The answer is:
Sigma 960 (1/4)^i-1 ; the sum is 1280
Step-by-step explanation:We are given the first term of the geometric sequence as:
[tex]a_1=960[/tex]
Also, the common ratio of the terms in geometric sequence is: [tex]\dfrac{1}{4}[/tex]
We know that if the series is a geometric series than the sum of the terms is given by:
[tex]a+ar+ar^2+.....\\\\=a(1+r+r^2+....)\\\\=\sum ar^{n-1}[/tex]
where a is the first term of the series and r is the common difference.
Here a=960
and r=1/4
Hence,
The sum of the series is:
[tex]=\sum 960(\dfrac{1}{4})^{i-1}[/tex]
Now we know that the sum of the infinite geometric series is given by:
[tex]S=\dfrac{a}{1-r}[/tex]
where S is the sum of the series.
Hence, here the sum of the series is calculated by:
[tex]S=\dfrac{960}{1-\dfrac{1}{4}}\\\\\\S=\dfrac{960}{\dfrac{3}{4}}\\\\\\S=\dfrac{960\times 4}{3}\\\\\\S=1280[/tex]
Hence, the sum is: 1280
A ramp 24 ft long rises to a platform that is 20 ft off the ground. find x , the angle of elevation of the ramp. round your answer to the nearest tenth of a degree.
Final answer:
To determine the angle of elevation x, we use the sine function with the opposite side (20 ft) and the hypotenuse (24 ft) to get sin(x) = 20/24, and by taking the arcsin of the result, we find that x ≈ 56.4° when rounded to the nearest tenth of a degree.
Explanation:
To find x, the angle of elevation of the ramp, we can use trigonometric functions. Since we have the length of the ramp (hypotenuse) and the height of the platform (opposite side) in a right-angled triangle, we can use the sine function, which is defined as the ratio of the opposite side to the hypotenuse.
Using the sine function: sin(x) = opposite/hypotenuse = 20/24.
We first calculate the ratio: sin(x) = 20/24 = 0.8333.
To find the angle x, we take the inverse sine (also known as arcsine) of the ratio: x = arcsin(0.8333).
Using a calculator set to degree mode, we find that x ≈ 56.4° when we round to the nearest tenth of a degree.
The perimeter of a rectangle is 120 feet. The ratio of the width to the length is 2:3. Find the length and width.
If a sphere has a radius of 2 cm what is the approximate surface area of the sphere
Identify the terms and coefficients for4p+5-3g
A board is 27 inches long. How long is the board in centimeters? Use the following conversion: 1 inch is 2.54 centimeters.
1 inch = 2.54 cm
so multiply 27 by 2.54
27 x 2.54 = 68.58 cm long
To convert 27 inches to centimeters, multiply 27 by the conversion factor of 2.54 cm per inch, resulting in 68.58 centimeters.
The student has asked how long a board that is 27 inches long is in centimeters, given the conversion factor that 1 inch is equal to 2.54 centimeters. To convert inches to centimeters, you multiply the length in inches by the conversion factor. In this case, you would multiply 27 inches by 2.54 to find the length in centimeters.
Here is the calculation:
27 inches x 2.54 cm/inch = 68.58 centimeters.
Therefore, a board that is 27 inches long is 68.58 centimeters long.
All right triangles with the same acute angle θ are
2x-3y=8 solve for (y)?
How to find the midpoint formula
At the Atkins County Fair, the crowd for a concert by The Cooper Band cannot exceed a certain number. The limit for the number of people who can attend the concert is 0.04 people per square foot of space in the arena. If the arena is shaped as a rectangle with sides 220 feet by 200 feet, how many people can attend the concert?
I'm having a bit of trouble, could you help?
It takes Brian 15 hours longer to build a model car than it takes John. If they work together, they can build the model car in 4 hours. Using complete sentences, explain each step in figuring out how to determine the time it would take Brian to build the car on his own
It would take Brian 20 hours to build the model car on his own.
To determine the time it would take Brian to build the model car on his own, we can follow these steps:
1. Let's denote the time it takes John to build the model car as J hours.
2. Since it takes Brian 15 hours longer than John, we can express the time it takes Brian as J +15 hours.
3. If they work together, they can build the model car in 4 hours. This means that in 1 hour, they complete [tex]\(\frac{1}{4}\)[/tex] of the job together.
