We have 20 marbles in the bag. The probability of a marble that is not green is 20/20.
So the probability of the randomly that the marble is not green is 20/20 or just 20.
What is the equation through the points: (-7, -3), (1, 2)
ASAP please
The equation of the line passing through the points (-7, -3) and (1, 2) is [tex]y = \frac{5}{8}x + \frac{11}{8}[/tex].
What is the equation of the line?The formula for equation of line is expressed as;
y = mx + b
Where m is slope and b is y-intercept.
First, we determine the slope of the line.
Given the two points are (-7, -3) and (1, 2)
We can find the slope of the line by using the slope formula:
m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Substituting the values, we get:
m = (2 - (-3)) / (1 - (-7))
m = 5/8
Using the point-slope form, plug in one of the given points and slope m = 5/8 to find the equation of the line.
Let's use the point (-7, -3:
y - y₁ = m(x - x₁)
[tex]y - (-3) = \frac{5}{8}( x - (-7) ) \\\\y + 3 = \frac{5}{8}(x + 7 )\\\\y + 3 = \frac{5}{8}x + \frac{35}{8} \\ \\y = \frac{5}{8}x + \frac{35}{8} - 3\\\\y = \frac{5}{8}x + \frac{11}{8}[/tex]
Therefore, the equation of the line is [tex]y = \frac{5}{8}x + \frac{11}{8}[/tex].
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The approximate number of zombies in a certain city over time is given in the table below. Answer the questions below to determine what kind of function would best fit the data, linear or exponential.
Number of Hours Since Zombies First Spotted, x
1
2
3
4
Approximate Number of Zombies, f(x)
24
56
88
122
function would best fit the data because as x increases, the y values change
. Rounded to the nearest .5, the
of this function is approximately
A linear function would be the best fit for the given data because as x increase the y-values change linearly.
Rounded to the .5 the slope of the function is approximately 32.5
Which type of function models the population?A linear relation means that any increase of 1 unit in the independent variable (x) has always the same impact in the other variable (y).
In this particular case, the increases in the y-value are:
56 - 24 = 32
88 - 56 = 32
122 - 88 = 34
The mean of that, rounded to .5, is (32 + 32 + 34)/3 = 32.5
So the change is almost the same one, then this seems to be a linear relation.
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What is an equation of the line that passes through the point (5,-5)(5,−5) and is parallel to the line x+5y=20?
Answer:
[tex]y = -x/5 -4.[/tex]
Step-by-step explanation:
To simplify the line x + 5y = 20 into y = mx + b form:
x + 5y = 20.
5y = -x + 20.
y = -x/5 + 4.
The line parallel to the line y = -x/5 + 4 will have the same slope of -1/5.
We get the equation:
y = -x/5 + b.
To find b, we plug in the point (5, -5).
-5 = -5/5 + b.
-5 = -1 + b.
b = -4.
[tex]y = -x/5 -4.[/tex]
How do I solve this?
The 99% confidence interval estimate of the population standard deviation is [12.54, 54.62] mi/h.
What is a confidence interval?A confidence interval is the mean of your estimate plus and minus the variation in that estimate. This is the range of values you expect your estimate to fall between if you redo your test.
x = [64, 60, 56, 60, 52, 59, 58, 59, 70, 68]
s = 5.963
n = 10
CI = [(n-1)*s²/chi2_upper, (n-1)*s²/chi2 lower]
CI = [(9*5.963^m²)/19.022, (9*5.963²)/2.700]
CI = [12.54, 54.62]
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Ted made $56 in interest by placing $700 in a savings account with simple interest for 1 year. What was the interest rate?
The interest rate if Ted made $56 in interest for 1 year on a principle of $700 will be 8%.
Given, the interest amount is $56.
The principal amount is $700.
The time period of investment is 1 year.
Now, we have to find the interest rate.
The formula to find the simple interest is,
S.I = PTR/100.
Now, we need to substitute the values in the above formula.
S.I = PTR/100
56 = 700×1×R / 100
56 = 700R/100
5600 = 700R
R = 5600/700
R = 8
Therefore, the interest rate is 8%.
