The coordinates of the point are (7,0).
How to solve for the coordinatesdistance from (-3,-5) to (x,y) = 2 * distance from (x,y) to (9,-8)
Using the distance formula, we can write this equation as:
√[(x - (-3))^2 + (y - (-5))^2] = 2 * √[(9 - x)^2 + (-8 - y)^2]
Simplifying this equation, we get:
[tex](x + 3)^2 + (y + 5)^2 = 4[(9 - x)^2 + (-8 - y)^2][/tex]
Expanding and simplifying further, we get:
[tex]17x + 16y = 119[/tex]
So the coordinates of the point on the directed line segment from (-3,-5) to (9,-8) that partitions the segment into a ratio of 2 to 1 are:
x = (119 - 16y)/17
y = any value (since we can choose any value of y and then calculate x using the equation above)
For example, if we choose y = 0, then we get:
x = (119 - 16(0))/17 = 7
So the coordinates of the point are (7,0).
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In a ABCD Rhombus, B angle minus A equals 20 degrees. What degrees are all the angles of the Rhombus if B-A=20°?
All the angles of the Rhombus if B-A=20 is angle A = angle C = 80°, and angle B = angle D = 100°.
In a rhombus ABCD, if angle B minus angle A equals 20 degrees (B-A=20°), we can find the degree measures of all the angles.
Step 1: Recognize that in a rhombus, opposite angles are equal. Therefore, angle A = angle C and angle B = angle D.
Step 2: Remember that the sum of the angles in any quadrilateral is 360 degrees. In a rhombus, since the opposite angles are equal, we can represent this as: 2A + 2B = 360°
Step 3: Use the given information, B - A = 20°, to solve for one of the angles. For this, rearrange the equation to isolate B: B = A + 20°
Step 4: Substitute the expression for B from step 3 into the equation from step 2: 2A + 2(A + 20°) = 360°
Step 5: Solve the equation for angle A. 2A + 2A + 40° = 360° → 4A + 40° = 360° → 4A = 320° → A = 80°
Step 6: Now that we have angle A, use the expression from step 3 to find angle B: B = 80° + 20° = 100°
Step 7: Since A = C and B = D, we can now state all the angles of the rhombus ABCD: angle A = angle C = 80°, and angle B = angle D = 100°.
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A cereal company packages their granola in cylindrical containers that have a diameter of 20 cm and a height of 17. 4 cm. Approximately how much granola will a container hold?
The volume of granola a cylindrical container can hold 5451.6 cubic centimeters if containers have a diameter of 20 cm and a height of 17. 4 cm.
The number of unit cubes (cubes of unit length) that can fit inside a cylinder determines its volume.
Identifying the radius (r) and height (h)
r = 10 cm
h = 17.4 cm
Calculating the volume (V) using the formula V = πr²h
V = π × (10 cm)² × 17.4 cm
V ≈ 3.14 × 100 cm² × 17.4 cm
V ≈ 5451.6 cm³
Approximately, a container will hold 5451.6 cubic centimeters of granola.
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The math team wants to visit the Museum of Mathematics
to celebrate Pi Day. They have $210 to spend. They need to
buy 14 student tickets and 1 adult ticket. A student ticket
costs $12, and an adult ticket costs $17. The team also
wants to buy sugar-free fruit pies. Each pie costs $6. How
many whole pies can the team buy? Show your work.
Answer:
4
Step-by-step explanation:
14 student tickets times $12 = 168
168 + $17 = 185
210-185=25
6*4=$24
so they can buy 4 pies with 1 dollar left over
sorry if I am wrong
How to get 51 by using all four numbers 8 5 6 7 once.
To get 51 using the numbers 8, 5, 6, and 7 exactly once each, you can use the following mathematical expression:
(8 x 6) - 7 + 5 = 51
How it works:
1. Multiply 8 by 6 to get 48: (8 x 6) = 48
2. Subtract 7 from 48 to get 41: 48 - 7 = 41
3. Add 5 to 41 to get 51: 41 + 5 = 51
Therefore, (8 x 6) - 7 + 5 = 51.
