The function (g∘f)(x) expressed as a polynomial in standard form is 9x² - 36x + 40
What is a function?A function can be defined as an equation, a rule or a law that shows or rather explain the relationship between two variables in an equation.
The variables are known as;
The independent variableThe dependent variableWe have the function given;
fx) = - 3x + 5g(x) = x² + 2x + 15To determine the composite function (g ∘f)(x), we have to substitute the value of x as fx) in the function g(x), we have;
(g∘f)(x) = (- 3x + 5)² + 2( -3x + 5) + 5
expand the bracket
(g∘f)(x) = 9x² - 15x - 15x + 25 - 6x + 10 + 5
Now, collect like terms, we get;
(g∘f)(x) = 9x² - 15x - 15x - 6x + 25 + 10 + 5
Add or subtract the like terms, we get;
(g∘f)(x) = 9x² - 36x + 40
The polynomial is written as; 9x² - 36x + 40
Hence, the value is 9x² - 36x + 40
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A thermostat in a house regulates when the heater turns on and off. The heater turns on then
the house is 4° below the set temperature and turns off when the house is 4° above the set
temperature. The temperature is set at 68°. At 10:15am the heater turns on, and at 10:30am the
heater turns off. Write a sinusoid that models the change in temperature in the house.
The sinusoid that models 4° = A cos B(x - C) and 68° = A cos B(x - C) the change in temperature in the house.
The heater turns on when the house is 4° below the set temperature and turns off when the house is 4° above the set temperature. The temperature is set at 68°. At 10:15 am the heater turns on, and at 10:30 am the heater turns off.
Temperature fluctuations can be modeled using the sine and cosine functions throughout the day.
The equations that can be used to model these data is of the form:
At 10:30 am the heater turned off,
⇒ 4° = A cos B(x - C)
At 10:15 am the heater turned on,
⇒68° = A cos B(x - C)
where A, B, and C are constants, y is the temperature in °C and x is the time
Thus, The sinusoid that models 4° = A cos B(x - C) and 68° = A cos B(x - C) is the change in temperature in the house.
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the sum of 12 terms of an arithmetic series is 186 and the 20th term is 83. find the sum of 40 terms of the series.
The Sum of 40 terms of the Arithmetic Sequence is 3,420..
Arithmetic Series is defined as a sum of the terms of arithmetic progression or sequence.
An Arithmetic progression is a sequence of terms in which the difference between two consecutive terms is a constant number.
We have following information about series
sum of 12 terms of an arithmetic series (S₁₂) = 186
the 20th term of arithmetic sequence = 83
i.e., a₂₀ = 83
As we know that nth term of arithmetic progression is aₙ= a + (n-1)d
where , a---> first term of sequence
n ----> nth term
d -----> constant difference between two consecutive terms
put n= 20 in above formula we get,
83 = a + (20-1 )d = a+ 19d ----(1)
Now , Using the sum of n arithmetic terms formula, Sₙ = n/2 ( 2a + (n-1)d)
so, for n = 12
186 = 12/2 ( 2a + 11d ) => 2a + 11d = 31 ---(2)
Solving equations (1) and (2) we get,
27d = 135 => d = 5
putting the value of d in equation (1)
83 = a + 19×5 => a = 83 - 95 = -12
thus sequence is like as -12, -7 , -2 , ------
Sum of first 40 terms of arithmetic sequence (S₄₀) = 40/2 ( 2× (-12) + 39d ) = 20 ( -24 + 39×5)
= 20( 171) = 3,420
So, required result is 3,420..
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What is an equation of the line that passes through the points (-2, -7) and
(-1, -3)?
m= (6- 3)/ (-5 -1)
2x - y + 11= 0
force
area
A block exerts a force of 9 N on the ground.
Calculate the pressure the block exerts on the ground, in N/m², when
it is
a) flat on the ground, so that the area of its base is 0.04 m².
b) standing up on its side, so that the area of its base is 0.015 m².
pressure =
A block exerts a force of 9 N on the ground.
