By the quotient and product rules,
[tex]\left(\dfrac fg\right)'(0) = \dfrac{g(0) f'(0) - f(0) g'(0)}{g(0)^2} = 1[/tex]
[tex](f\times g)'(0) = f(0) g'(0) + f'(0) g(0) = 21[/tex]
Given that [tex]f'(0)=5[/tex] and [tex]g'(0)=3[/tex], we have the system of equations
[tex]\dfrac{5g(0) - 3f(0)}{g(0)^2} = 1 \implies 5g(0) - 3f(0) = g(0)^2[/tex]
[tex]3f(0) + 5g(0) = 21[/tex]
Eliminating [tex]f(0)[/tex] gives
[tex]\bigg(5g(0) - 3f(0)\bigg) + \bigg(3f(0) + 5g(0)\bigg) = g(0)^2 + 21[/tex]
[tex]10g(0) = g(0)^2 + 21[/tex]
[tex]g(0)^2 - 10g(0) + 21 = 0[/tex]
[tex]\bigg(g(0) - 7\bigg) \bigg(g(0) - 3\bigg) = 0[/tex]
[tex]\implies \boxed{g(0) = 7 \text{ or } g(0) = 3}[/tex]
Solve for [tex]f(0)[/tex].
[tex]3f(0) + 5g(0) = 21[/tex]
[tex]3f(0) + 35 = 21 \text{ or } 3f(0) + 15 = 21[/tex]
[tex]3f(0) = -14 \text{ or } 3f(0) = 6[/tex]
[tex]\implies \boxed{f(0) = -\dfrac{14}3 \text{ or } 3f(0) = 2}[/tex]
Write an expression for the number of tiles Bruce used and an expression for the number of tiles Felicia used. Use x to represent the number of tiles in a box.
An expression for the number of tiles used by Bruce and Felicia are 5x + 2 and 5x + 5 respectively.
What is an expression?An expression refers to a mathematical equation which is used to show the relationship existing between two or more variables and numerical quantities.
Note: x represents the number of tiles in a box.
For Bruce, we have:
Expression = 3x + 2x + 2
Expression = 5x + 2.
For Felicia, we have:
Expression = 5x + 5.
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Complete Question:
Bruce retiled his kitchen. Originally, he bought three boxes of tile with the same number of tiles in each. He ran out of tile and had to go back to get three extra boxes of tile. However, he only used two tiles from the last box to finish the job.
Felicia also retiled her kitchen. She bought five boxes of tile, and each box contained the same number of tiles as each of Bruce’s boxes. She also ran out of tile. She had to go back to the store and get five more tiles to finish the job.
Write an expression for the number of tiles Bruce used and an expression for the number of tiles Felicia used. Use x to represent the number of tiles in a box.
Answer:
Bruce: y = 5x + 2
Felicia: y = 5x + 5
Prove that two non-zero vectors are collinear if and only if one vector is a scalar multiple of the other.
The Prove that two non-zero vectors are collinear if and only if one vector is a scalar multiple of the other is given below.
What are the proves?1. To know collinear vectors:
∧ ⁻a ║ ⁻a
If ⁻b = ∧ ⁻a
then |⁻b| = |∧ ⁻a|
So one can say that line ⁻b and ⁻a are collinear.
2. If ⁻a and ⁻b are collinear
Assuming |b| length is 'μ' times of |⁻a |
Then | 'μ' ⁻a| = | 'μ' ⁻a|
So ⁻b = 'μ' ⁻a
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An economist wants to estimate the mean per capita income (in thousands of dollars) for a major city in Texas. He believes that the mean income is $31.3, and the standard deviation is known to be $9.4. How large of a sample would be required in order to estimate the mean per capita income at the 85% level of confidence with an error of at most $0.46
The sample size which is required in order to estimate the mean per capita income at the 85% level of confidence with an error of at most $0.46 is 866.