4. In 1 hour, John completes [tex]\(\frac{1}{J}\)[/tex] of the job, and Brian completes [tex]\(\frac{1}{J + 15}\)[/tex] of the job.
5. So, the equation representing the rate at which they work together is:
[tex]\[ \frac{1}{J} + \frac{1}{J + 15} = \frac{1}{4} \][/tex]
6. Now, we solve this equation to find J, the time it takes John to build the model car on his own.
7. Once we find J, we can find J+15 which represents the time it takes Brian to build the model car on his own.
So, the equation representing the rate at which they work together is:
[tex]\[ \frac{1}{J} + \frac{1}{J + 15} = \frac{1}{4} \][/tex]
To solve this equation, we can multiply both sides by the least common denominator, which is [tex]\( 4J(J + 15) \)[/tex], to clear the fractions:
[tex]\[ 4(J + 15) + 4J = J(J + 15) \][/tex]
Expanding and simplifying:
[tex]\[ 4J + 60 + 4J = J^2 + 15J \]\[ 8J + 60 = J^2 + 15J \]\[ J^2 + 15J - 8J - 60 = 0 \]\[ J^2 + 7J - 60 = 0 \][/tex]
Now, we have a quadratic equation. We can solve this equation using factoring, completing the square, or the quadratic formula. Let's use factoring:
[tex]\[ (J - 5)(J + 12) = 0 \][/tex]
Setting each factor equal to zero:
[tex]\[ J - 5 = 0 \] or \( J + 12 = 0 \)[/tex]
Solving each equation:
[tex]For \( J - 5 = 0 \), we get \( J = 5 \).For \( J + 12 = 0 \), we get \( J = -12 \).[/tex]
Since time cannot be negative, we discard the negative solution.
Now that we have found J=5 hours, which represents the time it takes John to build the model car on his own, we can find [tex]\( J + 15 = 5 + 15 = 20 \)[/tex] hours.
What is the graph of the function?
f(x) = 2x 2
Answer:
For f(x)=2x² graph, the answer is A.
Step-by-step explanation:
For x = -2; y = 8For x = -1; y = 2For x = 0; y = 0For x = 1; y = 2For x = 2; y = 8Create a set of numbers where the mode is equal to 10 the median is equal to 12 and the average is 12.
Please I really need help!
A set of n = 15 pairs of X and Y scores has SSX = ...A set of n = 15 pairs of X and Y scores has SSX = 10,SSY = 40, and SP = 30. What is the slope for the regression equation for predicting Y from X?
g(a) = 3^3a-2 . Find g(1)
what is the last line of proof
A student has some 1 and 5 bills in his wallet. he has a total of 14 bills that are worth 50 how many of each type of bill does he hae
x = ones
y = fives
x+y = 14 total bills
rewrite as x = 14-y
1x +5y = 50
1(14-y) +5y = 50
14-1y +5y = 50
14 +4y = 50
4y = 36
y = 36/4 = 9
x = 14-y = 14-9 = 5
9+5 = 14
9x5 = 45 +5x1 = 50
he had 9 fives and 5 ones
Answer:
There were 5 bills of '1' and 9 bills of '5'.
Step-by-step explanation:
let the number of '1' bill be x
let the number of '5' bill be y.
Total number of bills = 14
x + y = 14 ..[1]
Worth of 14 bills = 50
So,[tex]x\times 1+y\times 5=\$50[/tex]
[tex]x+5y=50[/tex]..[2]
x + y = 14
x = 14 - y
Putting value of x in [2]:
[tex]14- y +5y=50[/tex]
[tex]y=\frac{50-14}{4}=9[/tex]
y = 9
x = 14-y = 14 - 9 = 5
There were 5 bills of '1' and 9 bills of '5'.
To which subset of real numbers does -18 not belong?
We have that the -18 is a rational quantity due to the fact it can be expressed as a fraction of two integers. It's additionally an integer.
Rational number & Integer
Real numbersGenerally
The subsets of the actual numbers are as follows
Irrational numbersRational numbersWhole numbersNatural numbers etc-18 is a rational quantity due to the fact it can be expressed as a fraction of two integers. It's additionally an integer.