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How do you find area?
The area of a shape is calculated by calculating the amount of space on the shape
How do you find area?By definitinon, the area of a shape is the amount of space on the shape
using the above as a guide, we have the following:
Area of rectangle = Length * WidthArea of square = Length²Area of triangle = 1/2 * base * heightArea of circle = π * radius²Area of parallelogram = base * heightThere are several formulas to calculate area
The formula to use is dependent on the shape whose area is being calculated
This means that the area of trapezoid cannot be calculated using the area of parallelogram
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Unit seven geometry and measurement homework to perimeter area of rectangles and parallelograms answer key
The required maximum area of the rectangle is 17.5m
How to explain the perimeterThe perimeter of a rectangle is = 70m.
We have to find The maximum area of rectangle .
A rectangle will have the maximum possible area for a given perimeter when all the sides are the same length. Since every rectangle has four sides, if you know the perimeter, divide it by four.
Maximum area of rectangle is = 70 / 4
= 17.5
The required value of maximum area of the rectangle is 17.5m.
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the perimeter of a rectangle is 70m. What are the dimensions that will produce the maximum area of such a rectangle
Can someone help pleaseeee
The result of the row 1 multiplication by 1/4 is determined as [¹/₂ 0 ].
What is the result of the row multiplication?The resultant of the row multiplication in the Matrice is calculated by applying the following method;
row 1 in the given matrices = [2 0]
To multiply row by 1/4, we will multiply each entity by 1/4 as shown below;
[2 0] x 1/4 = 1/4(2 0)
= [¹/₂ 0 ]
Thus, the result of the row multiplication is determined by multiplying each entry in row 1, that is 2, and 0 by 1/4.
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Please help quickly I'm begging!
Assignment details: In this assignment, you will create two graphs and answer questions about Bond's Gym, the business you are supporting.
The functions are f(x) = 600 - 10x & g(x) = 20/3x, and the graphs are added as attachments
Creating the functions of the Bond's GymFrom the question, we have the following parameters that can be used in our computation:
The table of values
For the membership application, we have the following features
Initial value = 600Constant rate of change = (450 - 600)/(15 - 0) = -10So, the function is
f(x) = 600 - 10x
For the membership available, we have the following features
Initial value = 0Constant rate of change = (100 - 0)/(15 - 0) = 20/3So, the function is
g(x) = 20/3x
So, the functions are f(x) = 600 - 10x & g(x) = 20/3x
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PLS HURRY I AM GIVING BRAINLIEST!!!
the question is in the photo!!
A. Using the formula for area of rectangle, the length of the rectangular space is: 2x - 3
B. The expression to prove this is: (2x - 3) * 4x = 8x² - 12x
What is the Area of a Rectangle?To find the area of a rectangle, multiply the width and the length together. This means: Area = length * width.
Part A: We are given that area of the rectangular space is expressed as 8x² - 12x, if the width is 4x, then:
Length of the rectangular space = (8x² - 12x) / 4x = 4x(2x - 3)/4x
Length = 2x - 3
Part B: To prove this, we have:
length * width = 8x² - 12x
Substitute:
(2x - 3) * 4x = 8x² - 12x
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A company makes paper labels for paint cans. As shown below, each can is in the shape of a cylinder with a height of 5 in and a diameter of 6in . The paper label is wrapped around the can and covers only the side of the can (not the top or bottom). If the company has a total of 4521.6in^2 of paper available, how many labels can be made? Use 3.14 for n , and do not round your answer.
48 labels can be made if the company has a total of 4521.6in² of paper available.
We can calculate the number of labels by finding the surface area of the side of the can.
What is the surface area of the side of the can?The surface area of the side of the can is the circumference of the base times the height:
C = πd
C = 3.14 x 6
C = 18.84 in
Area = C x h
Area = 18.84 x 5
Area = 94.2 in²
Therefore, each label requires 94.2 in² of paper.
To find the number of labels that can be made, we divide the total amount of paper by the amount needed per label:
Number of labels = Total area of paper / Area per label
Number of labels = 4521.6 / 94.2
Number of labels = 48
Therefore, 48 labels can be made using a total of 4521.6in² of paper.
