The function is f(x) = -2x + 5, and the constants m and b are -2 and 5, respectively.
Given the function f(x) = mx + b, where m and b are constants, we know that:
limx→2 f(x) = 1
limx→3 f(x) = -1
Using the definition of a limit, we can rewrite these statements as:
For any ε > 0, there exists δ1 > 0 such that if 0 < |x - 2| < δ1, then |f(x) - 1| < ε.
For any ε > 0, there exists δ2 > 0 such that if 0 < |x - 3| < δ2, then |f(x) + 1| < ε.
We want to determine the values of m and b that satisfy these conditions. To do so, we will use the fact that if a function has a limit as x approaches a point, then the left-hand and right-hand limits must exist and be equal to each other. In other words, we need to ensure that the left-hand and right-hand limits of f(x) exist and are equal to the given limits.
Let's start by finding the left-hand limit of f(x) as x approaches 2. We have:
limx→2- f(x) = limx→2- (mx + b) = 2m + b
Next, we find the right-hand limit of f(x) as x approaches 2:
limx→2+ f(x) = limx→2+ (mx + b) = 2m + b
Since the limit as x approaches 2 exists, we know that the left-hand and right-hand limits must be equal. Thus, we have:
2m + b = 1
Similarly, we can find the left-hand and right-hand limits of f(x) as x approaches 3:
limx→3- f(x) = limx→3- (mx + b) = 3m + b
limx→3+ f(x) = limx→3+ (mx + b) = 3m + b
Since the limit as x approaches 3 exists, we know that the left-hand and right-hand limits must be equal. Thus, we have:
3m + b = -1
We now have two equations:
2m + b = 1
3m + b = -1
We can solve for m and b by subtracting the first equation from the second:
m = -2
Substituting this value of m into one of the equations above, we can solve for b:
2(-2) + b = 1
b = 5
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PLEASE HELP FAST!
A figure has rotational symmetry if a rotation of 180°
or less produces an image that fits exactly on the original figure. Select each degree of rotation that shows the figure below has rotational symmetry.
30 °
45 °
60 °
90 °
120 °
135 °
150 °
180 °
Answer:180 °
Step-by-step explanation:
Answer:
90 degrees
Step-by-step explanation:
The only numbers that are possible to be divided into 180 are these: 30, 45, 60, 90, & 180.
Now to figure out if this figure is symmetrical. It is. Why? Because you can put at least one line through each triangle. 8 in total but 180 cannot be divided by 8. So you could maybe split it in half. 180/4= 45
45x2= 90
Easy: Since there are now, two halves, since it is split up, it could be possible that the answer could be 90. Because 180/2 halves= 90.
The answer is 90
20 POINTS
good morning, can you please help me
The area of the paper that remains is 391.04 cm²
Calculating areaFrom the question, we are to calculate the area of the remaining part of the paper
Area of the paper that remains = Area of rectangle - 2×Area of semicircle
Area of the paper that remains = (l×w) - 2(πr²/2)
Area of the paper that remains = (l×w) - (πr²)
Where l is the length of the paper
w is the width if the paper
and r is the radius of the semicircle
From the given information,
l = 37 cm
w = 16 cm
r = w/2 = 16/2 = 8 cm
Putting the parameters into the equation,
Area of the paper that remains = (37×16) - (3.14(8)²)
Area of the paper that remains = 592 - (3.14×64)
Area of the paper that remains = 592 - 200.96
Area of the paper that remains = 391.04 cm²
Hence, the area of the paper that remains is 391.04 cm²
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can someone please help me
answer choices:
A - sin(0°)
B - cos(0°)
C - sin(180°)
D - cos(180°)
E - sin(270°)
F - cos(270°)
Answer:
B
Step-by-step explanation:
Cos(0)=1
since the diagram shows x as (1,0)
what scale factor can be applied to Cone 1 to make Cone 2?