Calculate the pressure the block exerts on the ground, in N/m², when it is
a) 225N/m^2
b) 600N/m^2
Define pressure.
Force applied per unit area is the definition of pressure. P=FA, where F is the force acting perpendicular to the surface area A, is the formula used to calculate pressure mathematically. Having only magnitude and no directional vector properties, pressure is a scalar quantity. The pascal (Pa), or one newton per square meter (N/m 2), is the industry standard for measuring pressure.
Solution Explained:
Given, Force = 9N
For each part of the question we will use the formula
Pressure = Force / Area (Since, Force = Pressure X Area)
a) Here, Area = 0.04m^2
So using the formula, we have
P = 9/0.04 = 225N/m^2
b) Here, Area = 0.015m^2
So using the formula, we have
P = 9/0.015 = 600N/m^2
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A seed company planted a floral mosaic of a national flag. The perimeter of the flag is 2,280 ft Determine the flag's width and length if the length is 400 ft greater than the
width
The length of rectangle is 1170 ft and width is 770 ft.
Define perimeter.The perimeter of a form is the space surrounding its edge. A closed path that covers, encircles, or outlines a one-dimensional length or a two-dimensional shape is called a perimeter. A circle's or an ellipse's circumference is referred to as its perimeter. We must sum up the lengths of the rectangle's four sides in order to get its perimeter. Since there are two of each side length, it is easy to accomplish this by simply adding the length and width and multiplying the result by two. The perimeter formula is defined as perimeter=(length + width)2.
Given,
The perimeter of the flag is 2,280 ft .
Length = 400 ft
Let l represent the size of the flag.
and
The width is w.
then in line with the preceding statement
l = w+400
Sphere size is 2280 feet.
The perimeter formula is provided by:
Perimeter = 2(Length + Width)
2280 = 2( w + w+ 400)
2280/2 = 2w - 400
1140 + 400 = 2w
1540 = 2w
width = 770
Length = 770 + 400
Length = 1170
The length of rectangle is 1170 ft and width is 770 ft.
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. suppose the sides of a rectangle are changing with respect to time. the first side is changing at a rate of 2 in./sec whereas the second side is changing at the rate of 4 in/sec. how fast is the diagonal of the rectangle changing when the first side measures 16 in. and the second side measures 20 in.? (round answer to three decimal places.)
Diagonal of rectangle is changing by the rate of 4.373 in. /sec.
What is rate of change ?Rate of change (ROC) is the rate at which a variable changes over a particular period of time
To find the average rate of change, divide the change in the y-values by the change in the x-values . Finding the average rate of change is especially useful for identifying changes in measurable values such as average speed or average velocity.
A ratio is a special ratio in which the two terms have different units.
Calculationthe calculation part is shown in the below picture attached .
so after seeing the solution we come to an answer of = 4.373 in./sec
therefore diagonal of rectangle is changing by the rate of 4.373 in. /sec
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Trevor and Michael are hosting a banquet. They plan to spend $400, plus or minus $50, at a cost of $25 per guest. Solve |25x-400| = 50 to find the maximum and minimum number of guests. If there can be up to 7 guests at each table, what is the minimum number of tables Trevor and Michael should reserve so that every guest has a seat?
1. The maximum and the minimum number of guests that Trevor and Michael should host at the banquet are 18 and 14, respectively.
2. The minimum number of tables that Trevor and Michael will need for the banquet is 2.
How are the numbers determined?We can employ the mathematical operations of addition, multiplication, division, and subtraction to determine the numbers.
Since the budget is limited to $450 or less, we can compute the maximum and the minimum number of guests for the event, using division operation.
Total budget for the banquet = $400 ± $50
Cost per guest = $25
Maximum number of guests to expect = 18 ($450/$25)
Minimum number of guests to expect = 14 ($350/$25)
The number of guests a table can accommodate = 7
The minimum number of tables to use for the banquet = 2 (14/7)
The maximum number of tables to use = 3 (18/7)
Thus, using division operation, Trevor and Michael should expect to host between 14 and 18 guests at the banquet.
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What is an
equation of the line that passes through the points (6, 0) and (3, 4)?
