On factorizing 3y⁻⁴ + y⁻⁵ with the given factor y⁻⁵ we get y⁻⁵(3y + 1 ).
Factorization of the polynomial refers to the process of expressing the polynomial as a product of two or more factors.
For Example ; the factors of x²-16 can be written as (x+4)(x-4).
In the given question
(3y⁻⁴ + y⁻⁵)
the common factor y⁻⁵ is given
we have to factorize using the common factor .
taking y⁻⁵ common from both the terms we get
= y⁻⁵(3y⁻⁴/y⁻⁵ + 1 )
=y⁻⁵(3y⁻⁴⁺⁵ + 1 )
=y⁻⁵(3y¹ + 1 )
=y⁻⁵(3y + 1 )
Therefore , On factorizing 3y⁻⁴ + y⁻⁵ with the given factor y⁻⁵ we get y⁻⁵(3y + 1 ).
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The annual attendance at the amusement park is initially 2 million people and is increasing at 3% per year. The park’s annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.2 million people per year.
a. Write the equation that represents the food supply. Write the equation represents the park attendance.
The equation that represents food supply is
b. Based on these assumptions, in approximately what year will the amusement park first experience shortages of food?
c. If the park doubled its initial food supply and maintained the rate of increase of 0.2 million people per year, would shortages still occur? In approximately which year?
d. If the park doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur? After how many years would the food supply run out?
Please show work. I will mark brainliest.
Answer:
[tex]\textsf{a)} \quad f(t) = 4+0.2t \:\: \textsf{ and }\:\:A(t)=2(1.03)^t[/tex]
b) 77 years
c) 86 years
d) 110 years
Step-by-step explanation:
Part (a)Given:
Initial food supply adequacy = 4 million peopleConstant annual growth rate = 0.2 million peopleAs the food supply grows at a constant rate of being adequate for an additional 0.2 million people per year, it can be expressed as a linear function:
[tex]f(t) = 4+0.2t[/tex]
where:
f(t) = annual food supply (in millions of people).t = time (in years).Given:
Initial attendance = 2 million peopleAnnual growth rate = 3%As the annual attendance increases by 3% per year, it can be expressed as an exponential function:
[tex]A(t)=2(1.03)^t[/tex]
where:
A(t) = annual attendance (in millions of people).t = time (in years).Part (b)Graph the two functions (see attached) and locate the value of t for which A(t) > f(t) for the first time.
From inspection of the graph, the two functions intersect at t ≈ 77. So after approximately 77 years the food supply will not be enough for the number of people attending the amusement park. Therefore, after approximately 77 years, the park will first experience shortages of food.
Part (c)If the park doubles its initial food supply and maintains the rate of increase of 0.2 million people per year, the new equation would be:
[tex]f(t) = 8+0.2t[/tex]
Again, graph the new function against A(t) and find the point where the two functions intersect. A(t) = f(t) at approximately t = 86 so the park will first experience food shortages after 86 years. So doubling the initial food supply delays the eventual food shortage by only c. 9 years.
Part (d)If the park doubled the rate at which its food supply increases in addition to doubling its initial food supply, the new equation would be:
[tex]f(t) = 8+0.4t[/tex]
Again, graph the new function against A(t) and find the point where the two functions intersect. A(t) = f(t) at approximately t = 110 so the park will first experience food shortages after 110 years. So doubling the initial food supply and doubling the rate delays the eventual food shortage by only c. 33 years compared to the initial parameters. Shortages would still occur, but it would be later in approximately 110 years' time.
If f(x) = 3x2 – 2x + 4 and g(x) = 5x2 + 6x – 8, find (f +g)(x).
Answer:
[tex](f+g)(x)=8x^{2} +4x-4[/tex]
Step-by-step explanation:
[tex](f+g)(x)=3x^{2} +5x^{2} -2x+6x+4-8=8x^{2} +4x-4[/tex]
Hope this helps
Let f (x) = 3x2 − 2x + 4. Evaluate f (a + h) and simplify your answer as much as possible.