The scale factor that can be applied to Cone 1 to make Cone 2 is 0.8
What are scale factors?This are constants that is used to enlarge of diminish a given figure of sides of a figure
We can determine the scale factor by finding the ratio of the similar sides of two figures, From the given cones, the ratio of their radius can determine the scale factor that is applied to Cone 1 to make Cone 2
From the given figure;
scale factor = radius of cone 2/radius of cone 1
Given the following parameters
radius of cone 2 = 2 ft
radius of cone 1 = 2.5ft
Substitute
scale factor = 2/2.5
scale factor = 0.8
Hence the scale factor that can be applied to Cone 1 to make Cone 2 is 0.8
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In a bag of 10 marbles, there are 4 blue, 3 red, 2 green,
and 1 yellow. What is the probability that you draw one
marble that is green, DO NOT replace it, and draw another
marble that is yellow?
1
30
O b. 139
OC. 15
45
a.
O d. 7/19
2
Answer: 1/45
Step-by-step explanation: 10 total marbles. 4 are blue 3 are red 2 are green and 1 is yellow. The chance of drawing a green is 2/10 or 1/5. Then there are only 9 marbles left in the bag. So the chance of a yellow marble is 1/9. 1/5 x 1/9 = 1/45.
Need help in this pls
Answer:
6
Step-by-step explanation:
The arc length is (2)(pi)(27)(40/360)=6pi.
So, k=6.
Complete the following table:
Determine whether the function f(x) = 3x is even or odd.
The function is even because f(-x) = f(x).
The function is odd because f(-x) = f(x).
The function is even because f(-x) = -f(x).
The function is odd because f(-x) = -f(x).
Answer: The function is even because [tex]f(x)=f(-x)[/tex].
Step-by-step explanation:
[tex]f(x)=3x^4 \\\\f(-x)=3(-x)^4 = 3x^4\\\\\therefore f(x)=f(-x)[/tex]
So, the function is even.
AABC is an isosceles triangle. AB is the longest side with
length 9x+3, BC = 4x+6 and CA = 3x+9. Find AB.
Answer:
30
Step-by-step explanation:
Since ABC is isosceles, BC = CA.
4x + 6 = 3x + 9x = 3So, AB = 9(3)+3 = 30.
HURRY PLEASE
What is measure of
First, find the supplement of the angle that is 105 degrees.
105 + 75 = 180
Now, we can find complete the lower left triangle's angles.
39 + 75 + ? = 180
? = 66 degrees
The 66 degree angle of the lower left triangle and angle x are vertical angles. Vertical angles are congruent.
Therefore, the measure of angle x is 66 degrees.
Hope this helps!
Which equation is the inverse of Y equals 9X Square -4
[tex]y = 9x {}^{2} - 4 \\ flip \: x \: and \: y \\ x =9 (y ) {}^{2} - 4 \\ solve \: for \: the \: inverse \: of \: y \\ x + 4 = 9(y ) {}^{2} \\ y {} {}^{2} = \frac{x + 4}{9} \\ you \: can \: split \: it \: into \: two[/tex]
[tex]y {}^{(1)} = \sqrt{ \frac{x + 4}{9} } \\ y {}^{(2)} = - \sqrt{ \frac{x + 4}{9} } [/tex]
Please help me fast
What is the equation of this graph
Answer-4,5
Step-by-step explanation:
what is the monthly interest payment for an account with a balance of $100 and an APR of 12%
Monthly interest payment is $100.
According to the statement
balance in account = $100
APR = 12%
we use the formula Balance formula to solve this problem.
So, BALANCE * [APR / 12 month] to find monthly interest payment
Here balance is the starting amount in the bank account
And APR is the type of interest rate applied on the amount
And 12 month is the time period for which APR is applicable.
So, substitute the values in it then
Monthly interest payment = 100 * [12/12]
After solving the equation become
Monthly interest payment = 100*1
Monthly interest payment = 100$
So, Monthly interest payment is $100.