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ACTIVITY 1: Find f−1 in each of the following.
The [tex]f^{-1}[/tex] inverse of f of each
1) (x + 1)/2
2) [tex]\sqrt{\frac{x+1}{4} } \\[/tex]
3) 3 - 4x
4) [tex](x+5)^{1/3}[/tex]
5) [tex]\sqrt{x + 4}[/tex]
1) function f(x) = 3x + 1
let f(x) = y
y = 3x + 1
y - 1 = 3x
(y - 1)/3 = x
x obtained is [tex]f^{-1}[/tex]
and y will be x
[tex]f^{-1}[/tex] = (x-1)/3
2) equation f(x) = 4x² - 1
let f(x) = y
y = 4x² - 1
y + 1 = 4x²
(y + 1 )/4 = x²
x = [tex]\sqrt{\frac{x + 1}{4} }[/tex]
x obtained is [tex]f^{-1}[/tex]
and y will be x
[tex]f^{-1}[/tex] = [tex]\sqrt{\frac{x + 1}{4} }[/tex]
similarly,
3) f(x) = 3-x/4
y = 3-x/4
4y = 3 - x
x = 3 - 4x
[tex]f^{-1}[/tex] = 3- 4x
4) y = x³ - 5
y + 5 = x³
x = [tex](y + 5)^{1/3}[/tex]
[tex]f^{-1}[/tex] = [tex](x+5)^{1/3}[/tex]
5) y = x² - 4
y + 4 = x²
x = √y + 4
[tex]f^{-1} = \sqrt{x + 4}[/tex]
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Ellie and her older brother, Shawn, just got new phones! Both phones are shaped like rectangular prisms that are 3/4 of a centimeter tall. Ellie went with the smaller model that is 12 centimeters long and 6 centimeters wide, while Shawn went with the larger model that is 15 centimeters long and 8 centimeters wide. How many cubic centimeters larger is Shawn's phone than Ellie's?
Shawn's phone is 36 cubic cm larger than Ellie's phone.
How many cubic cm larger is Shawn's phone?To get difference in volume between the two phones, we must calculate volume of each phone and subtract the volume of Ellie's phone from the volume of Shawn's phone.
The volume of a rectangular prism is: length*width* height. The height of both phones is 3/4 centimeters.
The volume of Ellie's phone is:
= length x width x height
= 12 cm x 6 cm x (3/4) cm
= 54 cubic centimeters
The volume of Shawn's phone is:
= length x width x height
= 15 cm x 8 cm x (3/4) cm
= 90 cubic centimeters
The difference in volume is:
= Volume of Shawn's phone - Volume of Ellie's phone
= 90 cubic centimeters - 54 cubic centimeters
= 36 cubic centimeters
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Draw n equally spaced marks between 0 and 1 on each of the x and y axes. Connect the first mark on the x-axis (closest to 0) to the last mark on the y axis (closest to 1), the second mark on the x axis to the second last on the y axis, and so on.
As n increases this process creates a curved boundary, C, as seen in this example diagram. Give an equation for this curve as n → ∞.
the equation for the curve C as n approaches infinity is y = x This is the equation of the diagonal line that passes through the points (0, 0) and (1, 1).
what is diagonal line ?
A diagonal line is a straight line that runs obliquely (at an angle) between two opposite corners of a geometric shape, such as a rectangle or a square. It is called "diagonal" because it connects two non-adjacent vertices of the shape, forming a diagonal across the interior of the shape.
In the given question,
As n increases, the process described in the question creates a set of line segments that connect the equally spaced points on the x-axis to the equally spaced points on the y-axis. The resulting curve is a piecewise linear function that converges to a continuous curve as n approaches infinity.