Answer:
Step-by-step explanation:
8 x (7 - 5) + 6 = 51
Answer the questions below to determine what kind of function is depicted below
Answer:
This function is an exponential function because the base is a constant and the exponent is a variable.
The probability that Trevor studies for at least 50 minutes and passes his Algebra test is 0.88. The probability that he studies for at least 50 minutes is 0.92.
Step-by-step explanation:
If we let A be the event that Trevor studies for at least 50 minutes, and let B be the event that he passes his Algebra test, then we know:
P(A and B) = 0.88
P(A) = 0.92
We want to find the probability that Trevor passes his Algebra test given that he studied for at least 50 minutes, or in other words, we want to find P(B|A).
We can use Bayes' theorem to find this probability:
P(B|A) = P(A and B) / P(A)
Substituting in our values, we get:
P(B|A) = 0.88 / 0.92
Simplifying this fraction, we get:
P(B|A) = 0.9565
Therefore, the probability that Trevor passes his Algebra test given that he studied for at least 50 minutes is approximately 0.9565.
Andi moved a game piece down 4 pieces
choose all of the expressions that could be used to indicate how many spaces the game piece was moved.
The expression 4 that could be used to find spaces in the game piece was moved.
Since Integers are the collection of negative, and positive numbers including zero. Then the positive integers lie on the right side of the zero on the number line and the negative lie on the left side of zero on the number line.
We are given that Andi moved a game piece down 4 pieces
The expression in option 1: |-4| which is equal to 4
The expression in option 2: |4| which is equal to 4
The expression in option 3: -|4| which is equal to -4
The expression in option 4: 4 which is equal to -4
∴The correct expression is option 4
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Express the negation of each of these statements in terms of quantifiers without using the negation symbol.
a) ∀x(x > 1)
b) ∀x(x ≤ 2)
c) ∃x(x ≥ 4)
d) ∃x(x < 0)
e) ∀x((x < −1) ∨ (x > 2))
f ) ∃x((x < 4) ∨ (x > 7))
The negation of each of these statements in terms of quantifiers without using the negation symbo
a) There exists at least one x such that x is not greater than 1.
b) There exists at least one x such that x is not less than or equal to 2.
c) For all x, x is less than 4.
d) For all x, x is greater than or equal to 0.
e) There exists at least one x such that either x is not less than or equal to -1 or x is not greater than 2.
f) For all x, x is not less than 4 and x is not greater than 7.
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A circle has a diameter of 4 inches. Which statement about the area and circumference of the circle is true?
O A comparison of the area and circumference of the circle is not possible because there is not enough information to
find both.
O The numerical values of the circumference and area are equal.
O The numerical value of the circumference is greater than the numerical value of the area.
The numerical value of the circumference is less than the numerical value of the area.
Answer:
The numerical values of the circumference and area are equal
Step-by-step explanation:
Circumference: 12.57
Area: 12.57
12.57=12.57
Hope this helps! :)
Directed Line Segments Given the points A(-1, 2) and B(7. 8), find the coordinates of the point Pon directed line segment AB that partitions AB in the ratio 1:3.
The coordinates of point P on the directed line segment AB, which divides AB in the ratio 1:3, are (5, 6.5).
To find the coordinates of the point P on the directed line segment AB that partitions AB in the ratio 1:3, we can use the concept of section formula.
Let's assume the coordinates of point P are (x, y). According to the section formula, the coordinates of P can be calculated as follows:
x = (3x2 + 1x1) / (3+1) = (37 + 1(-1)) / 4 = (21 - 1) / 4 = 20/4 = 5
y = (3y2 + 1y1) / (3+1) = (38 + 12) / 4 = (24 + 2) / 4 = 26/4 = 13/2 = 6.5
Therefore, the coordinates of point P on the directed line segment AB, which divides AB in the ratio 1:3, are (5, 6.5).
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All the dimensions of a cube increase by a factor 3/2 how many times greater is the surface area? explain
If all the dimensions of a cube increase by a factor of 3/2, the surface area will increase by a factor of 9/2.
If all the dimensions of a cube increase by a factor of 3/2, then the new dimensions of the cube will be 3/2 times the original dimensions.