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1) 8k-(6k-4)=10 show work please
2) -12x+8+5x=36 show work please
3) -2y+4=8y-6
1) 8k - (6k - 4) = 10
[tex]\mathsf{8k-6k+4=10}\\\\ \mathsf{2k=10-4}\\\\ \mathsf{2k=6}\\\\ \underline\mathsf{k=\dfrac{6}{2}=3}}[/tex]
2) -12x + 8 + 5x = 36
[tex]\mathsf{-12x+8+5x=36}\\\\ \mathsf{-12x+5x=36-8}\\\\ \mathsf{-7x=28}\\\\ \mathsf{x={28}{-7}}\\\\ \underline{\mathsf{x=-4}}[/tex]
3) -2y + 4 = 8y - 6
[tex]\mathsf{-2y+4=8y-6}\\\\ \mathsf{-2y-8y=-6-4}\\\\ \mathsf{-10y=-10}\\\\ \mathsf{y=\dfrac{-10}{-10}}\\\\ \underline{\mathsf{y=1}}[/tex]
A building manager installs sensors to see how often people turn off the lights when they leave a room. After a month, the manager has a sample size of 400, a sample mean of 47%, and a sample standard deviation of 4%. What is the confidence level for a confidence interval of 46.6% to 47.4%?
A. 68%
B. 85%
C. 99.7%
D. 95%
The answer is C.
The figures in your problem are inconsistent with
the data.
The sample std deviation is not and cannot be .04 (4%).
the sample std dev is sqrt[(.47)(1-.47)/400]=.02495.
The margin of error from your CI is .474 -.47= .004 (very
small).
This would represent .02495/.004= 6.24 std dev. from
expected, which would have a very
high level of confidence (99.97.. %). Something is likely
wrong with the figures in your problem.
Answer: 95%
Step-by-step explanation:
Complete the statement f(3) is
The corresponding output for an input value of 3 is -1. Hence f(3) is -1
from the mapping function given, the domain values are the input values of the function while the range is the output.
To get the value of f(3), we need to check the corresponding output when the input variable is 3.
From the given one-on-one mapping, we can see that the corresponding output for an input value of 3 is -1. Hence f(3) is -1
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A vehicle travels on a highway at a rate of 65 mi/h. How long does it take the vehicle to travel 25 mi?
How did Greek mathematician Pythagoras come up with the Pythagorean Theorem. Explain
A ladder is leaning up against a 17 foot Wall at an angle of elevation of 37°. How far is the foot of the latter from the wall? Round your answer to the nearest 10th of a foot
The mapping diagram shows a relation.
What is the domain of the relation?
{x| x = –4 , 0, 1, 2}.
{x| x = –7, –6, 2, 11, 3}.
{y| y = –4, 0, 1, 2}.
{y| y = –7, –6, 2, 11, 3}.
Answer:B. {x| x = –7, –6, 2, 11, 3}.
which description matches the transformations y=cosx undergoes to produce y= -2cos3x
A. Horizontal compression by factor 1/3, vertical stretch by factor 2, then a reflection through the x-axis
B. reflection through the y-axis, vertical shift of 2 units, horizontal shift right by 3 units
C.Horizontal shift 2 units, then vertical shift up by 3 units
D.Horizontal stretch by factor 2, reflection through the x-axis, then the vertical stretch by factor 3
Answer: A. Horizontal compression by factor 1/3, vertical stretch by factor 2, then a reflection through the x-axis
Step-by-step explanation:
Since, If there is a function, y= a cos bx
Then a shows vertical stretch while b shows horizontal compression.
Here, Given function, y= cos x
After transformation, It is giving y = -2 cos 3x
therefore here is a horizontal compression occurs with factor 1/3.
Also Negative sign shows there also did reflection through x-axis.
And, It also, stretched vertically by factor 2. ( also shown in the below graph)
Thus, Option A) is correct.
what is the simplified form of (6^2+4)-15
The solutions to the inequalities 0≤x≤2,0≤y≤1, and y≤−12x+32 form a polygonal region in the plane. How many sides does this polygon have?
Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s). Line A passes through points (-1, 5) and (3, -1). Line B passes through points (7, 2) and (6, -1). At what point does line A intersect line B?
In two supplemantary angles, the measure of one angles is 6 more than the twice the measure of the other.the measure of these two angles are