The equation that passes (6,0) and (3,4) is -3y=4(x-6)
What are the different forms of straight lines ?
The following information about the straight line is required if we wish to obtain the equation for it.
1. The y-intercept and slope
2. Two-point and slope
3. Two things.
4. the intersection of the x- and y-axes
the straight line that passes through (6,0) and (3,4).
The equation can be expressed in a point-slope form which is ,
y/x-6=4/-3
∵ -3y=4(x-6)
Hence,
The equation that passes (6,0) and (3,4) is -3y=4(x-6)
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PLEASE I REALLY NEED HELP I DON'T UNDERSTAND PLEASE THANK YOU GUYS FOR THE HELP!! I APPRECIATE IT!!!
The value of KM in triangle ΔKLM is 110.
What is similarity?Similarity of a triangle is the ratio of their corresponding sides or angles which are equal .
In the given triangle ΔHIJ,
HI=13, HJ=25,
And in triangle ΔKLM,
KL=57, KM=?
triangle ΔHIJ is similar to ΔKLM,
By the property of similarity, HI/HJ=KL/KM
Substitute the values here,
13/25=57/KM
KM=25×57/13
KM=109.6
The value of KM=110.
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(x-1)^2/x^2-1.
what is the x intercept
The x-intercept is 1 if the function is f(x) = (x-1)²/x²-1 after plugging f(x) = 0 or y = 0 the answer is 1.
What is a function?
It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
It is given that:
The function is:
f(x) = (x-1)²/x²-1
To find the x-intercept where y = 0
or f(x) = 0
0 = (x-1)²/x²-1
x ≠ ±1
x - 1 = 0
x = 1
Thus, the x-intercept is 1 if the function is f(x) = (x-1)²/x²-1 after plugging f(x) = 0 or y = 0 the answer is 1.
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Find a formula for the nth term in this
arithmetic sequence:
a1 = 7, a2 = 4, a3 = 1, a4 = -2, ...
The formula of nth term is = 10 - 3n
What is AP?
A series of numbers called an arithmetic progression or arithmetic sequence (AP) has a constant difference between the terms. Take the numbers 5, 7, 9, 11, 13, and 15 as an example. . . is a sequence of numbers having a common difference of two.The n-th term of the sequence is given by:, if the beginning term of an arithmetic progression is and the common difference between succeeding members is, thenIf the AP contains m phrases, then denotes the final term, which is given by:The term "finite arithmetic progression" or "arithmetic progression" refers to a finite segment of an arithmetic progression. An arithmetic series is the total of a finite arithmetic progression.Acc to our question-
For the nth term in an algebraic seriesU(n) = a + (n - 1)dthe number of terms is n.The first term is a.d is the typical differenceFrom the preceding sequencea = 7d = 4 - 7 = - 3The nth term's formula isU(n) = 7 + (n - 1)-3= 7 - 3n + 3The ultimate solution is= 10 - 3nHence,The formula of nth term is = 10 - 3n
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What is the value of y?
y =
Find the following angle measures.
mAngleJKM = °
mAngleMKL = °
Applying the definition of linear pair angles, the measures are:
y = 10
m∠JKM = 106°
m∠MKL = 74°
What is the Linear Pair?A linear pair involves two angles that lie adjacent to each other on a straight line. The sum of their measures is equal to 180 degrees, this is because linear pair angles are supplementary to each other.
Given the following:
Measure of angle JKM = (10y + 6)°
Measure of angle MKL = (8y - 6)°
Therefore:
m∠JKM + m∠MKL = 180 [linear pair angles are supplementary]
Substitute
10y + 6 + 8y - 6 = 180
Combine like terms
18y = 180
18y/18 = 180/18
y = 10
Plug in the value of y to find the measures:
Measure of angle JKM = (10y + 6)° = 10(10) + 6 = 106°
Measure of angle MKL = (8y - 6)° = 8(10) - 6 = 74°
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Evaluate f(x) = log₂ (x) for f(8).
The value of f(8) is 3.
What is logarithm?