The expression f(x) = 3 · x² - 2 · x + 4 evaluated at x = a + h is equal to the expression f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4.
How to evaluate and simplify a quadratic equation
In this question we find a quadratic equation, which has to be evaluated at x = a + h and simplified afterwards. Then, the complete procedure is shown below:
f(x) = 3 · (a + h)² - 2 · (a + h) + 4
f(x) = 3 · (a² + 2 · a · h + h²) - 2 · a - 2 · h + 4
f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4
The expression f(x) = 3 · x² - 2 · x + 4 evaluated at x = a + h is equal to the expression f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4.
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Solve.
⎧⎩⎨⎪⎪2x−y+2z=−6−3y+z=−22x−3z=4
Enter your answer, in the form (x,y,z), in the boxes in simplest terms.
The solution is (x, y, z) = (-1, 0, -2)
Given equations are:
2x−y+2z=−6 ---------------(1)
−3y+z=−2 ------------------(2)
2x−3z=4 -------------------(3)
From (2), z = -2+3y
Substitute this into (1) and (3)
(1)⇒ 2x-y+2(-2+3y) = -6
⇒ 2x+5y-4=-6
⇒2x+5y = -2
and
(3) ⇒ 2x-3(-2+3y)=4
⇒ 2x-9y+6 = 4
⇒ 2x-9y=-2
Now solve 2x+5y = -2 and 2x-9y=-2
Subtracting the equations above,
14y = 0
⇒y=0
Then from 2x-9y=-2, x = -1
From (2),z = -2+3y
⇒ z = -2
So the solution is (x, y, z) = (-1, 0, -2)
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thank you so much for the help!!
The independent variable in these equations based on how they are written is x, since y is a function of x here so y depends on the value of x, while x can be changed freely.
So, for each of these questions, plug in the given value of the independent variable for x.
Independent variable equal to 10: 0.1*10 + 5 = 6
Independent variable equal to 50: 0.1*50 + 5 = 10
Independent variable equal to 120: 0.1*120 + 5 = 17
The temperature at
7:00 P.M is 18°F. At
7:00 A.M. the next
day, the temperature
is -12°F. What integer
represents the change
in temperature?
Answer: -30 degrees fahrenheit
Step-by-step explanation: 18 + (-12) = -30
write an expression to show the total number of songs on Iris's playlist after w weeks
120 + 4w is the expression to show the total number of songs on Iris's playlist after w weeks.
She total ahs 120 songs already
She downloads or adds 4 songs every week
An expression to show the total number of songs on Iris's playlist after w weeks is:
120 + 4w
What is an mathematical expression?An expression or mathematical expression is a limited combination of symbols that is well-formed according to context-dependent norms. To help identify the sequence of operations and other features of logical syntax, mathematical symbols can denote numbers (constants), variables, operations, functions, brackets, punctuation, and grouping.
Many authors differentiate between an expression and a formula, with the former referring to a mathematical item and the latter referring to a statement about mathematical objects.
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22 out of 25
math students
bring their
calculators to
class how many
students would
have a
calculafors in a
group of 80
math students
HELP PLEASE
A point A (x, y) is translated 7 units to the left and 2 units up. What is the mapping to the new point A’?
(x+2, y+7)
(x-7, y+2)
(x+2, y-7)
(x+7, y+2)
Reason:
7 units to the left means x becomes x-7
2 units up means y becomes y+2
Therefore, the old preimage (x,y) translates or shifts to the new image (x-7, y+2)
Example:
[tex](\text{x},\text{y}) = (3,5)\\\\(\text{x},\text{y})\to(\text{x}-7,\text{y}+2)\\\\(3,5)\to(3-7,5+2)\\\\(3,5)\to(-4,7)\\\\[/tex]
The preimage (3,5) moves to (-4,7) after shifting 7 units left and 2 units up.