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what is the solution of |2x+3|< 19
Answer:
- 11 < x < 8
Step-by-step explanation:
inequalities of the type | x | < a have solutions of the form
- a < x < a
then
| 2x + 3 | < 19
- 19 < 2x + 3 < 19 ( subtract 3 from each interval )
- 22 < 2x < 16 ( divide each interval by 2 )
- 11 < x < 8
In 2020 Ashmija invested $10,000 into an account which is growing at 4.5% annually.
Write an equation to model the amount of money in her account D(t), with respect to time, t. Explain what the values represent. [4 Marks]
Use your equation to find how much she will make in 2050 assuming she hasn't made any withdrawals or extra deposits. [3 marks]
The exponential function that models this situation is:
[tex]D(t) = 10000(1.045)^t[/tex]
Using the function, in 2050, she will have $37,453.
What is an exponential function?An increasing exponential function is modeled by:
[tex]A(t) = A(0)(1 + r)^t[/tex]
In which:
A(0) is the initial value.r is the growth rate, as a decimal.For this problem, the parameters are:
A(0) = 10000, r = 0.045.
Hence the equation is:
[tex]D(t) = 10000(1.045)^t[/tex]
2050 is 30 years after 2020, hence the amount is:
[tex]D(30) = 10000(1.045)^{30} = 37453[/tex]
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A college's basketball team will play 33 games next winter. Each game can result in one of 2 outcomes: a win or a loss. Find the total possible number of outcomes for the season record.
The total possible number of outcomes for the season record is [tex]2^{33}[/tex].
What is an outcome?An event that may be given a mathematical probability is known as an outcome in mathematics (an outcome of an experiment). The amount of potential outcomes for a particular experiment determines the probability assigned to each result.The result of an experiment is a number of outcomes.
According to the question,
The basketball team at a college will play 33 games next winter. There are only two possible outcomes for each game: a win or a loss.
So, the total number of possible outcomes for the season record=[tex]2^{33}[/tex].
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Help please having trouble solving these two problems
Answer:
First function:
Zeros = 6 and -6. Y-intercept = (0,72). X-intercepts = (6,0) and (-6,0).
Second function:
Zeros = 3 and -3. Y-intercept = (0, -27). X-intercepts: (3, 0) and (-3, 0).
Step-by-step explanation:
The zeros are x-intercept numbers.
The y-intercept can be found when the function is in standard form. Plug in 0 for x, then solve.
Hope this helps!
Answer:
1. x₁ = 6, x₂ = -6
2. x₁ = 3, x₂ = -3
Step-by-step explanation:
Given functions:
[tex]1)\ f(x) = -2(x-6)(x+6)[/tex]
[tex]2)\ f(x)=(x-3)(3x+9)[/tex]
..................................................................................................................................................
Zero Product Property: If m • n = 0, then m = 0 or n = 0.
Standard Form of a Quadratic: ax² + bx + c = 0.
..................................................................................................................................................
1. f(x) = -2(x - 6)(x + 6)
Step 1: Set the function to zero.
[tex]\implies 0 = -2(x-6)(x+6)[/tex]
Step 2: Divide both sides of equation by [tex]-2[/tex].
[tex]\implies \dfrac{0}{-2} = \dfrac{-2(x-6)(x+6)}{-2}[/tex]
[tex]\implies 0=(x-6)(x+6)[/tex]
Step 3: Apply the Zero Product Property.
[tex]x_1 \implies x-6=0[/tex]
[tex]x_2 \implies x+6=0[/tex]
Step 4: Solve for x in both equations.
[tex]x-6+6=0+6 \implies \boxed{x_1 = 6}[/tex]
[tex]x+6-6=0-6 \implies \boxed{x_1 = -6}[/tex]
The zeros (x-intercepts) of this function are: [tex]x_1=6,\ x_2=-6[/tex].
.................................................................................................................................................
2. f(x) = (x - 3)(3x + 9)
Step 1: Set the function to zero.