To derive an equation for this curve, we can start by considering the line segment connecting the first mark on the x-axis to the last mark on the y-axis. Let's call this point (0, 1) and the nth mark on the x-axis (1, 0). The slope of the line connecting these two points is given by:
m = (0 - 1) / (1 - nth mark) = (1 - nth mark)^(-1)
where the nth mark on the x-axis is given by:
nth mark = 1 - (1/n)
Substituting this expression for nth mark into the equation for m, we get:
m = (n/(n-1))
The y-intercept of the line is given by:
b = 1 - m = 1 - (n/(n-1)) = 1/(n-1)
Therefore, the equation for the line segment connecting the first mark on the x-axis to the last mark on the y-axis is:
y = (n/(n-1))x + 1/(n-1)
Similarly, the equation for the line segment connecting the second mark on the x-axis to the second last on the y-axis is:
y = (n/(n-2))x + 1/(n-2)
Continuing this process for all n line segments, we get:
y = (n/(n-k))x + 1/(n-k)
where k = 1, 2, ..., n.
To get the equation for the curve C, we need to take the limit as n approaches infinity. As n goes to infinity, the value of k becomes insignificant compared to n, and we can approximate the equation for the curve as:
y = x
Therefore, the equation for the curve C as n approaches infinity is:
y = x
This is the equation of the diagonal line that passes through the points (0, 0) and (1, 1).
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HELP PLEASE I REALLY NEED HELP ON THIS QUESTION HELP!
The number for which the theoretical probability matches the experimental probability is 2.
What is experimental and theoretical probability?
Experimental probability and theoretical probability are two different ways to determine the likelihood of an event occurring.
Experimental probability is determined by performing an experiment or conducting observations and collecting data to see how often an event occurs. This method involves counting the number of times an event occurs and dividing that number by the total number of trials or observations.
Theoretical probability, on the other hand, is determined by using mathematical principles and calculations to determine the likelihood of an event occurring. It is based on the assumption that all outcomes are equally likely.
In the given table;
For theoretical, P(2) = 1/4
For experimental, P(7) = 7/28 = 1/4
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my bike tire has a radius of 12 inches how much distance will i cover after 50 rotations
Therefore , the solution of the given problem of unitary method comes out to be 50 tyre rotations, you will have covered around 376.99 inches (or about 31.41 feet).
What is a unitary method?The well-known simple approach, real variables, and any crucial elements from the very initial And specialised inquiry can all be used to finish the work. Customers may then be given another chance to try the product in response. If not, significant impacts on our understanding of algorithms will vanish.
Here,
The circumference of the circle the bike tyre forms is equal to the distance travelled in one rotation of the tyre. The following formula determines a circle's circumference:
=> C = 2πr
where the mathematical constant is roughly equivalent to 3.14159 and r is the circle's radius.
Your bicycle tyre's radius in this instance is 12 inches. The tyre's circumference is as follows:
=> C = 2π(12) = 24π inches
You may easily calculate how far you will travel after 50 spins by multiplying the tyre's circumference by the number of turns:
=> Covered distance: 50 × 24π ≈ 376.99 inches
As a result, after 50 tyre rotations, you will have covered around 376.99 inches (or about 31.41 feet).
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- x²= +x +12=0 .....................................
Therefore, the solutions of the equation -x² + x + 12 = 0 are x = -3 and x = 6.
What is equation?In mathematics, an equation is a statement that shows the equality between two expressions, typically separated by an equal sign (=). An equation can contain one or more variables, which are symbols that represent unknown or varying values. The value of the variable(s) can be found by solving the equation.
Here,
The given equation is:
-x² + x + 12 = 0
To solve for x, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
In this case, we have:
a = -1, b = 1, and c = 12
Substituting these values into the quadratic formula, we get:
x= (-1 ± √(1² - 4(-1)(12))) / 2(-1)
x = (-1 ± √(1 + 48)) / (-2)
x = (-1 ± √(49)) / (-2)
x = (-1 ± 7) / (-2)
There are two solutions:
x = (-1 + 7) / (-2)
= -3
x = (-1 - 7) / (-2)
= 6
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Please help confused thank you
The solution to the inputs of the given function are:
a) f(0) = 13
b) f(2) = -1
c) f(-2) = 27
d) f(1) + f(-1) = 26
How to solve Function Inputs?The set of input values of a function is referred to as the domain of the function. Then, the set of output values of the function is referred to as the range of the function. Thus, if we possess a set of ordered pairs, we can find the domain by listing all of the input values, which are the x-coordinates.