Name a point non-coplanar to plane K
W is the point which is non coplanar to plane k .
WHAT IS A COPLANAR/NON-COPLANAR POINT ?A group of points that are in the same plane are said to be coplanar. A coplanar line connects any two or three points. It's possible that four or more points are coplanar.
The coplanar points A, B, C, and D are displayed on the left side of the above figure. Additionally, there are other sets of coplanar points in the box to the right. For instance, the left side of the box's left side is the plane that contains the coplanar points P, Q, X, and W. There are four coplanar points on each of the box's six faces, however they are not the only coplanar point groups. Points Q, X, S, and Z, for instance, are coplanar despite the fact that the plane containing them isn't depicted.
Non-coplanar points are a collection of points that are not all located on the same plane.
Points P, Q, X, and Y in the previous figure are not coplanar. Q, X, and Y are located on the box's top, and P, Q, and X are located on the box's left side, however no flat surface has all four points.
CALCULATIONsince from the figure we can find out that except two points W , Y all the points are on the plane K .
so according to question and given figure we conclude that one of the point which in non coplanar to K is W.
points highlighted in the figure is non coplanar to point k .
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Find the average rate of change of the function f(x) = x² + 4x from x₁ = 3 to x₂ = 5.
The average rate of change of the given function is 12.
The average rate of change of function f on interval [a,b] is:
f(b) -f(a)/b-a
we have interval [x₁,x₂]
[x₁,x₂] = [3,5]
f(3) = x²₊4x
f(3) = 3²+4(3)
f(3) = 21
f(5) = x²₊4x
f(5) = 5²+4(5)
f(5) = 45
f(x₂) -f(x₁)/x₂-x₁
⇒ 45-21/5-3
= 24/2
= 12
The average rate of change of function f(x) = x²+4x on interval [3,5]
is 12.
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4x + 3 = 18 - x help please
Answer:
the answer is x=3
Step-by-step explanation:
Answer:
Please mark my answer as the brainliest, it would mean a lot to me
x = 5
Step-by-step explanation:
4x + 3 = 18 - x
3x + 3 = 18
3x = 15
x = 5
XZ is the perpendicular bisector of segment WY. Solve for m. Enter a NUMBER only.
2(14m-8)=180
24m-16=180
24m=196
m= 8 and 1/6
The graph of the function y = x^3 + ax^2 + b has a local minimum point at (4, -11). Find the coordinates of the local maximum. Answer: (0, 21)
Answer:
(0, 21)
Step-by-step explanation:
Stationary points occur when the gradient of a graph is zero.
Therefore, to find the x-coordinate(s) of the stationary points of a function, differentiate the function, set it to zero and solve for x.
Differentiate the given function:
[tex]\begin{aligned}y & = x^3+ax^2+b\\\implies \dfrac{\text{d}y}{\text{d}x}&=3 \cdot x^{3-1}+2 \cdot ax^{2-1}+0\\\implies \dfrac{\text{d}y}{\text{d}x}& = 3x^2+2ax\end{aligned}[/tex]
Set the differentiated function to zero:
[tex]\begin{aligned} \dfrac{\text{d}y}{\text{d}x}& =0\\ \implies 3x^2+2ax & = 0 \\ \implies x(3x+2a)&=0\end{aligned}[/tex]
Therefore the stationary points occur when:
[tex]\implies x=0[/tex]
[tex]\implies 3x+2a=0 \implies x=-\dfrac{2}{3}a[/tex]
There is a stationary point at (4, -11), therefore substitute x = 4 into the expression for x and solve for a:
[tex]\begin{aligned}x & = 4\\\implies -\dfrac{2}{3}a & = 4\\a & = -\dfrac{3}{2}(4)\\a & = -6\end{aligned}[/tex]
Substitute the found value of a and the point (4, -11) into the function and solve for b:
[tex]\begin{aligned}y & = x^3 + ax^2 + b\\\implies -11 & = (4)^3+(-6)(4)^2+b\\ -11 & = 64-96+b\\b & = -11-64+96\\ b & = 21\end{aligned}[/tex]
Therefore, the function is:
[tex]\boxed{y=x^3-6x^2+21}[/tex]
The other stationary point is when x = 0. Therefore, to find the coordinates of this point, substitute x = 0 into the function:
[tex]\begin{aligned}y & =x^3-6x^2+21\\x=0\implies y & =(0)^3-6(0)^2+21\\ y & =21 \end{aligned}[/tex]
Therefore, the coordinates of the other stationary point are (0, 21).