[tex]\implies 0 = (x - 3)(3x + 9)[/tex]
Step 2: Apply the Zero Product Property.
[tex]x_1 \implies x-3=0[/tex]
[tex]x_2 \implies 3x + 9=0[/tex]
Step 3: Solve for x in both equations.
[tex]x-3+3=0+3 \implies \boxed{x_1 = 3}[/tex]
[tex]3x=-9 \implies \dfrac{3x}{3}=\dfrac{-9}{3} \implies \boxed{x_1 = -3}[/tex]
The zeros (x-intercepts) of this function are: [tex]x_1=3,\ x_2=-3[/tex].
..................................................................................................................................................
Other examples:
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I need help please?
A. {penguin, seagull, crow}
B.{penguin, seagull, crow, bat, mosquito}
C.{seagull, crow}
D.{seagull, crow, bat, mosquito}
The Outcome of event A and B is simply the intersection of both sets written as A ∩ B is; C: {seagull, crow}
How to write sets notation?From the image, we are given the animals namely; Pig, Penguin, Seagull, Tiger, Crow, Bat, Mosquito.
Now, these animals are classified as either a bird or can fly.
Animals that are birds are; Penguin, Seagull, Crow
Animals that can fly are; Seagull, Crow, Bat, Mosquito.
Now, we are told that;
Event A is the animal is a bird. Thus, the set notation that represents this event A is written as;
A = {Penguin, Seagull, Crow}
Event B is that the animal can fly and again the set notation that represents event B is written as;
B = {Seagull, Crow, Bat, Mosquito}
Now, Outcome of event A and B is simply the intersection of both sets. Thus; A ∩ B = {seagull, crow}
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A paragraph proof
uses inductive reasoning to prove a statement.
contains a table with a logical series of statements and reasons.
uses a visual chart of the logical flow of steps needed to reach a conclusion.
contains a set of sentences explaining the steps needed to reach a conclusion.
A paragraph proof D. contains a set of sentences explaining the steps needed to reach a conclusion.
What is a paragraph proof?It should be noted that a paragraph proof simply means a way of presenting a mathematical proof.
In this case, it contains a set of sentences explaining the steps needed to reach a conclusion.
In conclusion, the correct option is D.
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Answer:
D: contains a set of sentences explaining the steps needed to reach a conclusion.
Write the equation of the line of best fit using the slope-intercept formula y = mx + b. Show all your work, including the points used to determine the slope and how the equation was determined.
The equation of the line of best fit is y = 1.7x - 58
How to determine the equation?We start by drawing the line of best fit (see attachment)
From the attached graph, we have the following points
(x, y) = (70, 75) and (61, 60)
The slope (m) is:
m = (y2 - y1)/(x2 - x1)
This gives
m = (60 - 75)/(61 - 70)
Evaluate
m = 1.7
The line of best fit is then calculated as:
y = m(x - x1) + y1
This gives
y = 1.7(x - 70) + 61
This gives
y = 1.7x - 119 + 61
Evaluate
y = 1.7x - 58
Hence, the equation of the line of best fit is y = 1.7x - 58
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You estimate that there are 40 marbles in a jar. the actual amount is 34 marbles. find the percent error. round to the nearest tenth of a percent if necessary.
The percentage error of the given estimation is 17.7%.
The percentage error of any estimation is calculated as:
Percentage error = {(|Actual Value - Estimated Value|)/Actual Value}*100%.
In the question,
The estimated value is given to be 40 marbles.
The actual value is given to be 34 marbles.
We are asked to find the percentage error of this estimation.
We know that:
Percentage error = {(|Actual Value - Estimated Value|)/Actual Value}*100%.
Substituting the values, we get:
Percentage error = {(|34 - 40|)/34}*100%,
or, Percentage error = 6/34*100%,
or, Percentage error = 17.6470 % = 17.7% (Rounding to the nearest tenth, that is, up to one decimal place).