We are given the function:
f(x) = (-x)³ - x² - x + 13
a) f(0) = (-0)³ - (0)² - 0 + 13
f(0) = 13
b) f(2) = (-2)³ - (2)² - 2 + 13
f(2) = -1
c) f(-2) = (2)³ - (-2)² + 2 + 13
f(-2) = 27
d) f(1) = (-1)³ - (1)² - 1 + 13
f(1) = 10
f(-1) = (1)³ - (-1)² + 1 + 13
f(-1) = 16
f(1) + f(-1) = 10 + 16 = 26
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Write the linear equation that gives the rule for this table.
X
4
5
6
7
Y
24
30
36
42
Write your answer as an equation with y first, followed by an equals sign.
The equation of the line passing through the given points is y = 6x
Given that are the values of x and y coordinates we need to find the equation of the line using them,
So, considering the points (4, 24) and (5, 30),
We know that the equation of a line passing through points (x₁, y₁) and (x₂, y₂) is =
y-y₁ = y₂-y₁ / x₂-x₁ (x-x₁)
Here (x₁, y₁) and (x₂, y₂) are (4, 24) and (5, 30),
Therefore, the required equation is =
y-24 = 30-24/5-4 (x-4)
y-24 = 6(x-4)
y-24 = 6x-24
y = 6x
Hence, the equation of the line passing through the given points is y = 6x
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Use the quadratic formula to solve -2x^2+2x+1
The solutions for -2x²+2x+1 are x = 1 + √(3)/2 and x = 1 - √(3)/2.
To use the quadratic formula to solve -2x²+2x+1, we first identify the coefficients a, b, and c of the quadratic equation in standard form: ax²+bx+c=0. In this case, a=-2, b=2, and c=1. We then plug these values into the quadratic formula
x = (-b ± √(b²-4ac)) / 2a
Substituting the values, we get
x = (-2 ± √(2²-4(-2)(1))) / 2(-2)
Simplifying the expression inside the square root
x = (-2 ± √(4+8)) / (-4)
x = (-2 ± √12) / (-4)
Simplifying the radical
x = (-2 ± 2√3) / (-4)
Reducing the fraction
x = (1 ± √3/2)
Therefore, the two solutions are x = (1 + √3/2) and x = (1 - √3/2).
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ACTIVITY 1: Complete the table below by computing the unknown component of simple interest.
Answer:
what arecrosses on line 2
Can someone help me with this please
Answer:
36 °
Step-by-step explanation:
The markings imply equal sides*
So both of the angles are 27° there
To find angle a we do 180 - (27+27) and obtain 126°
Then to find angle y, we do 180-angle a which is 180-126 and we get 54°
We now have reached the triangle where you're supposed to find x right?
So x is equal to 180 - both other angles, which is 180 - (90 and angle y) = 180- (90+54) = 180 - 144 = 36 which is your final answer
in the diagram below A, B, C are points in the same horizontal plane with AC =BC=22.3 meters.AD is a vertical tower which is anchored at B and C.The angle of elevation of D from B is 25.4. BDC=58.6 and ACB= 50.8.
The length of AB is 27.3m and AD is 113 m.
Using Trigonometry in Triangle ABC
tan 50. 8 = P / B
tan 50.8 = AB / 22.3
1.2261 = AB/ 22.3
AB = 27.34 m
Now, again using Trigonometry
tan 25.4 = P / B
tan 50.8 = AD / 27.34
0.474 = AD/ 27.34
AD = 12.95916 m
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Oak and Maple streets are parallel to each other. Main Street has a traffic
light at (5, 1) and is perpendicular to Oak and Maple. What is the equation
of the line representing Main Street?
Oak St.: y =
A. Y
1
22
+3 Maple St.: y = 22
C. y = -2x + 11
B.y=2x-9
D. y = -2x + 15
+6
The equation of the line representing Main Street is option C. y = -2x + 11
How did we arrive at this equation?The slope of the parallel lines is ½. The Main Street is perpendicular to Oak and Maple. So, the slope of Main Street is -2.