To determine if a stationary point is minimum or maximum, differentiate the function again:
[tex]\begin{aligned}y & =x^3-6x^2+21\\\implies \dfrac{\text{d}y}{\text{d}x} & = 3x^2-12x\\\implies \dfrac{\text{d}^2y}{\text{d}x^2} & = 6x-12\\\end{aligned}[/tex]
Substitute the x-coordinate of the stationary point into the second derivative:
[tex]\begin{aligned} \dfrac{\text{d}^2y}{\text{d}x^2} & = 6x-12\\x=0 \implies \dfrac{\text{d}^2y}{\text{d}x^2} & = 6(0)-12= -12 \\\end{aligned}[/tex]
[tex]\textsf{As $\dfrac{\text{d}^2y}{\text{d}x^2} < 0$, stationary point $(0,21)$ is a maximum.}[/tex]
Differentiation Rules
[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Differentiating $ax^n$}\\\\If $y=ax^n$, then $\dfrac{\text{d}y}{\text{d}x}=nax^{n-1}$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{4cm}\underline{Differentiating a constant}\\\\If $y=a$, then $\dfrac{\text{d}y}{\text{d}x}=0$\\\end{minipage}}[/tex]
the estimate closest to the length of a pen: 30cm, 160mm, or 16 mm.
Answer:
160 mm.
Step-by-step explanation:
the average pen is 149mm. the average pen is 14.9cm. 160mm is closest.
It would have to be the 160mm.
This is because store-bought pens average 130 and 140mm/ 13 – 14 cm. This is closely followed by pens in the 140 – 150mm range.
Hope this helps!
Have a great day and God bless! :)
A student does an experiment in a lab on the boiling point of water. In the 3 trials, the student gets the temperatures of 99.2, 99.5, 100.2 degrees C. Are these measurements:
Question 2 options:
Accuracy
Neither precision nor accuracy
Both precision and accuracy
Precision
The measurements are precise. Therefore, the correct option is D.
What is precision?It should be noted that precision simply means how close a test is when it's repeated.
On the other hand, it should be noted that accuracy simply means how close the results are to the standard value.
Therefore, when the student does an experiment in a lab on the boiling point of water. In the 3 trials, the student gets the temperatures of 99.2, 99.5, 100.2 degrees, they're precise.
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a bricklayers assistant starts building a wall laying 50 bricks per hour. Two hours later the master bricklayer joins the assistant and both lay bricks. The master bricklayer lays 80 bricks per hour. write an equation stating that the assistant and the master have laid the same number of bricks. how long has each one worked when they have laid the same number?
Answer:
80t = 50(t+2)master: 3 1/3 hoursassistant: 5 1/3 hoursStep-by-step explanation:
Given an assistant bricklayer laying 50 bricks an hour has worked 2 hours longer than a master bricklayer laying 80 bricks an hour, you want an equation stating they have laid the same number of bricks. You also want to know how long each has worked.
Rate and outputThe output of each bricklayer is the product of the rate at which they lay bricks and the time for which they do it. If t is the time spent by the master bricklayer, their output will be 80t. The time spent by the assistant will be (t+2) hours, and their output will be 50(t+2).
Here is the equation that says their output is the same:
80t = 50(t+2)
TimeSolving the equation for t, we have ...
80t = 50t +100
30t = 100 . . . . . . . . . subtract 50t
t = 3 1/3 . . . . . . . divide by 30
The master bricklayer has worked 3 1/3 hours when they have laid the same number of bricks.
