The form of the partial fraction decomposition is [tex]\frac{At + B}{(t^2 + 4)} + \frac{Ct + D}{(t^2 + 2)} + \frac{Et + F}{(t^2 + 2)^2}[/tex]
How to decompose the function?The fraction is given as:
[tex]\frac{t^4 + t^2 + 1}{(t^2 + 4)(t^2 + 2)^2}[/tex]
To decompose the above, we make use of the following rule:
[tex]\frac{px^n + qx^m +....+ r}{(x^2 + a)(x^2 + b)^2} = \frac{Ax + B}{(x^2 + a)} + \frac{Cx + D}{(x^2 + b)} + \frac{Ex + F}{(x^2 + b)^2}[/tex]
So, we have:
[tex]\frac{t^4 + t^2 + 1}{(t^2 + 4)(t^2 + 2)^2} = \frac{At + B}{(t^2 + 4)} + \frac{Ct + D}{(t^2 + 2)} + \frac{Et + F}{(t^2 + 2)^2}[/tex]
Hence, the form of the partial fraction decomposition is [tex]\frac{At + B}{(t^2 + 4)} + \frac{Ct + D}{(t^2 + 2)} + \frac{Et + F}{(t^2 + 2)^2}[/tex]
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Write the equation of the line with slope -5 and y-intercept 300.
The slope-intercept form is y=mx+b
m=slope=-5
y-int/b = 300
so, y = -5x+300 is the equation of the line
Dalvin drove 190 miles in 5 hours. If he continued at the same rate, how far would he travel in 8 hours?
Explanation:
Dalvin drives 190 miles in 5 hours. His speed is 190/5 = 38 mph.
From here we would say the following
distance = rate*time
distance = (38 mph)*(8 hrs)
distance = 304 miles
Which linear inequality is represented by the graph?
O y > ²/ ²x-3/1/2 -X 5
O y²³x+²/ 1
O y ≤ ² x + 1/1/0
O y< ²/x - / / -X 5
The linear inequality depicted by the graph is: [tex]y \leq \frac{2}{3}x + \frac{1}{5}[/tex]
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.For this line, when x = 0, y = 0.2, hence the y-intercept is of b = 0.2. When x = 3, y = 2.2, hence the slope is given as follows:
[tex]m = \frac{2.2 - 0.2}{3 - 0} = \frac{2}{3}[/tex]
Hence the equation of the line is:
[tex]y = \frac{2}{3}x + 0.2[/tex]
[tex]y = \frac{2}{3}x + \frac{1}{5}[/tex]
The inequality is the values below(less) than the line, also including it, as it is not dashed, hence:
[tex]y \leq \frac{2}{3}x + \frac{1}{5}[/tex]
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How to Answer this i dont know please like it was not in the book im so confused
3y−13≥−9 and 3y−13≤−1
The solution to the system of inequalities given is: [tex]\frac{4}{3} \leq y \leq 4[/tex]
What is the solutions to a system of inequalities?The solution is the set of values that respect all conditions of the system.
The first condition is:
[tex]3y - 13 \geq -9[/tex]
[tex]3y \geq 4[/tex]
[tex]y \geq \frac{4}{3}[/tex]
The second condition is:
[tex]3y - 13 \leq -1[/tex]
[tex]3y \leq 12[/tex]
[tex]y \leq \frac{12}{3}[/tex]
[tex]y \leq 4[/tex]
The intersection of the conditions, which is the solution to the inequality, is:
[tex]\frac{4}{3} \leq y \leq 4[/tex]
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Art store sells packages to different sized square Picture frames the side length of the larger frame SX is modeled by the function SX equals three square root X minus one where X is the area of the smaller frame in square inches which graph shows us
Option A graph represents S(x).
What is a Function ?Function is a statement that relates tow variables.
It is given that
the side length of the larger frame S(x)
S(x) = [tex]\rm 3 \sqrt{x -1}[/tex]
X is the area of the smaller frame in square inches
A square root function is given by
s(x) = [tex]\rm \sqrt{x}[/tex]
when the graph is shifted towards right and stretched by 3 , due to which the graph is taller as compared to the parent graph and
Therefore Option A graph represents S(x).
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A whale swimming against an ocean current traveled 140 mi in 7 h. Swimming in the opposite direction, with the current, the whale was able to travel the same distance in 3.5 h. Find the speed of the whale in calm water and the rate of the ocean current.