The opposite of exponentiation in mathematics is the logarithm. The exponent to which b must be raised in order to obtain a number x is hence the logarithm of a number x to the base b. For instance, since 1000 = 103, log10 of 1000 is 3, or [tex]log_1_0=3[/tex].
Let, [tex]f(x) = log_2(x)[/tex]
We have to find the value of f(8).
Plug x = 8 in the above equation.
[tex]f(8) =log_2(8)[/tex]
Here 8 = 2 x 2 x 2 = 2^3
So, replace 8 with 2^3 in the above equation.
[tex]f(8) = log_2(2^3)[/tex]
Using the log rule: [tex]log_a(a^b) = b[/tex]
Here a = 2, b = 3
So, [tex]log_2(2^3) = 3[/tex]
Hence, f(8) = 3.
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x^2-36=0
solve by graphing
The solutions for the given equation is (6, -6). The graph for the given equation is plotted below.
The given equation is x²-36=0.
What is the graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points. A graph consists of some points and lines between them. The length of the lines and position of the points do not matter.
We need to graph the given equation
x²-36=0
⇒ x²=36
⇒ x=±√36
⇒ x=±6
So, solution is (6, -6)
The graph for equation x²-36=0 is given below
The solutions for the given equation is (6, -6). The graph for the given equation is plotted below.
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Explain how it is possible to determine whether the graphs of 5x−3y=2 and 5x-3y=8 are parallel without doing any calculations
The slope of line 1 and line 2 are equal. Then the lines are parallel to each other.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The slope of the parallel lines will be equal.
The equations are given below.
5x − 3y = 2 ...1
5x − 3y = 8 ...2
The slope of line 1 is given as,
5x − 3y = 2
y = (5/3)x − 2/3
The slope of line 1 is 5/3.
The slope of line 2 is given as,
5x − 3y = 8
y = (5/3)x − 8/3
The slope of line 2 is 5/3.
The slope of line 1 and line 2 are equal. Then the lines are parallel to each other.
The graph is given below.
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The line graphed below has a slope equal to -2. The point (3, 28) lies on
this line as shown. Determine the y-value on this line when the x-value is
equal to 11.
Answer:
y = 12
Step-by-step explanation:
We'll use the form of an equation of a straight line of y=mx+b, where m is the slope and b the y-intercept (the value of y when x=0).
We are told the slope, m, is -2.
That leads to: y = -2x + b
y = -2x + b
B, the y-intercept, can be found by using the given point (3,28) in the equation (it falls on the line, so must be a valid solution to the equation. Use the given x and y in the equation and solve for b:
y = -2x + b
28 = -2*(3) + b
28 = -6 + b
b = 34
The equation is y = -2x + 34
Use this to find y when x = 11:
y = -2(11) + 34
y = -22 + 34
y = 12
y is 12 when x is 11
See the attached graph.
Factor the expression
27 + 33 how do work it out
The factor of the expression 27 + 33 is 3.
What is meaning of factor?
A factor is a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, then the numbers we are adding are factors of the product because the product is divisible by them.
Factors can be found using either division or multiplication. Rules of divisibility can also be applied.
What meaning of expression?
A mathematical expression is a phrase that has a minimum of two numbers or variables and at least one mathematical operation.
Given expression is 27 + 33
The factors of 27 are 1, 3, 9, 27.
The factors of 33 are 1, 3, 11, 33.
The common factor of 27 and 33 is 3.
Taking common 3 from the given expression:
27 + 33 = 3 (9 + 11)
The factor of the given expression is 3.
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The area, Acm², of a patch of mould is measured daily. The area, n days after the measurements started, is given by the formula A = Asub0b^n". When n = 2, A = 1.8 and when n = 3, A = 2.4. Find the value of b.