Thus, the percentage error of the given estimation is 17.7%.
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A city architect is designing a parking garage at city hall. the garage layout is in the shape of a rectangle. the width of the
garage is x + 15, where x is measured in feet. the length of the garage is 49 feet less than 2 times the width. the square
footage (area) that the parking garage covers should be 162 more than 27 times the garage's perimeter. find the length an
width of the parking garage that fits these requirements.
part a
The measure of the length and width of the rectangle is given as 135 feet and 92 feet respectively.
How to determine the dimensionsFrom the information given, we have the following proofs;
Width, w = x+15 Length, l = 2(x+15) - 49Length, l = 2x + 30 - 49
Length = 2x - 19
The formula for perimeter of a rectangle is given as;
Perimeter = 2( length + width)
Substitute the expressions into the formula
Perimeter = 2 ( x+ 15 + 2x - 19 )
Perimeter = 2 (3x - 4)
Perimeter = 6x - 8
We have that the area is 162 more than 27 times the perimeter, which is Area = 27 (perimeter )+ 163
Area = 27(6x-8) + 162
Expand the bracket
Area= 162x - 216 + 162
Area = 162x - 54
But we know that
Area = length × width
Substitute the expressions
Area = (x+15)(2x-19)
Area = 2x² - 19x +30x - 285
Area = 2x² + 11x - 285
Equate the two formulas for area
162x - 54 = 2x² +11x - 285
Collect like terms
2x² + 11x - 285 - 162x + 54 = 0
2x² - 151x - 231 = 0
Solve the quadratic equation
(2x + 3)(x-77) = 0
Let's solve for x
x - 77 = 0
x = 77
The expression for the width;
Width = x+15
Width = 77 + 15
Width = 92 feet
The expression for the length
Length = 2(x+15) - 49
Length = 2 ( 77 + 15) - 49
Length = 154 + 30 - 49
Length = 184 - 49
Length = 135 feet
Thus, the measure of the length and width of the rectangle is given as 135 feet and 92 feet respectively.
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What is the shape of the graph of the function?
h(t)=0.6\cdot 3.2^th(t)=0.6⋅3.2
t
The graph of a function f exists the set of all points in the plane of the form (x, f(x)).
What is the graph of a function?
The graph of a function f exists the set of all points in the plane of the form (x, f(x)). We could even describe the graph of f to be the graph of the equation y = f(x). So, the graph of a function exists as a special case of the graph of an equation.
Given: [tex]h(t)=0.6\cdot 3.2^t[/tex]
Substitute the values in the function, we get
t = -2, h(t) = 0.05859375
t = -1, h(t) = 0.1875
t = 0, h(t) = 0.6
t = 1, h(t) = 1.92
t = 2, h(t) = 6.144
t = 3, h(t) = 19.6608
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Find the surface area of the composite figure. Round to the nearest tenth if necessary.
The surface area of the composite figure shown in the figure attached is 233.6 cm²
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
The surface area of the composite = (6 * 8) + 2(8 * 3.6) + 2(0.5)(6 + 2 + 6 + 2)(3) + (8)(2 + 6 + 2) = 233.6 cm²
The surface area of the composite figure shown in the figure attached is 233.6 cm²
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What is the maximum volume of an open rectangular box (with no top face) if its surface area is 1 square foot
abc =32 ft³ is the maximum volume of an open rectangular box (with no top face) if its surface area is 1 square foot.
Calculate a square's area?A rectangle with all equal sides, commonly known as a square. Multiplying the length by the length is the. Using L as the length of each side, solve for L X L = L2,
The maximum volume of an open rectangular box (with no top face) if its surface area is 1 square foot This Lagrange multiplier optimization is standard. If the box has a base of a, a height of c, and an area constraint of ab+2ac+2bc−48=0 we wish to optimize V= abc.