The whale exists swimming at 30 mph in calm water and the current exists flowing at 10 mph.
How to estimate the speed of the whale in calm water and the rate of the ocean current?Let x be the speed of a whale in calm water
and y be the speed of the current
In 7 hours the whale swims against the current and travels 140 miles:
7(x - y) = 140
x - y = 140/7
x - y = 20 ...........(1)
In 3.5 hours the whale swims the same distance with the current:
3.5(x + y) = 140
x + y = 140/3.5
x + y = 40 ...........(2)
Adding (1) and (2), we get the value of x:
x - y = 20 + ...........(1)
x + y = 40 ...........(2)
-----------------
2x = 60
x = 60/2
x = 30
Using this value for "x" in either (1) or (2) we can solve for "y":
x - y = 20 ...........(1)
substitute for "x"
30 - y = 20
10 = y
Therefore, the whale exists swimming at 30 mph in calm water and the current exists flowing at 10 mph.
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graph Y=-2x-3 and y=-x+6
See the graph in the attached image.
A shipping company must design a closed rectangular shipping crate with a square base. The volume is 2880ft3. The material for the top and sides costs $3 per square foot and the material for the bottom costs $7 per square foot. Find the dimensions of the crate that will minimize the total cost of material.
The dimensions of the crate that will minimize the total cost of material are 10, 10, and 29.
How to compute the dimension?The volume will be:
= Length × Width × Height
= lwh
For the square base, w = l
2880 = w²h
The cost function will be:
C = 12wl + 4wh + 4lh
Put l = w.
C = 12w² + 8wh
C = 12w² + 8w(2880/w²)
C = 12w² + 23040/w
Differentiate with respect to w
C'(w) = 0
24w = 23040/w²
24w³ = 23040
w³ = 960
w = 10 approximately
The length is 16 as well as they are the same. The height will be:
= 2880/w²
= 2880/10²
= 28.8 = 29
Therefore, the dimensions are 10, 10, and 29.
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Please help word problems challenging for me
Answer:
36 Adult Tickets. 352 Student Tickets
Step-by-step explanation:
Firstly, we have to create 2 expressions to solve a problem like this. If we use the variable s to mean student and a to mean adult, we can create the expressions:
3s + 6a = 1272 to represent how much they made with all ticket sales
s = 10a - 8 to represent how many student tickets there are.
Now, we plug our definition of s into the first expression to get:
3(10a - 8) + 6a = 1272
30a - 24 + 6a = 1272
36a - 24 = 1272
36a = 1296
a = 36
We have determined the number of adult tickets so we can move on to the number of students:
s = 10(36) - 8
s = 360 - 8
s = 352
Question Solved
A die is tossed three times. let X be the sum of the three numbers obtained.
Determine whether the random x has a binomial distribution if so take the number of trials n. If it doesn’t explain why.
Since for each trial there are more than two outcomes, the random variable x is not a binomial variable.
What is required for a binomial variable?
Three things are required:
In this problem, for each trial, there are six possible outcomes for the sum, hence the random variable x is not a binomial variable.
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A meal consists of an appetizer, a main dish, a side dish, and a drink. There are 3 appetizer choices, 6 main dish choices, 5 side dish choices, and 4 drink choices. Describe two ways you could find the number of possible combinations. Choose one of the methods to find the answer, and explain your choice.
1) You can find the number of possible methods through a tree diagram or the fundamental counting principles.
2) The total number of combinations is 360
What is the tree diagram and the fundamental counting principles all about in finding the total number of combination?To find the total number of combinations,
A Tree Diagram is a the various possibilities in a pictorial way. The diagram begins with a single point, which represents the initial decision, then branches out to indicate all possible outcomes for each successive option.
The Fundamental Counting Principle tells us that if there are 'n' ways doing a thing and 'm' ways to do another, there are n x m ways to do both. Which means that the total number of outcomes equals the product of the number of ways each job can completd.
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a wholesaler allows a discount of 25kobo for every 25 naira goods purchased . find the percentage discount allowed.