Answer:
b = (4/3), with rounding
Step-by-step explanation:
Area = A, in cm^2
n = days
A(n) = [tex]A_{0}[/tex][tex]b^{n}[/tex]
------------------
Data:
n = 2, A = 1.8
A(n) = [tex]A_{0}[/tex][tex]b^{n}[/tex]
1.8 = [tex]A_{0}[/tex][tex]b^{2}[/tex]
n = 3, A = 2.4
2.4 = [tex]A_{0}[/tex][tex]b^{3}[/tex]
Although we are not given the initial area, we can still make the calculation since we have two data points and each has the same initial area, [tex]A_{0}[/tex]. Lets take the first equation and multiply it by b, in order to bring the [tex]b^{2}[/tex] to [tex]b^{3}[/tex]
1.8 = [tex]A_{0}[/tex][tex]b^{2}[/tex]
b(1.8 = [tex]A_{0}[/tex][tex]b^{2}[/tex])
1.8b = [tex]A_{0}[/tex][tex]b^{3}[/tex]
We can know from the second equation that 2.4 = [tex]A_{0}[/tex][tex]b^{3}[/tex] . We can eliminate this term in the above relationship by substituting 2.4:
1.8b = [tex]A_{0}[/tex][tex]b^{3}[/tex]
1.8b = 2.4
b = 1.3333
or (4/3)
Check:
Does the value of (4/3) for b work for both data points? To check this, we do need to assume an initial starting area. Lets assume the starting area, [tex]A_{0}[/tex] = 1.
n = 2, A = 1.8, b = (4/3)
A(n) = [tex]A_{0}[/tex][tex]b^{n}[/tex]
1.8 = 1*[tex](4/3)^{n}[/tex]
1.8 = 1.78 Close enough?
n = 3, A = 2.4, b = (4/3)
2.4 = [tex]A_{0}[/tex][tex](4/3)^{n}[/tex]
2.4 = 1*[tex](4/3)^{3}[/tex]
2.4 = 2.37 Close enough?
help meeeeeeeeeeeeeeeeeee pleaseeeeeeeeee!!!!!!!!!!!!!!!!!!!!!
Answer:
2. 25
max = 88 feet
See the attachment photo!
Solve the equation 750 +0.05x = 1,100+ 0.025x for x.
O x = 140
O x = 350
O x = 1,850
O x = 14,000
Answer:
a
Step-by-step explanation:
The value of x will be; x = 14,000
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
We are given that the equation 750 +0.05x = 1,100+ 0.025x
Here, we need to solve for x;
750 +0.05x = 1,100+ 0.025x
combine the like terms;
750 - 1,100= -0.05x + 0.025x
-350 = - 0.025x
x = 14,000
Therefore, the solution will be as;
x = 14,000
The value of x will be; x = 14,000
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you have seven different video games. how many different ways can you arrange the games side by side on a shelf?
The number of different ways to arrange 7 video games is = 7!
The question requests for the number of arrangements; That means we're dealing with permutation
the number of video games is 7;
To arrange 7 games, we make use of the following permutation formula;
[tex]^{n}P_{n} = \frac{n!}{(n-n)!}[/tex]
So, we have
n=7
No. of ways we can arrange is = [tex]\frac{7!}{(7-7)!}[/tex] = 7!
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help me please please please
Answer:
SA = 148 cm²
Step-by-step explanation:
the opposite faces of the cuboid are congruent , then
SA = 2(6 × 5) + 2(4 × 5) + 2(6 × 4)
= 2(30) + 2(20) + 2(24)
= 60 + 40 + 48
= 148 cm²
Gwen opens a savings account with $50.00. Over the next year, she plans to deposit
$10.00 each month into the account. Write a linear function for the amount A, in
dollars, in Gwen's account after x months.
paing
working out the following additions
√8 + √18
[tex]\fbox{7.07...}[/tex]
First, we need the square of both numbers (8 and 18)
square of 8:
[tex]\sqrt{8}[/tex]
[tex]=2.828[/tex]
square of 18:
[tex]\sqrt{18}[/tex]
[tex]=4.242[/tex]
Now, let's add both numbers:
[tex]2.828+4.242[/tex]
[tex]=7.07[/tex]
A goat is tied by a rope 2.5 m to a peg
in some grass. The goat eats 1 m² of
grass in
28 min. How long does it take to eat all that
it can reach?