Given mean income of $31.3 and standard deviation of $9.4.
We have to calculate the sample size of confidence interval of 85% and at an error of $0.46.
Margin of error is basically the difference between actual values and calculated values.
Mean =$31.3
Standard deviation=$9.4
Confidence level=85%
Margin of error=$0.46
Margin of error=z*σ/[tex]\sqrt{n}[/tex]
0.46=1.44*9.4/[tex]\sqrt{n}[/tex]
[tex]\sqrt{n}[/tex]=1.44*9.4/0.46
[tex]\sqrt{n}[/tex]=29.42
squaring both sides
n=865.53
Round off and we get the following:
n=866.
Hence the sample size needed to estimate the mean per capita income at the 85% level of confidence is 866.
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5. Please answer the question on exponential growth or decay. Round to
the nearest whole number or penny if the question is concerning money if
necessary
For tax purposes, a car rental company
assumes each car in their fleet depreciates
by 7% per year. If the initial value of a
car is $32,200.00, what will the value be
when the car is 9 years old?
The value of this car when it is 9 years old is equal to $16,757.00.
How to calculate the value?In order to determine the value of this car when it is 9 years old, we would use an exponential function which also relates to compound interest as follows:
A = P(1 - r)^t
Substituting the given parameters into the formula, we have;
A = 32,200(1 - 0.07)^9
A = 32,200(0.93)^9
A = 32,200 × 0.5204
A = $16,757.00.
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3. Determine if the following triangle is a right triangle or not using the Pythagorean Theorem Converse. Triangle with side lengths of: 37 ft, 35 ft, and 12 ft
False, these measures do not form a right triangle.
True, these measures do form a right triangle.
Answer:
false
Step-by-step explanation:
these measure do not form a right triangle
Find the surface area of the composite figure.
20 in.
11 in.
5 in.
19 in
12 in.
12 in.
SA
11 in.
5 in.
20 in.
[?] in. 2
By using the area formulae for triangles and rectangles and the concept of surface area of the composite figure is equal to 1108 square inches.
What is the surface area of the composite figure?
The surface area is the area of all faces of a solid. In this case, we must sum the areas of seven rectangular and two triangular faces to determine the surface area:
A = 2 · (0.5) · (12 in) · (9 in) + 2 · (11 in) · (20 in) + 2 · (5 in) · (20 in) + 2 · (5 in) · (12 in) + (12 in) · (20 in)
A = 1108 in²
By using the area formulae for triangles and rectangles and the concept of surface area of the composite figure is equal to 1108 square inches.
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What will be the answer for 25 out of 100?
Hello !
[tex]\frac{25}{100}=0.25[/tex]
Have a nice day ;)
Answer:
25
Step-by-step explanation:
25
Answer: 25% of 100 is 25.
Whats the easiest way to solve a modulu problem. For example we can use 2^100/3. Whats the easiest way to find the remainder? Without using shortcuts that only work with specific numbers.
The easiest way to solve a modulu problem is by
Start by taking the initial number (before carrying our the modulo operation). Then select the divisor. Then one have to divide one number by the other and then round down.Then Multiply the divisor using the quotient. Then subtract the above number from one's initial number (dividend).The number gotten is the outcome of the modulo operation.2^100/3
=3√2^100
=3√2^99×2
=2^33×3√2
The [tex]\sqrt{2}[/tex] is the reminder.
What is the easiest way to find remainders?The easiest way to find the remainder that only work with specific numbers is that this can be done when one solve the division by the use of a calculator by:
First if you answer is in decimal form, you have to subtract the whole number and later multiply the decimal value that is said to be left by the divisor of the original problem. The outcome is your remainder.
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Add the following polynomials, then place the answer in the proper location on the grid.