The percentage discount allowed by the wholesaler is 1%
Percentage1 naira = 100 Kobo25 naira = 25 × 100
= 2500 Kobo
Percentage discount allowed = 25 / 2500 × 100
= 0.01 × 100
= 1%
Therefore, the percentage discount allowed is 1%
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jake has 32 quarters on his desk, and molly has q qaurters in her purse. Which algebraic expression represents the total numbers of quarters that jake and molly have.
Answer:
32+q=x
Step-by-step explanation:
32(jakes) + q (molly) = x(total)
Answer:
32+Q= total number of quarters.
Step-by-step explanation:
Using the driver's speed in feet per second, how far did her car travel during her reaction time?Round your answer to two decimal places
The distance traveled by the driver during her reaction time based on her speed was 44 feet.
How far did the driver get during her reaction time?find the speed in feet per second to be:
= (40 x 5,280 feet per hour) / Number of seconds in an hour
= 211,200 / 3,600
= 58.67 feet per second
With a reaction speed of 3/4 seconds, the distance traveled is:
= 58.67 x 3/4
= 44 feet
Full question is:
The driver of a car traveling at a speed of 40 miles per hour had to stop the car quickly. Using the driver's speed in feet per second, how far did her car travel during her reaction time of 3/4 of a second?Round your answer to two decimal places
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A frozen yogurt store charges C dollars for w ounces of frozen yogurt, and C can be modeled by C = 49w+80/100. According to the model, for every increase of 1 ounce in frozen yogurt, by how many dollars will the cost increase?
A function assigns the values. For every increase of 1 ounce in frozen yogurt the cost will increase by $49.
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
Given the frozen yogurt store charges C dollars for w ounces of frozen yogurt, C can be modelled by C = 49w+80/100. Therefore, for every increase of 1 ounce in frozen yogurt the cost will increase by $49.
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The difference of a number and its reciprocal is 35/6 Find the number.
Answer:
[tex]\huge\boxed{\sf x = 6 \ \ \ OR \ \ \ x = -1/6}[/tex]
Step-by-step explanation:
Let the number be x
Given condition:[tex]\displaystyle x - \frac{1}{x} =\frac{35}{6} \\\\Subtract \frac{35}{6} \ to \ both \ sides\\\\x - \frac{1}{x} -\frac{35}{6} = 0\\\\Multiply \ 6x \ to \ both \ sides\\\\x \times 6x - \frac{1}{x} \times 6x -\frac{35}{6} \times 6x = 0 \times 6x\\\\6x^2-6-35x=0\\\\6x^2-35x - 6 = 0[/tex]
Applying mid term break formula
We can use the factor -36 and +1 to break the mid term (-35x) because multiplying -36 and +1 gives -36 which is the product of side terms (-6 * +6 = -36)[tex]6x^2-36x+x-6=0\\\\Take \ common\\\\6x(x-6)+1(x-6)=0\\\\Take \ (x-6) \ common\\\\(x-6)(6x+1)=0[/tex]
Either:
x - 6 = 0 OR 6x + 1 = 0
x = 6 OR 6x = -1
x = 6 OR x = -1/6
17.3152 to the nearest hundredth=
Answer:
17.32
Step-by-step explanation:
since there is a 5 in the thousandths, we round the 1 in the hundredths up and get 17.32
show a graph that represents y>-6x+5?
The slope m = -6 and y-intercept = ( 0, 5 ).
The graph is a dash line, shaded area above the boundary line since y is greater than -6x + 5.
What is the graph that represents the given equality?Given that;
y > -6x + 5
We use the slope intercept form y = mx + b to determine the slope (m) and the y-intercept (b).
y > -6x + 5
y = mx + b
We can see that;
m = -6 and b = 5
Hence, slope = -6 and y-intercept = ( 0, 5 ).
The graph is a dash line, shaded area above the boundary line since y is greater than -6x + 5.
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A grain silo has a cylindrical shape. Its diameter is 19ft, and its height is 40ft. What is the volume of the silo? Use the value 3.14 for π, and round your answer to the nearest whole number. Be sure to include the correct unit in your answer.
Step-by-step explanation:
formula to get the volume of a cylindrical shape =πr²h
π=3.14 , r=diameter /2=19/2= 9.5, h= 40
: πr²h= 3.14 ×9.5²×40
3.14×90.25×40
=11,335.4
=11,335
Mia has 24 yards of ribbon. She gave 10 yards to her sister. Mia now wants to work on a project that requires 2 1/4 yards of ribbon per project. What is the greatest number of projects Mia can complete with this ribbon.