Time taken to eat all = 549.5 minutes.
What is unitary method?
The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units. What can be values and units.Let's say you go to the store to buy six apples. You are informed by the shopkeeper that he is offering 10 apples for Rs 100. In this instance, the value and the units are the price of the apples.Recognizing the units and values is crucial when using the unitary technique to a problem.Always write the items that need to be computed on the right side and the things that are known on the left side to simplify things. We are aware of the quantity of apples and the amount of money in the aforesaid problem.acc to our question-
The goat can rototate in a circule the radius of the curcle is 2.5 mSo the area of circle is;A=(pi) (r^2)A=(3.14)(2.5)^2A=19.625 m^2As 28 minutes for 1 m^2 so;Time taken to eat all = (28)(19.625)=549.5 minutesHence,Time taken to eat all = 549.5 minutes.
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n a recent survey in a statistics class, it was determined that only 60% of the students attend class on fridays. from past data it was noted that 98% of those who went to class on fridays pass the course, while only 20% of those who did not go to class on fridays passed the course. a. what percentage of students is expected to pass the course?
The percentage of students expected to pass the course is 66.8%
The given data represents;
In a recent survey in a statistics class, it was determined that only 60% of the students attend class on Fridays. from past data, it was noted that 98% of those who went to class on Fridays passed the course, while only 20% of those who did not go to class on Fridays passed the course.
We need to find the percentage of students who are expected to pass the course.
Let 'A' be the event of passing,
'B' be the event of attending on Fridays.
P (A I B) = 0.98 * P (A I B') = 0.2
P (B) = 0.6
Then, P (B') = 1 - 0.6
= 0.4
→ P (A ∩ B) = P (A I B) * P (B)
= 0.98 * 0.6
= 0.588
→ P (A ∩ B') = P (A I B') * P (B)
= 0.2 * 0.4
= 0.08
∴ P(A) = P (A ∩ B) + P (A ∩ B')
= 0.588 + 0.08
= 0.668
Hence, the percentage of students expected to pass the course is 66.8%
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Francisco just got his driver's license and wants to rent a car. The platinum car rental company charges $180 plus $92 per hour to rent a car. The plastic car rental company charges $250 plus $75 per hour. For what number of hours would the companies charge the same amount?
6900 hours, make a list of multiples until lcm is found
Consider the expressions x+4, x-3, and 2x+1
Create a new function, p(x) , that is the product of these three expressions.
Explain how the function demonstrates the Fundamental Theorem of Algebra.
The new function p(x) = 2x³+3x²-23x-12 is the product of expressions x+4, x-3, and 2x+1
How to create a new function and state how it demonstrates the Fundamental Theorem of Algebra?A function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and codomain or range. A function is generally denoted by f(x) where x is the input
Given: the expressions x+4, x-3, and 2x+1
The product of the expressions is:
p(x) = (x+4)(x-3)(2x+1)
= (x²-3x+4x-12)(2x+1)
= (x²+x-12)(2x+1)
= 2x³+2x²-24x+x²+x-12
= 2x³+3x²-23x-12
Therefore, the new function is p(x) = 2x³+3x²-23x-12.
Since the new function is a polynomial, it demonstrates the Fundamental Theorem of Algebra which states that every polynomial equation of degree n with complex number coefficients has n roots, or solutions, in the complex numbers
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3 is divided by the difference of 4 and a number
Answer:
[tex] \frac{3}{4 - x} [/tex]
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The equation of the parabola is f(x) = 7(x-3)^2+5.
Given:
The equation of the parabola that has the same shape has f(x) = 7x^2 but with vertex(3,5) in the form of f(x) = a(x-h)^2 + k.
If the parabola has same shape then its a values are same.
so f(x) = 7x^2
here a = 7
so the equation with vertex (3,5) also has a value is 7.
(h,k) = (3,5)
f(x) = a(x-h)^2+k
= 7(x-3)^2 + 5.
Therefore the equation of the parabola is f(x) = 7(x-3)^2+5.
Learn more about the vertex here:
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