Let's say the original side length of the cube was "s". Then the new side length would be (3/2)*s.
The surface area of a cube is given by the formula 6s^2, where s is the side length.
So the original surface area of the cube would be:
6s^2
And the new surface area of the cube would be:
6(3/2s)^2
= 6(9/4)s^2
= 27/2 s^2
To find how many times greater the new surface area is compared to the original surface area, we can divide the new surface area by the original surface area:
(27/2 s^2) / (6s^2)
= (9/2)
So the new surface area is 9/2 times greater than the original surface area.
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the following table shows the number of miles a hiker walked on a trail each day for 6 days. day 1 2 3 4 5 6 number of miles 8 5 7 2 9 8 what was the mean number of miles the hiker walked for the 6 days? responses 3.5 3.5 4.5 4.5 6.5 6.5 7.5 7.5 8
The mean number of miles the hiker walked for the 6 days was 6.5 miles.
To calculate the mean or average of a set of numbers, we add up all the numbers and then divide the sum by the number of items in the set. In this case, we have the number of miles the hiker walked on each of the six days. To find the total number of miles the hiker walked, we simply add up all the numbers
8 + 5 + 7 + 2 + 9 + 8 = 39
Next, we divide the total number of miles by the number of days (which is 6) to get the average or mean number of miles the hiker walked per day:
Mean number of miles = Total number of miles / Number of days
= 39 / 6
= 6.5
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please help with question 9
The value of the function g(x) = 4x⁴ - 4x³ - 10x - 51
What are functions?Functions are defined as those expressions or equations showing the relationship between two variables.
From the information given, we have the functions;
f(x) = 4x³ + 3x² - 12x - 32
h(x) = 4x⁴ - 3x² + 2x - 19
(f + g(x) = h(x)
To determine the function, let us follow the expression
f(x) + g(x) = h(x)
Make g(x), the subject of formula
g(x) = h(x) - f(x)
Substitute the expressions
g(x) = 4x⁴ - 3x² + 2x - 19 - 4x³ + 3x² - 12x - 32
Now, collect the like terms, we get;
g(x) = 4x⁴ - 4x³ - 3x² + 3x² + 2x - 12x - 19 - 32
Add or subtract the values
g(x) = 4x⁴ - 4x³ - 10x - 51
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2a=−2+4(a+3)
a =
−4b=−5(3−b)+6
b =
Answer:
a = - 5 , b = 1
Step-by-step explanation:
2a = - 2 + 4(a + 3) ← distribute parenthesis
2a = - 2 + 4a + 12 ( subtract 4a from both sides )
- 2a = 10 ( divide both sides by - 2 )
a = - 5
-------------------------------------------
- 4b = - 5(3 - b) + 6 ← distribute parenthesis
- 4b = - 15 + 5b + 6 ( subtract 5b from both sides )
- 9b = - 9 ( divide both sides by - 9 )
b = 1
Step-by-step explanation:
2a=6(a+3)
2a=6a+18
2a-6a=18
-4a=18
-4a/-4=18/-4
a=-4.5
Stacy's time in her 50-meter freestyle race as measured with a stopwatch was 32. 4 seconds. The more precise electronic touchpad measured her time as 32. 36 seconds. What is the percent error for the stopwatch's measurement?
To find the percent error for the stopwatch's measurement of Stacy's time in her 50-meter freestyle race, we'll use the following formula:
Percent Error = (|(Measured Value - Actual Value)| / Actual Value) * 100
Here, the Measured Value is the stopwatch's time (32.4 seconds), and the Actual Value is the electronic touchpad's time (32.36 seconds).
Step 1: Calculate the absolute difference between the measured and actual values:
|32.4 - 32.36| = 0.04
Step 2: Divide the absolute difference by the actual value:
0.04 / 32.36 = 0.001236
Step 3: Multiply the result by 100 to get the percentage:
0.001236 * 100 = 0.1236%
The percent error for the stopwatch's measurement of Stacy's time in her 50-meter freestyle race is approximately 0.124%.
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The radian measure -1.7 pi is equivalent to -306 degrees.