(Optional, just please write down step by step explanation) PLEASE ANSWER WITH EXPLANATION AS IF YOU WERE WRITING DOEN ON SCRATCH PAPER
Answer:
6 projects
Step-by-step explanation:
Original ribbon length = 24 yards
After 10 yards given to sister, Mia is left with 24-10 = 14 yards of ribbon
Each project requires 2 1/4 yards of ribbon.
2 1/4 is 9/4 yards as an improper fraction (2 * 4 + 1 is the numerator and the denominator is unchanged)
We now have to figure out how many times 9/4 yards goes into 14 yards.
For this,
14 ÷ 9/4.
To do this, flip the numerator and denominator of the fraction and multiply
14 ÷ 9/4 = 14 × 4/9 = 56/9 . We ignore any remainder.
9 goes into 56 6 times with remainder of 2 (ignore remainder)
In other words the greatest number of projects = 6
In the space provided, describe the similarities and differences between the seasonal changes of the oak tree in Florida and the oak tree in Maine make it Conclusions
subje
As a result of climatic peculiarities the seasonal changes of oak tree in Florida and oak in Maine are distinguished such that the differences are remarkably satisfactory.
What are the seasonal changes of the oak tree in Florida and the oak tree in Maine?We know that many oaks are native to Florida and some of the Florida oaks include;
Shumard oakWillow oakSouthern live oak.Northern red oaks can also be found in northern Florida. Some other oak trees in Florida include;
Black oakBluff oakChapman oakChinkapin oakLaurel oak.From the information above, we can conclude that as a result of climatic peculiarities the seasonal changes of oak in Florida and oak in Maine are distinguished such that the differences are remarkably satisfactory.
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MULTIPLICATION! QUESTIONS!
a) 9x2
b)16x2
c)14x9
d)20x10
e)10x3
Answer:
a) 9×2=18
b)16×2=32
c)14×9=126
d)20×10=200
e)10×3=30
The average price of a personal computer (PC) is $949. If the computer prices are approximately normally distributed and standard deviation is $100, what is the probability that a randomly selected PC costs more than 1200? The least expensive 10% of personal computers cost less than what amount?
The probability that a randomly selected PC costs more than 1200 will be 0.006. The least expensive 10% of personal computers cost less than what amount will be 820.8.
What is a normal distribution?The Gaussian distribution is another name for it. The most significant continuous probability distribution is this one. Because the curve resembles a bell, it is also known as a bell curve.
The z-score is given as
z = (x – μ) / σ
Where μ is the mean, σ is the standard deviation, and x is the sample.
The average price of a personal computer (PC) is $949.
If the computer prices are approximately normally distributed and standard deviation is $100.
Then the probability that a randomly selected PC costs more than 1200 will be
z = (1200 – 949) / 100
z = 2.51
Then the probability will be
P(x > 1200) = P(z > 1200)
P(x > 1200) = 1 – P(z < 1200)
P(x > 1200) = 1 – 0.99396
P(x > 1200) = 0.0060366
The least expensive 10% of personal computers cost less than what amount will be
z-value for 10% will be -1.282.
– 1.282 = (x – 949) / 100
– 128.2 = x – 949
x = 820.8
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3. The total weight of the luggage on the flight is 4,338 pounds . What is the weight written in tons and pounds ？
Hello!
The weight is 4,338 pounds and 2.169 tons.
Answer: 2 tons and 338 pounds
Step-by-step explanation:
1 ton is 2000 pounds
4338 has 2 tons and 338 pounds
What is the inverse of the equation y=3x ?
Answer:
y = x/3
Step-by-step explanation:
y = 3x
x = 3y
3y = x
y = x/3
⊱_____________________________________________________⊱
Answer:
[tex]y=\dfrac{x}{3}}[/tex]Step-by-step explanation:
[tex]\large\displaystyle\begin{gathered} \rm{To \ find \ the \ inverse, \ switch \ the \ places \ of \ x \ and \ y} \\ \sf{ y=3x} \\ \sf{x=3y} \\ \rm{Now \ solve \ for y} \\ \sf{x/3=y} \\ \sf{\dfrac{x}{3}=y \end{gathered}[/tex]
done !!