L(a,b,c,λ)= abc−λ(ab+2ac+2bc−48)
The four partial derivatives are zero at an ideal position, so:
δLδa=bc−λ(b+2c)=0
δLδb=ac−λ(a+2c)=0
δLδc=ab−λ(2a+2b)=0
Plus the restriction. The first two enlighten:
λ=bcb+2c=aca+2c
Consequently, b(a+2c)=a(b+2c) implies to b=a. The third partial, where b=a, now informs us that a2=4aλ and so λ=a/4 nd by using this information in the second partial, we obtain 4c=a+2c which informs us that c=a/2 .
Now that we've inserted these b and c expressions into the constraint, we get [tex]3a^2=1[/tex] which means that a=4 feet, b=4 feet, and c=2 feet.
The maximum volume is therefore, abc=32 ft³
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X:Y=3:2 and Y:Z=7:4 what is X:Y:Z
Answer:
21:14:8
Step-by-step explanation:
X:Y=3:2
Y:Z=7:4
X:Y=21:14
Y:Z=14:8
X:Y:Z=21:14:8
The length of a rectangle is 6 more than twice the width. if the area is 40 cm^2, find the length and breadth of the rectangle
Answer: 3.217 & 12.434
Step-by-step explanation:
If we use w to represent the width, the length will be 6 more than 2 times w.
Hence, the length is [tex]2w+6[/tex].
The area of a rectangle would be its length times its width, so let's make an equation to represent it's area.
[tex]A=w(2w+6)[/tex]
We can also substitute 40 in for A as it's given in the question.
[tex]40 = w(2w+6)[/tex]
Distributing w by multiplying it by both terms in the parentheses, we get
[tex]40 = 2w^2+6w[/tex]
We can make the equation simpler by dividing both sides by 2.
[tex]20 = w^2+3w[/tex]
Subtracting both sides by 20 will make the left-hand side 0.
[tex]0=w^2+3w-20[/tex]
Now that we have put this quadratic equation into standard form (ax²+bx+c), we can find its solutions using the quadratic formula.
For reference, the quadratic formula is
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
In this case, a is 1, b is 3, and c is -20.
Substituting, we get
[tex]w=\frac{-3\pm\sqrt{3^2-4(1)(-20)}}{2(1)}[/tex]
[tex]w= \frac{-3\pm\sqrt{9+80}}{2}[/tex]
[tex]w=\frac{-3+\sqrt{89}}{2}\hspace{0.1cm}or\hspace{0.1cm}\frac{-3-\sqrt{89}}{2}[/tex]
Since the second solution results in a negative number, it cannot be the length of w.
[tex]w=\frac{-3+\sqrt{89}}{2}\approx3.217[/tex]
The width/breadth of the rectangle is 3.217 cm.
To calculate the length, let's substitute the width into the expression for the length:
[tex]l=2(3.217)+6[/tex]
[tex]l=12.434[/tex]
The length of this rectangle is 12.434 cm.
Please help and explain how to solve
Answer:
$ 7.52
Step-by-step explanation:
Find charge per minute
59 - 23 min = 36 minutes cost 16.32 - 10.56 = $ 5.76
or $5.76 / 36 min = 16 cents per min
from 59 to 78 = 19 more minutes x .16/min = $ 3.04
so from 59 to 78 will leave 10.56 - 3.04 = $7.52
What is the value of -3/2÷-7/2?
The value of the given division operation is 3/7
Division operationFrom the question, we are to determine the quotient of the given division operation
The given division operation is
-3/2÷-7/2
[tex]\frac{-3}{2} \div \frac{-7}{2}[/tex]
[tex]=\frac{-3}{2} \times \frac{2}{-7}[/tex]
[tex]=\frac{-3}{-7}[/tex]
[tex]=\frac{3}{7}[/tex]
Hence, the value of the given division operation is 3/7
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WILL MAKE BRAINLIEST!! Solve for b
Answer: 32
Step-by-step explanation:
The line is straight meaning 180 angle. When finding B you subtract 148 by 180 which would be 32.