanation:
From the table, we see that:
When: x = -5, Probability = 0.15
When: x = 0, Probability = 0.05
When: x = 5, Probability = 0.1
Therefore, the probability of -5 ≤ x ≤ 5 is obtained by the sum of the probabilities from -5 to 5, we have:
[tex]\begin{gathered} P=0.15+0.05+0.10 \\ P=0.30 \end{gathered}[/tex]Therefore, the probability is 0.30
What are the first two steps to graph y=4/5x + 3?
Answers
The function y=4/5x + 3 represents a linear equation.
Let's find some points.
Where x=1, then:
[tex]y=\frac{4}{5}x+3[/tex][tex]y=\frac{4}{5}(1)+3[/tex][tex]y=\frac{19}{5}[/tex]We find the point (1,19/5).
Where 19/5 = 3.8
Now, lest find the x-intercept and y-intercept.
To find the x-intercept, set y=0
[tex]0=\frac{4}{5}x+3[/tex]Solve for x:
[tex]-3=\frac{4}{5}x[/tex][tex]5\cdot-3=4x[/tex][tex]-15=4x[/tex]where
[tex]x=-\frac{15}{4}=-3.75[/tex]So, the x-intercept is the point (-15/4, 0)
To find the y-intercept, set x=0. Then:
[tex]y=\frac{4}{5}x+3[/tex][tex]y=\frac{4}{5}(0)+3[/tex][tex]y=3[/tex]So, the y-intercept is the point (0,3)
Use this information to graph the line.
Hence, the graph for y=4/5x + 3 is given by:
helppppppppppppppppppppppppppppp
Answers
Answer:
(see attached image)
Step-by-step explanation:
Imagine that there is a line drawn at y=x, when a problem wants you to "show the inverse of a function", imagine that y=x acts as a mirror and you have to make the "reflection" of your given function across that mirror.
Write a division equation that represents the equation, How many 3/4 are in 10/9?
Answers
Given:
The number of 3/4 in 10/9.
To find the division equation that represents the given problem:
That is a number that is multiplied by 3/4 to obtain 10/9.
We need to find the number.
[tex]x\times\frac{3}{4}=\frac{10}{9}[/tex]Thus, the division equation will be,
[tex]x=\frac{10}{9}\div\frac{3}{4}[/tex]All changes saved16. Suppose you invest $7500 at an annual Interest rate of 4.2% compounded continuously. How much will you have in the account after 2 years? Round the solution to the nearest dollar.$8158$17,372$17,373$8157
Answers
We have to use the continuous compound interest formula
[tex]A=P\cdot e^{rt}[/tex]Where P = 7500, r = 0.042, t = 2. Let's replace and solve
[tex]\begin{gathered} A=7500\cdot e^{0.042\cdot2} \\ A\approx8,157 \end{gathered}[/tex]Hence, the answer is $8,157.A colony of 4,050 bacteria doubles in size every 297 hours. What will the population be 594 hours from now?
Answers
Answer:
About 280,000
Step-by-step explanation:
Answer:
16,200
Step-by-step explanation:
594 ÷ 297 = 2
4,050 × 2 × 2 = 16,200
another way: 4,050 × 2^2 = 16,200
A motor scooter travels 22 mi in the same time that a bicycle covers 8 mi. If the rate of the scooter is 6 mph more than twice the rate of the bicycle, find both rates.The scooter’s rate is ____ mph. (Type an integer or a decimal)
Answers
Let's use the variable x to represent the speed of the scooter and y to represent the speed of the bicycle.
For a same time t, the scooter travels 22 mi and the bicycle travels 8 mi, so we can write the following equation:
[tex]\begin{gathered} distance=speed\cdot time\\ \\ 22=x\cdot t\\ \\ t=\frac{22}{x}\\ \\ 8=y\cdot t\\ \\ t=\frac{8}{y}\\ \\ \frac{22}{x}=\frac{8}{y} \end{gathered}[/tex]Then, if the rate of the scooter is 6 mph more than twice the rate of the bicycle, we have the following equation:
[tex]x=2y+6\\[/tex]Using this value of x in the first equation, let's solve it for y:
[tex]\begin{gathered} \frac{22}{2y+6}=\frac{8}{y}\\ \\ 22y=8(2y+6)\\ \\ 22y=16y+48\\ \\ 6y=48\\ \\ y=8\text{ mph} \end{gathered}[/tex]Now, calculating the value of x, we have:
[tex]\begin{gathered} x=2y+6\\ \\ x=16+6\\ \\ x=22\text{ mph} \end{gathered}[/tex]Therefore the scooter's rate is 22 mph and the bicycle's rate is 8 mph.