- 4.1x^2 + 0.9x - 9.8
1.7x^2 - 2.4x - 1.6
Answer: - 2.4x^2 - 1.5x - 11.4
Step-by-step explanation:
- 4.1x^2 + 0.9x - 9.81.7x^2 - 2.4x - 1.6 = - 2.4x^2 - 1.5x - 11.4
Find the two inequalities equations graphed below please.
The two inequalities are:
a) y ≤ -2x + 5
b) y < 3x - 14
How to find the two graphed inequalities?
On the first graph, we can see that the shaded region is below the line and that the line is solid, so the inequality is of the form:
y ≤ line.
Now, remember that if the line passes through (x₁, y₁) and (x₂, y₂).
Then the slope is:
[tex]a = \frac{y_2 - y_1}{x_2 -x_1}[/tex]
We can see that the line passes through (2, 1) and (-1, 3)
Then the slope is:
[tex]a = \frac{1 - 3}{2 - 1} = -2[/tex]
So the line is:
y = -2*x + b
To find the value of b, we use the point (2, 1), then we get:
1 = -2*2 + b
1 = -4 + b
1 + 4 = b = 5
Then the first inequality is:
y ≤ -2x + 5
Now let's go to the other line equation, again the shaded region is below the line and this time the line is dashed, then:
y < line.
Two points on the line are (5, 1) and (4, -2)
Then the slope is:
[tex]a = \frac{1 - (-2)}{5 - 4} = 3[/tex]
y = 3*x + b
To find the value of b, we use the point (5, 1)
Then:
1 = 3*5 + b
1 = 15 + b
1 - 15 = b = -14
Then the inequality is:
y < 3x - 14
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Do anyone know thissss
Answer: False
Step-by-step explanation:
This is not a function because the single input [x] of 6 has outputs [y] of 2 and 4, and the single input [x] of 12 has outputs [y] of 2 and 4.
A function must have only one output [y-value] for every input [x-value].
let f(x) = x^3-3x^2+2 and g(x) = x^2 -6x+11 Enter the value of x such that f(x)=g(x)
The value of x such that f(x) = g(x) is x = 3
Quadratic equationGiven the following expressions as shown
f(x) = x^3-3x^2+2 and;
g(x) = x^2 -6x+11
Equate the expressions
x^3-3x^2+2 = x^2 -6x+11
Equate to zero
x^3-3x^2-x^2+2-11 = 0
x^3-3x^2-x^2 + 6x - 9 = 0
x^3-4x^2+6x-9 = 0
Factorize
On factorizing the value of x = 3
Hence the value of x such that f(x) = g(x) is x = 3
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A graph titled Cost of a Lunch in a School's Cafeteria has Days since school started on the x-axis and cost of lunch (dollars) on the y-axis. A horizontal line is at y = 2.5.
Which is correct about the graph’s slope?
Its slope is positive.
Its slope is negative.
Its slope is zero.
It has no slope.
The true statement about the graph's slope is (c) Its slope is zero.
How to determine the true statement?From the question, we have:
A horizontal line is at y = 2.5.
The horizontal line implies that the cost of lunch do not change
Horizontal lines have a slope of 0
Hence, the true statement about the graph's slope is (c) Its slope is zero.
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The diagram shows AKLM. Which term describes point N?
K
L
A. Orthocenter
N
B. Circumcenter
C. Centroid
D. Incenter
The term that describes point N is centroid. The correct option is C. Centroid
Centroid of a triangleThe centroid of a triangle is the point in which the three medians of the triangle intersect.
The median is a line that joins the midpoint of a side and the opposite vertex of the triangle.
In the given diagram, we can observe that the lines join the midpoints of each side to the corresponding opposite vertex.
Point N is the point where the lines intersect.
Hence, the term that describes point N is centroid. The correct option is C. Centroid
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Answer:
B. Circumcenter
Have an Amazing day/night!!
Assume that they all agree on the sample size and the sample mean use the pairs below to select the team that has the smallest sample standard deviation?