[The product of slopes of perpendicular Lines is -1.)
y - 1 = -2 (x -5)
y - 1 = -2x +10
y = -2x + 10 + 1
y = -2x + 11
Therefore, the best choice is option C. y = -2x + 11
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Since Main Street is perpendicular to Oak and Maple, its slope will be the negative reciprocal of the slope of Oak and Maple. Let's first find the equation of Oak Street.
We don't have enough information to determine the equation of Oak Street, so we need to make an assumption about its equation. Let's assume that Oak Street passes through the point (0,3) and has a slope of 2 (since the options suggest that Oak Street has a positive slope).
Using the point-slope form of a line, we have:
y - 3 = 2(x - 0)
y - 3 = 2x
y = 2x + 3
Now we can find the slope of Maple Street by noticing that it is parallel to Oak Street. Since parallel lines have the same slope, Maple Street also has a slope of 2.
To find the equation of Main Street, we need to use the fact that it passes through the point (5,1). Using the point-slope form of a line, we have:
y - 1 = -1/2(x - 5)
y - 1 = -1/2x + 5/2
y = -1/2x + 7/2
Therefore, the equation of the line representing Main Street is y = -1/2x + 7/2. The answer is (D).
Seven days a year, Tiger Stadium becomes the fifth largest city in the state of Louisiana. Over 92,000 fans pack the stadium to watch the Tigers play. After the game, if the fans leave at a rate of 10% per minute, how long will it take before the stadium is half empty?
1. Find the data using at least 10 numbers in the x column.
2. Create a scatter plot. Label the graph and show increments.
3. Write an exponential equation.
4. Interpret the meaning of the "a" and "b" in your function y=ab^x including the units.
5. Find out how long it will take before the stadium is half empty and all the way empty.
Consider the following statements:
Which of the statement(s) regarding the "limit" concept in
Mathematics is/are TRUE and which is/are FALSE?
I
A limit is a point or value that a pattern, function or sum of
a series, approaches as the number of terms in the series
increases.
II A limit is a value that a function approximates for a given
input value.
III A limit is a value we get closer and closer to, but never
quite achieve.
I: TRUE - A limit is a value that a function, pattern, or sum of a series approaches as the number of terms or input value increases.
What is a Limit in Mathematics based on the given context?II: TRUE - A limit can also be a value that a function approximates for a specific input value, especially when dealing with continuity or differentiability.
III: FALSE - Although a limit might seem like a value that is never quite achieved, it can be reached or even exceeded, depending on the function and its behavior.
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1. A sample of 250 high-school students in a city results in an average number of text messages sent per month of 172.6, with a margin of error of + 4.8. If there are 3000 high-school students in the city, what is the estimated number of text messages sent in a month? between______ and____text messages
Answer:
Step-by-step explanation:
1) Find the upper bound of the confidence interval:
Upper bound = sample mean + margin of error
= 172.6 + 4.8
= 177.4
2) Find the lower bound of the confidence interval:
Lower bound = sample mean - margin of error
= 172.6 - 4.8
= 167.8
3) Estimate the total number of text messages sent in a month:
Total number of text messages sent = (number of students in the population / number of students in the sample) x sample mean
= (3000 / 250) x 172.6
= 2071.2
Therefore, the estimated number of text messages sent in a month is 2071.2, with a 95% confidence interval between 167.8 and 177.4 text messages.
Answer:
Between 503400 and 532200 text messages----------------------------
Determine the average number of text messages sent per student within the margin of errorFind lower bound:
172.6 - 4.8 = 167.8 text messagesFind upper bound:
172.6 + 4.8 = 177.4 text messages Estimate the total number of text messages for all 3000 studentsLower bound:
167.8 text messages × 3000 students = 503400 text messagesUpper bound:
177.4 text messages × 3000 students = 532200 text messagesThe estimated number of text messages sent in a month for all 3000 high-school students in the city is:
Between 503400 and 532200 text messagesFind the surface area of each composite figure. Use 3.14 for π. Round to the nearest tenth. 12m. 6cm. 6cm. 4cm.