9Which is the best name for a quadrilateral with vertices at A(5,-2), B(2,2), C(1,-5), and D(-2,-1)?A parallelogramB squarerhombusD rectangle
Answers
Parallelogram. Option A is correct
Explanations:In order to determine the best name for a quadrilateral with the given vertices, we will find the measure of the distance AB, BC, CD, and AD using the distance formula as shown;
[tex]D=\sqrt[]{(x_2-x_1)^2+(y_2-y^{}_1)^2}[/tex]For the measure of AB with coordinates A(5,-2), B(2,2);
[tex]\begin{gathered} AB=\sqrt[]{(5-2)^2+(-2-2^{}_{})^2} \\ AB=\sqrt[]{3^2+(-4)^2} \\ AB=\sqrt[]{9+16} \\ AB=\sqrt[]{25} \\ AB=5 \end{gathered}[/tex]For the measure of BC with coordinates B(2,2) and C(1, -5)
[tex]\begin{gathered} BC=\sqrt[]{(2-1)^2+(2-(-5^{}_{}))^2} \\ BC=\sqrt[]{1^2+7^2} \\ BC=\sqrt[]{50} \\ BC=5\sqrt[]{2} \end{gathered}[/tex]For the measure of CD with coordinates C(1,-5), and D(-2,-1);
[tex]\begin{gathered} CD=\sqrt[]{(1-(-2))^2+(-5-(-1^{}_{}))^2} \\ CD=\sqrt[]{3^2+(-4)^2} \\ CD=\sqrt[]{9+16} \\ CD=\sqrt[]{25} \\ CD=5 \end{gathered}[/tex]For the measure of AD with coordinates A(5, -2), and D(-2,-1);
[tex]\begin{gathered} AD=\sqrt[]{(5-(-2))^2+(-2-(-1^{}_{}))^2} \\ AD=\sqrt[]{(5+2)^2+(-2+1)^2} \\ AD=\sqrt[]{7^2+(-1)^2} \\ AD=\sqrt[]{50} \\ AD=5\sqrt[]{2} \end{gathered}[/tex]For the slopes;
Check if the length AB is perpendicular to AD
[tex]\begin{gathered} m_{AB}=\frac{2+2}{2-5} \\ m_{AB}=-\frac{4}{3} \end{gathered}[/tex]For the slope of AD
[tex]\begin{gathered} m_{AD}=\frac{-1+2}{-2-5} \\ m_{AD}=-\frac{1}{7} \end{gathered}[/tex]Since AB is not perpendicular to AD, hence the quadrilateral is not a rectangle and also not a square or rhombus since all the sides are not equal.
From the given distances, you can see that opposite sides are equal (AB = CD and BC = AD ), hence the best name for a quadrilateral is a parallelogram.
Write an equation in point slope form for the line given the slope of 4,and a point on the line (1,2)
Answers
[tex]\begin{gathered} \text{ the equation of a line in slope-point form is} \\ y=mx+b,\text{ we know that m=4, and that (1,2) is on the line, so} \\ 2=4(1)+b \\ 2=4+b \\ b=2-4 \\ b=-2 \\ \\ \text{ Thus, the equation has the form} \\ y=4x-2 \\ \end{gathered}[/tex]
Answer:
[tex]y-2=4(x-1)[/tex]
Step-by-step explanation:
Pre-SolvingWe are given that a line has a slope (m) of 4, and that it contains the point (1,2).
We want to write the equation of this line in point-slope form.
Point-slope form is given as [tex]y-y_1=m(x-x_1)[/tex], where m is the slope and [tex](x_1, y_1)[/tex] is a point, hence the name.
Since we are already given the slope, we can immediately plug it into the formula.
Substitute 4 for m.
[tex]y-y_1=4(x-x_1)[/tex]
Now, let's label the values of (1,2) to avoid confusion while substituting.