Step-by-step explanation:
A.Confidence Level: 99.7%; Confidence Interval: 42 to 48
B.Confidence Level: 95%; Confidence Interval: 43 to 47
C.Confidence Level: 68%; Confidence Interval: 44 to 46
D.Confidence Level: 95%; Confidence Interval: 44 to 46
Suppose water was being drained at from a conical tank. The radius is decreasing at a rate of 2 inch per minute and the height is decreasing at 4 inches per minute. Find the change in the volume when the height is 5 inches and the radius is 6 inches.
The change in volume is -88π
Volume?The volume is nothing but the amount of space occupied by a three-dimensional figure as measured in cubic units.
What is volume change?Volume Change is defined as the variation in the existing volume of the service type which is provided by the Supplier under the Agreement
Given:The radius is decreasing at 2 inches per minute
The height is given as decreasing at 4 inches per minute
So let [tex]\frac{dr}{dt}=-2[/tex] and
[tex]\frac{dh}{dt}=-4[/tex]
and we have radius r=6,
height h=5
The volume of the cone is represented by V as V=[tex]\frac{1}{3}\pi r^{2}h[/tex]
Then
[tex]\frac{dv}{dt}=\frac{\pi }{3}[r^{2}\frac{dh}{dt} +h*2r\frac{dr}{dt}][/tex]
[tex]=\frac{\pi }{3}[6^{2}(-4)+5*2(6)(-2)][/tex]
[tex]=\frac{\pi }{3}[36(-4)+5(12)(-2)]\\ =\frac{\pi }{3}[-144+(-120)]\\ =\frac{-264}{3}\pi \\ =-88\pi[/tex]
So the volume change is [tex]-88\pi[/tex]
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What is the solution to the equation 3 +6=2
Answer: no solution
Step-by-step explanation:
if we simplify this equation, we derive that 9=2.
this is a false statement, which means this equation has no real solutions.
If f(x) = 5x - 7, what is {3)?
O A. 22
O B. 2
O c. 1
• D. 8
Work Shown:
f(x) = 5x - 7
f(3) = 5(3) - 7 .... every x replaced with 3
f(3) = 15 - 7
f(3) = 8
[tex] \: [/tex]
Given :
f(x) = 5x - 7x = 3To Find :
What is the value of x ?Calculating the value of x :
[tex]\sf : \implies \: 5x - 7[/tex]
[tex]\sf : \implies \: 5(3) - 7[/tex]
[tex]\sf : \implies \: 15 - 7[/tex]
[tex]\bf : \implies \: 8[/tex]
Hence,
[tex] \implies \boxed{ \bf \: \therefore \: The \: value \: of \: x \: is \: 8 \: } \\ \\ [/tex]
A Virus is 30 Nanometres long.
Using the information below, calculate the length, in millimetres, of the virus.
1 metre = 1,000,000 Nanometres.
Answer:
0.00003mm
Step-by-step explanation:
We know that 1mm=1000000nm
To convert small unit to big we divide
So 30÷1000000=0.00003nm
Nosaira solved an equation. Her work is shown below:
3(2x + 1 ) = 2(x + 1) + 1
6x + 3 = 2x + 2 + 1
6x + 3 = 2x + 3
4x = 0
x = 0
She determines the equation has no solution.
Which best describes Nosaira’s work and answer?
Her work is correct, but there is one solution rather than no solution.
Her work is correct and her interpretation of the answer is correct.
Her work is incorrect. She distributed incorrectly.
Her work is incorrect. She moved terms across the equals sign incorrectly.
Her work is correct, but there is one solution rather than no solution.
Solving linear equationsLinear equations are equations that has a degree of 1. Given the equation below;
3(2x + 1 ) = 2(x + 1) + 1
Expand
3(2x) + 3(1) = 2(x) + 2(1) + 1
Simplify
6x + 3 =2x + 2 + 2
6x + 3 = 2x + 3
Collect the like term
6x - 2x = 3 - 3
4x = 0
x =0
Hence we can conclude that Her work is correct, but there is one solution rather than no solution.