How does sleep affect memory retention?To find the percent error of the stopwatch's measurement, we need to compare it to the more precise electronic touchpad measurement. The formula for percent error is:
percent error = (|measured value - actual value| / actual value) x 100%
In this case, the measured value is 32.4 seconds, the actual value is 32.36 seconds, and the absolute difference between them is 0.04 seconds. Plugging these values into the formula, we get:
percent error = (|32.4 - 32.36| / 32.36) x 100% = 0.124%
Therefore, the percent error for the stopwatch's measurement is 0.124%. This means that the stopwatch's measurement was very close to the actual value, with an error of only 0.124% of the actual value.
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If the probability of an event is 88/83 what is the probability of the event not happening? 88' Write your answer as a simplified fraction.
The probability of the event not happening is 5/83.
Here, probability refers to the likelihood of a given event occurring and that the inequality f(x) > 3g(x) holds for all x > 0.
If the probability of an event happening is 88/83, then the probability of the event not happening is 1 minus the probability of the event happening. This can be expressed as:
1 - 88/83
To simplify this expression, we can first find a common denominator for 1 and 88/83, which is 83/83:
83/83 - 88/83
-5/83
Therefore, the probability of the event not happening is 5/83.
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Which equations have the same value of x as 3/5 (30 x minus 15) = 72? Select three options.
A. 18 x - 15 = 72
B. 50 x -25 = 72
C. 18 x - 9 = 72
D. 3 (6 x - 3) = 72
E. x = 4.5
The equations that have the same value of x as 3/5 (30 x - 15) = 72 are C, D, and E.
Choosing the equations that are equivalentTo solve for x in 3/5 (30 x - 15) = 72, we can first simplify the left side by distributing the 3/5:
3/5 (30 x - 15) = 18 x - 9
Now we can solve for x by setting the right side equal to 72:
18 x - 9 = 72
Adding 9 to both sides:
18 x = 81
Dividing by 18:
x = 4.5
So we know that option E is one of the correct answers.
To check which of the other options have the same value of x, we can substitute x = 4.5 into each equation and see if it simplifies to 72:
A. 18 x - 15 = 72
18(4.5) - 15 = 72
81 - 15 = 72 (not equivalent)
B. 50 x - 25 = 72
50(4.5) - 25 = 200 - 25 = 175 (not equivalent)
C. 18 x - 9 = 72
18(4.5) - 9 = 72 (equivalent)
D. 3 (6 x - 3) = 72
3(6(4.5) - 3) = 3(24) = 72 (equivalent)
Therefore, the equations that have the same value of x as 3/5 (30 x - 15) = 72 are C, D, and E.
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A scientist uses a submarine to study ocean life.
She begins at sea level, which is an elevation of 0 feet.
She travels straight down for 112 seconds at a speed of 0.8 feet per second.
She then travels directly up for 120 seconds at a speed of 0.6 feet per second.
After this 232-second period, how much time, in seconds, will it take for the scientist to travel back to sea level at 3.5 feet per second? If necessary, round your answer to the nearest tenth of a second.
The length of time, in seconds, that it will take for the scientist to travel back to sea level at 3.5 feet per second will be 5.0 seconds.
How to calculate the amount of timeTo calculate the amount of time, we will begin by calculating the distance traveled from sea level in all of the instances.
1. 112 seconds × 0.8 feet per second = 89.6 feet
2. 120 seconds × 0.6 feet per second = 72 feet
The distance from sea level is now: 89.6 feet - 72 feet
= 17.6 feet
The time, in seconds, that it will take for the scientist to travel back to sea level at 3.5 feet per second will be:
17.6 feet ÷ 3.5 feet per second
5.0 seconds to the nearest tenth.
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Find the radius of the circle with equation x² + y² = 19²
r=0
Submit Answer
The radius of the circle of the equation is 19 units
Finding the radius of the circle of the equationFrom the question, we have the following parameters that can be used in our computation:
x² + y² = 19²
The equation of a circle is represented as
(x - a)² + (y - b)² = r²
Where
Center = (a, b)
Radius = r
using the above as a guide, we have the following:
Center = (a, b) = (0, 0)
Radius = r = 19
Hence, the radius of the circle of the equation is 19 units
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Lines c and d are perpendicular. The equation of line c is y=−1/2x+1 . What is the equation of line d ?