[tex]x_1=1\\y_1=2\\[/tex]
Substitute these values into the formula.
[tex]y-2=4(x-1)[/tex]
Topic: point-slope form
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1. Which of the following would be considered a statistical question? (1) Who was the highest paid athlete in 2020? (2) How much does a gallon of gasoline cost? (3) How many people voted for president in 2016?(4) What are the different types of maple trees?
Answers
Looking at the sentences, we have that (1), (2) and (3) are questions that require just one specific answer (For (1) it's a name of a person, for (2) it's a value and for (3) it's also a single value).
But in sentence (4), there are more than one type of maple tree, so we can answer with each type of maple tree, stating the percentage of each type for example.
So the sentence that would be considered a statistical question is (4).
I need help doing this it’s the homework but I need to understand it for the test
Answers
By similar triangle, we have:
[tex]\begin{gathered} Let\text{ the unknown measurement be x} \\ \text{Thus, we have:} \\ \frac{14}{x}=\frac{8}{15} \\ \text{cross}-\text{multiply} \\ 8x=210 \\ x=\frac{210}{8} \\ x=26.25\text{ f}eet \end{gathered}[/tex]Hence, the unknown measurement of the plan is 26.25 feet
The equation for a straight line (deterministic model) is y = Bo + B,X.If the line passes through the point ( - 10,3), then x = - 10, y = 3 must satisfy the equation; that is, 3 = Bo + B1(-10).Similarly, if the line passes through the point (11,4), then x = 11, y = 4 must satisfy the equation; that is, 4 = Bo+B1(11).Use these two equations to solve for Bo and By; then find the equation of the line that passes through the points (-10,3) and (11,4)...Find Bo and B,B1 =Bo(Simplify your answers. Type integers or simplified fractions.)
Answers
To find the equation of the line we just need to find the beta constants. In order to do this we have (they provided us with) the following system of equations:
[tex]\begin{cases}3=\beta_0+\beta_1(-10) \\ 4=\beta_0+\beta_1(11)\end{cases}[/tex]Let us subtract the second equation to the first one:
This implies that
[tex]\beta_1=\frac{-1}{-21}=\frac{1}{21}[/tex]Now, let us replace this value we just got into the second equation to find beta_0:
[tex]\begin{gathered} 4=\beta_0+\frac{1}{21}\cdot11, \\ 4=\beta_0+\frac{11}{21}, \\ \beta_0=4-\frac{11}{21}=\frac{4\cdot21}{21}-\frac{11}{21}=\frac{84-11}{21}=\frac{73}{21} \end{gathered}[/tex]At last,
[tex]\beta_1=\frac{1}{21},\beta_0=\frac{73}{21}[/tex]Then, the equation of the line is just
[tex]y=\frac{73}{21}+\frac{1}{21}x[/tex]the difference between the number c and the quotient of a and b in a mathematical expression.
Answers
Answer:
no difference
step by step explanations
because a/b=c
these means c(b) and a(1)
cb=a this means
cb/b=a/b
b cancle by b
and c=a/b
Division Properties of Exponents HW.
Answers
Given the expressions:
[tex]\begin{gathered} \frac{4^5}{4^2} \\ \text{and} \\ \frac{4^2}{4^5} \end{gathered}[/tex]we can use the following property for exponents in quotients:
[tex]\frac{a^n}{a^m}=a^{n-m}[/tex]in this case, we have the following:
[tex]\begin{gathered} \frac{4^5}{4^2}=4^{5-2}=4^3 \\ \text{and} \\ \frac{4^2}{4^5^{}}=4^{2-5}=4^{-3} \end{gathered}[/tex]then, the difference between both expressions is that when they are simplified, they get opposite signs on their exponents.
rewrite using a single exponent. 9 4, 9 4
Answers
We need to represent the product of two exponents as one single exponent. To do that we need to calculate their product, when the bases are equal we can conserve the base and add the exponents. We will do this below:
[tex]9^49^4=9^{4+4}=9^8[/tex]If two lines intersect and one of the angles formed has a measure of 67°, which of the following statements are true? Explain your answers.
Answers
Intersecting Lines
When two lines intersect, four angles are formed at the point of intersection.
Two pairs of angles are vertical, i.e., they have the same measure.
Two pairs of angles are complementary (or linear) therefore their sum adds up to 180°.