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A model without any simplifying assumptions a.provides simplified solutions to complex problems. b.is highly complex and likely unworkable. c.excludes important predictive variables. d.is very helpful for solving tough, real-world problems.
A model without any simplifying assumptions b.is highly complex and likely unworkable.
As their call implies, simplifying assumptions are assumptions that are protected within the version to simplify the evaluation as tons as feasible. While a simplified version now does not predict the conduct of the actual issue within perfect bounds, too many simplifying assumptions have been made.
While a scientific version enables us to make predictions, it's more valued. As clinical models are representations of simplified explanations, they do not are looking for to explain each scenario or every detail. This means that medical models often aren't equal to the 'real international' from which they are derived.
Mathematical models can help college students apprehend and discover the that means of equations or valuable relationships.
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Simplify fully 4m/20m leaving your answer in index form
The index form of the expression is 5^-1m
How to determine the index formNote that index form means in a power form
We have that;
= 4m/20m
Find the similar factor
= 1/5m
The inverse of a number, is that number to the power of -1
Then,
1/5m = 5^-1m
Thus, the index form of the expression is 5^-1m
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According to the partial construction shown, what is the next step in constructing a line perpendicular to a point on the line?
O Place compass at point D, and draw an arc that intersects the arc above DF.
1
O Place compass at point F, and draw an arc that intersects the arc above DF-
O Place compass at point P, and draw an arc above DF
O Place compass at point P, and draw an arc below DF
The next step in constructing a line perpendicular to a point on the line is A. Place a compass at point D, and draw an arc that intersects the arc above line DF.
To make a perpendicular line:
Draw a line segment like DFKeep the point on D and make an arc by increasing the length of the compassKeep the point on F and make an arc like same as beforeName the point as any meeting on the drawn arcConnect it to the point with PTherefore, the next step would be to Place a compass at point D, and draw an arc that intersects the arc above line DF.
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a+a+b=14 b+b+c=08 a+b+c=18 b=
The value of b in the expression is -2.
How to find the variable in an expression?a + a + b = 14
b + b + c = 8
a + b + c = 18
Therefore,
2a + b = 14
2b + c = 8
c = 8 - 2b
Hence,
a + b + 8 - 2b = 18
a - b = 18 - 8
a - b = 10
2a + b = 14
a - b = 10
3a = 24
a = 8
8 - b = 10
b = 8 - 10
b = -2
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Answer:
a = 8
b = -2
c = 12
Step-by-step explanation:
a + a + b = 14
2a + b = 14 ⇒ ( 1 )
b + b + c = 8
2b + c = 8 ⇒ ( 2 )
a + b + c = 18 ⇒ ( 3 )
First, let us make a as the subject in equation 1.
2a + b = 14
2a = 14 - b
[tex]a=\frac{14-b}{2} \\a = 7-\frac{b}{2}[/tex] ⇒ ( 4 )
Let us make c as the subject in equation 2.
2b + c = 8
c = 8 - 2b ⇒ ( 5 )
Now it is clear for us that, we can replace a & c with ( 7 - b / 2 ) & ( 8 - 2b ) respectively.
And now let us solve the equation 3 to find the value of b.
[tex]a+b+c=18\\\\7-\frac{b}{2} +b+8-2b=18\\\\Combine\: \:like\:\: terms.\\\\[/tex]
[tex]15-\frac{b}{2} -b=18\\\\\frac{15*2}{1*2} -\frac{b}{2}-\frac{b*2}{1*2}=18\\\\\frac{30}{2} -\frac{b}{2}-\frac{2b}{2}=18\\\\\frac{30-3b}{2} =\frac{18}{1} \\\\Use\:\: cross\;\: multiplication[/tex]
[tex]30-3b=18*2\\\\30-3b=36\\\\30-36=3b\\\\-6=3b\\\\-2 = b[/tex]
And now let us find the value of a
For that, let us use the equation 1. There, we can replace b with ( -2 ) to find the value of a.