The assistant bricklayer has worked 5 1/3 hours at that time.
__
Additional comment
Each will have laid (80)(3 1/3) = 266 2/3 bricks.
Simplify the expression
Answer:
-8
Step-by-step explanation:
2x/7 =
2(12)/-3 =
24/-3 =
-8
Answer:
-8
Explanation:
Given expression:
[tex]\rightarrow \sf \dfrac{2x}{y}[/tex]
when x = 12, y = -3
Substitute values inside expression:
[tex]\rightarrow \sf \dfrac{2(12)}{-3}[/tex]
multiply
[tex]\rightarrow \sf \dfrac{24}{-3}[/tex]
simplify
[tex]\rightarrow \sf -8[/tex]
A total of 200 and a difference of 72
According to the solving the total and the difference of the numbers:
total = 136 + 64
difference = 64
Difference in Math Definition?Mathematical difference is the outcome of one of the key operations, which is the subtraction of two numbers. It reveals the margin of difference between two numbers.
How do we find the difference?Subtract the largest number from the lowest number to determine the difference between two integers. The difference between these two numbers is the product of this sum. As a result, there are 55 between the numbers 45 and 100.
According to the given data:Let the numbers be x and y
Given,
x+y = 200_____(1)
x-y = 72_____(2)
Add above 2 eq.
2x= 272
x= 136
Substitute the value of x in eq 1,we get
136+y = 200
y = 64
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Find the nth term, the fifth term, and the 100th term, of the arithmetic sequence determined by a = 2 and d = 3
Answer: an=3n-1 a₅=14 a₁₀₀=299
Step-by-step explanation:
a=2 d=3
[tex]\boxed {a_n=a_1+d(n-1)}\\\\a_n=2+3(n-1)\\\\a_n=2+3n-3\\\\a_n=3n-1\\\\a_{100}=3(100)-1\\\\a_{100}=300-1\\\\a_{100}=299\\\\a_5=3(5)-1\\\\a_5=15-1\\\\a_5=14[/tex]
Please help I will give anyone brainliest
Answer:
1672.50 = 37.75x + 1068.50
x = 16
There were 16 players brought to the tournament.
Step-by-step explanation:
y = mx + b
y= the total cost
m = cost of each meal
x = the number of players
b = cost of the bus
Plug in what you know and solve for x
1672.50 = 37.75x + 1068.50 Subtract 1068.50 from both sides.
604 = 37.75x Divide both sides by 37.75
16 = x
Equation: 1068.50 + 37.75x =1,672.50
Answer: X= 16
If 1,672.50 is our total it will go behind the equal sign. 37.75x and 1068.50 must be added together to help us receive our total.
Our written equation should look like this
1068.50 + 37.75x =1,672.50
X is the number of players on the team.To find out we must first subtract the amount of money spent on the bus, from the total amount spent.
1,672.50-1068.50 is 604. This means our equation now looks like this
37.75x=604. The next and final step is to divide 37.75 into 604.
604/ 37.75 is 16. Which means that there are 16 players on the team.
I hope this helps & Good luck.
Decide whether the table shows a proportional relationship between x and y.
x y
17 2 5/6
5/6 5/36
6 1/6
19 3 1/6
The table shows a proportional relationship between x and y.
In this question, we have been given a table which contains values of x and y.
We need to decide whether the table shows a proportional relationship between x and y.
x y
17 2 5/6
5/6 5/36
6 1/6
19 3 1/6
First we convert the improper fraction in the y-column to proper fraction.
2 5/6 = 17/6
and 3 1/6 = 19/6
so, given table would be,
x y
17 17/6
5/6 5/36
6 1/6
19 19/6
Consider the ratio x/y of each row:
x y x/y
17 17/6 1/6
5/6 5/36 1/6
6 1/6 1/6
19 19/6 1/6
As the ratio x/y is constant, x and y are directly proportional.