The Surface area of the composite figure is calculated as approximately: 234 sq. cm
How to Find the Surface Area of a Composite Figure?The surface area of the composite figure is the area surrounding the faces of the solid as a whole. Therefore, we have:
Surface area (SA) = Surface area of the square prism + surface area of the square pyramid - 2(area of base)
Area of base = area of square = 6 * 6 = 36 sq. cm.
Surface area of the square prism = 2a² + 4ah
a = 6 cm
h = 4 cm
Plug in the values:
Surface area of the square prism = 2(6²) + 4*6*4
= 72 + 96
= 168 sq. cm.
Surface area of the square pyramid = 2bs + b²
b = side length = 6 cm
s = slant height = √(8² + 3²) = 8.5 cm
Plug in the values:
Surface area of the square pyramid = 2 * 6 * 8.5 + 6² = 138 sq. cm.
Surface area of the composite figure = 168 + 138 - 2(36) = 234 sq. cm
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PLS HELP ME WITH THIS QUESTION PLS
PLS SHOW YOUR WORKING OUT
The value of p is -3/2, the value of q is -1/(t+1), and the value of r is 2.
How did we get these values?Let the first term of the arithmetic series be a, and the common difference be d = 3. Then, we have:
a = 2t + 1
n-th term = a + (n-1)d = 2t + 1 + 3(n-1) = 3n + (2t - 2)
(Notice that the second equation can be found by substituting the expression for a into the formula for the n-th term and simplifying.)
We also know that the n-th term is given by (14t - 5), so we can equate the two expressions:
3n + (2t - 2) = 14t - 5
Simplifying and solving for n, we get:
n = (12t + 3)/3 = 4t + 1
So, the n-th term can also be expressed as:
3n + (2t - 2) = 3(4t + 1) + (2t - 2) = 14t - 5
Simplifying, we get:
14t - 5 = 14t - 5
This confirms that our expressions for the first term, common difference, and n-th term are all consistent with each other.
Now, we can use the formula for the sum of an arithmetic series to find the sum of the first n terms:
S_n = (n/2)(2a + (n-1)d) = (n/2)(4t + 4t + 1 + 3n - 3) = (3/2)n^2 + (5/2)t - 3n/2 + 1/2
We want to rewrite this expression in the form p(qt - 1)^r. To do this, we can try to complete the square in the n term, like this:
S_n = (3/2)[n^2 - 2n(t+1) + (t+1)^2] + (5/2)t - (3/2)(t+1)^2 + 1/2
S_n = (3/2)[n - (t+1)]^2 - (1/2)(t+1)^2 + (5/2)t + 1/2
Let u = n - (t+1), so that:
S_n = (3/2)u^2 - (1/2)(t+1)^2 + (5/2)t + 1/2
We want to rewrite this in the form p(qt - 1)^r, so let's try to match the terms:
p = -3/2
q = -1/(t+1)
r = 2
Therefore, the value of p is -3/2, the value of q is -1/(t+1), and the value of r is 2.
Note that the assumption that t is greater than 0 was not necessary for the derivation of the sum formula, but it is necessary for the existence of the arithmetic series (since otherwise the first term would be negative).
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The text format of the question in the picture:
22. The first term of an arithmetic series is (2t + 1) where t is > 0 The nth term of this arithmetic series is (14t - 5)
The common difference of the series is 3
The sum of the first n terms of the series can be written as p(qt - 1)^r where p, q and r are integers.
Find the value of p, the value of q and the value of r Show clear algebraic working.
In circle
V
V,
V
W
=
2
VW=2 and the length of
⌢
=
2
9
π
WX
⌢
=
9
2
π. Find m
∠
W
V
X
∠WVX.
Answer:
Step-by-step explanation:
In a circle, the measure of an inscribed angle is equal to half the measure of its intercepted arc. In this case, the inscribed angle is ∠WVX and its intercepted arc is ⌢WX.
Since the length of ⌢WX is (9/2)π, its measure in degrees is (9/2)π * (180/π) = 810/2 = 405 degrees.
Therefore, the measure of ∠WVX is half the measure of its intercepted arc, which is 405/2 = 202.5 degrees.