We are given one of the angles that has a measure of 67°.
Then, another angle also measures 67° (the vertical peer).
One of the other angles is 180° - 67° = 113°
The other angle also measures 113° (the other vertical peer).
According to the facts found above, the following statements are true:
* Vertical angles are congruent, therefore another angle must equal 67°
* The lines form linear pairs
* The lines form complementary angles
* Two of the angles formed measure 113°
* Two of the angles formed will have a sum of 180°
Note: The last statement should read "Two pairs of angles formed..."
School: Practice & Problem Solving 7.1.PS-18 Question Help A rectangle and a parallelogram have the same base and the same height. How are their areas related? Provide an example to justify your answer The areas equal. A rectangle has dimensions 5 m by 7 m, so its area is m² A parallelogram with a base of 5 m and a height of 7 m has an area of (Type whole numbers.)
Answers
The image shown below shows the relationship between areas of rectangle and parallelogram
It can be seen that the areas are equal when they have the same sides or dimension
A rectangle has dimensions 5 m by 7 m, so its area is 5m x 7m = 35m²
A parallelogram with a base of 5 m and a height of 7 m has an area of 5m x 7m = 35m²
t, Decimal, Fractions a. 100 is what percent of 80? = 125% b. What is 1/20 as a decimal? What is that decimal as a percent? –5%
Answers
Explanation
Step 1
a) 100 is what percent of 80?
you can solve this by using a rule of three
Let
x represents the percent
then
[tex]undefined[/tex]if a triangle has side lengths of 4x + 6, 6, and 5x; what is the perimeter
Answers
Complete the steps to find the value of x .
Answers
Find a function of the form y = A; A * sin(kx) + C or y = A * cos(kx) + C whose graph matches this one:I don’t understand how my answer is wrong
Answers
Recall that the graph of the cosine function is:
Now, notice that the given graph is the above graph but with a midline
[tex]y=1,[/tex]and amplitude equal to 4. The frequency is also different.
Therefore, the equation of the function given in the graph is:
[tex]y=4cos(\frac{\pi}{5}x)+1.[/tex]Answer: [tex]y=4cos(\frac{\pi x}{5})+1.[/tex]Find the value of x in this equation.
|2x − 3| − 11 = 0
Answers
Answer:7
Step-by-step explanation:
Answer:
x=−4
x=7
Step-by-step explanation:
1. Combine similar terms and use the equality properties to get the variable on one side of the equals sign and the numbers on the other side. Remember to respect the order of operations.
2. Add 11 to both sides of the equation.
3. Use the absolute value definition.
4. Add 3 to both sides of the equation.
5. Divide both sides by 2.
6. The answer is: x=7
x=−4
Sorry for the bad English, love from Vanuatu!
Created by Cortin Lyons riables on Both Sides #3 2.12r – 14 = 8x + 16 12x – 14 = 6x + 16 as with Distributive Property
Answers
Problem N 2
we have
12x-14=8x+16
solve for x
subtract 8x both sides
12x-14-8x=8x+16-8x
4x-14=16
Adds 14 both sides
4x-14+14=16+14
4x=30
divide by 4 both sides
x=30/4
x=7.5
Problem N 4
we have
12x-14=6x+16
solve for x
subtract 6x both sides
12x-14-6x=6x+16-6x
6x-14=16
adds 14 both sides
6x=16+14
6x=30
divide by 6 both sides
x=30/6
x=5
what is the area of a triangle with the legth of 8in,12in,6in
Answers
Area is
[tex]A=\frac{1}{2}bh=\frac{1}{2}\times6\times8=\frac{48}{2}=24[/tex]answer: 24 sq in
Rewrite the equation by completing the square. x^{2}-6x-16 = 0
Answers
Answer:
Step-by-step explanation:
x^2 - 6x - 16 = x^2 - 6x + 9 - 9 - 16 = (x - 3)^2 - 25
Can you Help me with this i cannot do ir
Answers
To translate f(x) 4 units down we subtract 4 from f(x):
[tex]f(x)-4.[/tex]Now, to reflect f(x)-4 over the x-axis we multiply by -1:
[tex]-1(f(x)-4)=-f(x)+4.[/tex]Answer:
[tex]g(x)=-\sqrt[3]{x-2}+4.[/tex]