Therefore, the table shows a proportional relationship between x and y.
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nate works at a grocery store. he has 7 1/2 pounds of rasins. heis splitting them into 1/8 pound packages. how many packages can he make
Answer:
Nate can make 60 packages.
Step-by-step explanation:
7 1/2 is equivalent to 7 4/8, or 60/8
Divide 60/8 by 1/8 and you get 60.
Help :)
Evaluate the following expression
|−3d−6|+|−5−d2| for d=−3
The value of the expression |−3d − 6| + |−5 − d²| when d = -3 is 17.
Given expression: |-3d - 6| + |-5 - d²|, and d = -3
Substitute the value of d: |-3(-3) - 6| + |-5 - (-3)²|
Calculate the values inside the absolute value signs:
|-(-9) - 6| + |-5 - 9|
Simplify the inside calculations:
|9 - 6| + |-5 - 9|
Continue simplifying:
|3| + |-14|
Calculate the absolute values:
3 + 14
Add the values together:
3 + 14 = 17
So, the value of the expression |−3d − 6| + |−5 − d²| when d = -3 is 17.
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Write
4/11
as a decimal.
O 0.36
0.36
0.036
0.36
Answer:
0.36
Step-by-step explanation:
Classify the system and identify the number of solutions.
3x + 5y + 4z = −11
2x − 4y − z = −11
4x + 3y − 2z = 11
Answer:
Hello,
Step-by-step explanation:
Only one solution: x=-2, y=3,z=-5
Answer:
consistent, independent; one
Step-by-step explanation:
The solution is (−2, 3, −5). Because the system has one solution, it is consistent and independent.
HELP PLEASE DUE TODAY
12) Z ∪ N = {Set of Integers}
13) H ∪ Q = R which is a set of real numbers
How to classify numbers?
We are told that;
R = U represents the set of real numbers
N is the set of natural numbers
W is the set of whole numbers
Z is the set of integers
Q is the set of rational numbers
H is the set of irrational numbers
12) We want to find Z ∪ N. This is a set union of the set of integers and the set of natural numbers.
A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero. Thus, an integer is also a rational number and all natural numbers are positive integers and as such;
Z ∪ N = {Set of Integers}
13) We want to find H ∪ Q. This is a set union of the irrational numbers and the set of rational numbers. Now, we know that all real numbers are classified as either rational or irrational numbers
Thus; H ∪ Q = R which is a set of real numbers
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19 more than four times a number is equal to the difference between -86 and three times the number find the number
Answer:
the number is -15
Step-by-step explanation:
if you turn the word problem into an equation it would be
4x+19=-86-3x
with x being the unknown number
the next step is to move the -3x to the other side by adding it to the 4x
your equation should now be 7x+19=-86
next you should move the 19 to the other side by subtracting it from the -86
this leaves you with the equation
7x= -105
lastly you must divide the -105 by seven which leaves you with the answer x= -15
find inequality : JoJo can eat no more than 25 carbs per meal. Which inequality describes JoJo's situation?
Answer:
Jojo is on a low carb diet
A t-shirt requires 1 3/8 yards of material. How many t-shirts can be made from 41 1/4 yards of material?
Applying the knowledge of fractions, the number of t-shirts that can be made is: 30 t-shirts.
How to Divide Fractions?In the problem given, we have the following information:
Material needed for one t-shirt = 1 3/8 yards
We want to find how many 1 3/4 we would find in 41 1/4.
Applying the knowledge of fraction, we are to divide 41 1/4 by 1 3/8. Convert both mixed fractions to improper fractions then divide:
41 1/4 ÷ 1 3/8
165/4 ÷ 11/8
165/4 × 8/11
165/4 × 8/11
15/1 × 2/1
= (15 × 2)/(1 × 1)
= 30/1
= 30.
Therefore, applying the knowledge of fractions, the number of t-shirts that can be made is: 30 t-shirts.
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