[tex]2a+b=14\\\\2a-2=14\\\\2a = 14+2\\\\2a=16\\\\a=8[/tex]
Now, let us find the value of c.
For that, let us take the equation 2. There also, we can replace b with -2 to find the value of c.
[tex]2b+c=8\\\\2*-2+c=8\\\\-4+c=8\\\\c=8+4\\\\c=12[/tex]
HELP PLEASEEEE
Zarea finds 3√6,000 and shows her work below.
Stephen has made a 91, 96, and 95 on the first three quizzes this six weeks. If he makes the same score on the last two quizzes, what grade does he need to make on each quiz to have a quiz average of EXACTLY 94 this grading period?
Answer:
A 94
Step-by-step explanation:
If we take the mean of 91, 96, and 95, we find that it is equal to 94. Thus, if he scores a 94 on his last two quizzes, his average will remain a 94.
What can be concluded about the line represented in the table? Select 3 options.. x y –6 –7 2 –3 8 0 The slope is 2. The slope is One-half. The y-intercept is –4. The y-intercept is 8. The points (–2, –5) and (8, 0) are also on the line. The points (–5, –2) and (1, 10) are also on the line.
The correct statements regarding the linear function are given as follows:
The slope is 1/2.The y-intercept is -4.The points (–2, –5) and (8, 0) are also on the line.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.In this problem, the line goes through points (-6,-7) and (8,0), hence the slope is given by:
[tex]m = \frac{0 - (-7)}{8 - (-6)} = \frac{7}{14} = \frac{1}{2} = 0.5[/tex]
Hence:
y = 0.5x + b
When x = 8, y = 0, hence the y-intercept is given as follows:
0 = 0.5(8) + b
b = -4.
When x = -2, we have that:
y = 0.5(-2) - 4 = -1 - 4 = -5.
The points (–2, –5) and (8, 0) are also on the line.
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Answer:
B,C,E
Step-by-step explanation:
took the test on edge and got 100
10. What is the perimeter of the triangle with sides 15 in, 20 in, and 25 in?
O 72 in
60 in
300 in
O 150 in
k
Answer:
P = 60 inches
Step-by-step explanation:
Perimeter is the distance around the outside of a shape. So, to find perimeter, add up all the sides.
The side lengths were given, add them:
15 + 20 + 25
= 60
The perimeter is 60 inches.
Polygon LMNOP was transformed to create polygon L'M'N'O'P'.2 pentagons have identical side lengths and angle measures. The points are labeled with the same letters.
Which angle corresponds to AngleN?
AngleN'
AngleO
AngleO'
AngleP'
The angle that corresponds to ∠N after transformation is: A. ∠N'.
What is Transformation?Transformation is a process of manipulating a figure to create an identical figure either by dilation, rotation, or translation.
In the transformation performed in the image shown below, the new image formed is congruent to the original figure. This means polygon LMNOP ≅ L'M'N'O'P.
Therefore, the angle that corresponds to angle N is: A. Angle N'
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Answer: A). Angle N'
Step-by-step explanation:
61. The true proportion p of people who give a favorable
rating to Congress is 8% with a margin of error of
1.5%. Describe this statement using an absolute
value equation.
Answer:
The expression for the statement using absolute value notation is [tex]$|p-0.08| \leq 0.015$[/tex].
Step-by-step explanation:
It is given in the question that the true proportion p of people who give a favourable rating to Congress is 8% with a margin of error of 1.5%. It is required to express the statement using absolute value notation.
To express the statement using absolute value notation, deduce the statement in the form of an equation.
The error margin for the hypothesis is 1.5% which implies the absolute difference between p and 8% should be less than or equal to 1.5%.[tex]$$|p-0.08| \leq 0.015$$[/tex]