⊰______________________________________________________⊱
CαllíɡɾαρhγIn a graphing program, graph the following 3 functions in the same coordinate plane.
y=1x
y=1x+3
y=1x – 2
How does k (+3 and -2 in functions b & c) change the rational graph? Select the best description from the words below. (Hint: one fits)
compresses-fatter
shifts right/left
reflects over x-axis
shifts up/down
stretches-steeper
reflects over y-axis
The values of k (i.e. +3 and -2) in functions b and c change the rational graph by (b) shifts right/left
How to determine the change in the rational graph?The functions are given as:
[tex]y = \sqrt x[/tex]
[tex]y = \sqrt {x + 3[/tex]
[tex]y = \sqrt {x - 2[/tex]
See attachment for the graphs of the three functions.
From the attached graph, we can see that [tex]y = \sqrt {x + 3[/tex] and [tex]y = \sqrt {x - 2[/tex] are at the left and right of the function [tex]y = \sqrt x[/tex]
Hence, the values of k in functions b and c change the rational graph by shifting it right/left
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Recursive formula of the arithmetic sequence 4,9,14,19
Answer:
4,9,14,19,24,29,34,39
Tommy starts walking from school to home at the same time that his Dad starts walking from home to school. They both depart at 3:00 p.m. Tommy is walking at a speed of 1.35 meters per second, and his Dad is walking at a speed of 1.65 meters per second. The distance between home and school is 720 meters. At what time will they meet?
Answer:
3:04
Step-by-step explanation:
see image
They will meet after 4 minutes from 3:00 p.m. So, they will meet at 3:04 p.m.
Let's find the time it takes for Tommy and his Dad to meet.
Let t be the time (in seconds) from 3:00 p.m. when they meet.
Tommy's distance covered (T) after t seconds is given by:
T = Tommy's speed * t
= 1.35 m/s * t
His Dad's distance covered (D) after t seconds is given by:
D = Dad's speed * t
= 1.65 m/s * t
Since they meet when the sum of their distances is equal to the total distance between home and school, we can write the equation:
T + D = 720 meters
Substitute the expressions for T and D:
1.35t + 1.65t = 720
t = 240 seconds
Since 1 minute is equal to 60 seconds, let's convert the time to minutes:
t = 240 seconds / 60 seconds/minute
t = 4 minutes
They will meet after 4 minutes from 3:00 p.m. So, they will meet at 3:00 p.m. + 4 minutes = 3:04 p.m.
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Which of the following number lines represents the solution set of > -6?
The number line first represents the solution to the inequality x/-1 > -6 option first is correct.
What is inequality?It is defined as the expression in mathematics in which both sides are not equal they have mathematical signs either less than or greater than known as inequality.
The complete question is:
Which of the following number lines represents the solution set of x/-1 > -6?
For the number lines shown in the attached picture, please refer to the picture.
We have an inequality:
x/-1 > -6
x < 6
x ∈ (-∞, 6)
Thus, the number line first represents the solution to the inequality x/-1 > -6 option first is correct.
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HELP
Often times you can use similar triangles in the real world. Sarah is attempting to find the height of a tree by using a mirror. By the laws of physics she knows that the angle of incidence is congruent to the angle of reflection (these are marked in red on the diagram). Explain why the two triangles are similar and if she knows her height and the distance from the mirror to her feet and also the distance from the mirror to the tree how could she calculate the height of the tree?
The two triangles are similar by the AA Similarity theorem.
The height of the tree can be calculated by figuring out the ratio between the distance between the mirror to her feet and the distance from the mirror to the tree
How to use the concept of similar Triangles?From Law of Reflection, we know that the angle of incidence and the angle of reflection are equal to each other.
Now, triangles can be proved similar by the AA, SAS, or SSS theorems. However, in this question, the triangles as seen in the attached image can be proved similar by the AA similarity theorem.
This is because both triangles have one congruent angle in common.
Sarah and the tree are standing straight and perpendicular to the ground and as such, the angles formed by Sarah and the tree are right angles.
The above tells us that the two triangles have two angles in common, making them similar triangles by the AA (Angle Angle) similarity theorem.
Since the triangles are similar, it means that the ratios of the sides of the triangles will be the same. Thus, if Sarah knows the distance from the mirror to her feet and the distance from the mirror to the tree, she can create the ratio between the two triangles.
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