Answer:
y=2x+3
Step-by-step explanation:
When a line is perpendicular to another line, it means that the slope is the opposite reciprocal of the other slope.
We can see that line C has a slope of -1/2, meaning that the opposite reciprocal is 2. This overall means that line D has a slope of 2.
We can also see that line d (from the graph) has a y-intercept of (0,3).
To write this equation:
y=2x+3
Hope this helps! :)
A food truck owner charges z dollars per burrito combo and $1. 50 for a side of guacamole. The expression 5 (x + 1. 50) represents the
total cost of 5 burrito combos and 5 sides of guacamole.
Which expression also represents the total cost of 5 burrito combos, which cost a dollars each, and 5 sides of guacamole, which cost $1. 50
each?
0
The expression that fits perfect for the given requirement is 5z + 7.50, under the condition that a food truck owner charges z dollars per burrito combo and $1.50 for a side of guacamole.
Here we have to apply the principles of solving algebraic equations, due to the expression provided.
From the given information, here the food truck owner charges z dollars per burrito combo along with $1.50 for a side of guacamole.
According to the information it is given that the expression is 5(z+1.50) which helps to state the total cost of 5 burrito combos and 5 sides of guacamole.
Lets now formulate the expression for the given required equation
= 5(z + 1.50)
= 5z + 7.50.
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A food truck owner charges z dollars per burrito combo and $1.50 for a side of guacamole. The expression 5(z+1.50) represents the total cost of 5 burrito combos and 5 sides of guacamole. What expression also represent the cost of 5 burrito combos and 5 sides if guacamole that cost 1.50 each.
A rectangular prism has a square
base with edge length (x + 1). Its
volume is (x + 1)2(x – 3). What
does the expression (x + 1)(x – 3)
represent?
area of the base
area of one side
height of the prism
surface area of the prism
The expression (x + 1)(x - 3) represents the Area of base of the prism.
What is Prism?a crystal is a polyhedron containing a n-sided polygon base, a respectable halfway point which is a deciphered duplicate of the first, and n different countenances, fundamentally all parallelograms, joining relating sides of the two bases. Translations of the bases exist in every cross-section that runs parallel to the bases.
According to question:
The volume of a rectangular prism is given by the formula V = Bh, where B is the area of the base and h is the height of the prism. In this case, the base is a square with edge length (x + 1), so its area is (x + 1)^2. The volume of the prism is given as (x + 1)^2(x - 3).
We can find the height of the prism by dividing the volume by the area of the base:
B = V/h = (x + 1)^2(x - 3)/(x + 1) = (x + 1)(x - 3)
Therefore, the expression (x + 1)(x - 3) represents the Area of base of the prism.
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Help pls! Find the area of the circle
Use π = 3.14 and round your answer to the nearest hundredth.
the area of the circle is 615. 4 m²
How to determine the area
The formula that is used to calculate the area of a circle is expressed with the equation.
We have the equation as;
A = πr²
Such that the parameters are given as;
A is the area of the circleπ takes the constant value of 22/7 or 3.14r is the radius of the circleFrom the diagram shown, we have that;
A = unknown
r = 14m
Now, substitute the values, we get;
Area = 3.14 ×14²
Find the square value
Area = 3.14(196)
Multiply the values
Area = 615. 4 m²
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Which ones are solutions to
7x+4y=-23
(-1,-4)
(2,6)
(-5,3)
(6,-7)
Hello!
In this question, we are asked to find which set of points are solutions to our equation: 7x + 4y = -23
In order to find which points are solutions to our equation, we will plug the values into our equation and solve. If both sides of the equation are equal, the point will be a solution.
Note: Our coordinate point is in the format of (x,y), so we will plug in the values according to its variable.
Solve:
(-1,-4):
Plug in coordinate.
7(-1) + 4(-4) = -23
Simplify.
-7 - 16 = -23
-23 = -23
Since it is equal, (-1,-4) is a solution.
(2,6):
Plug in coordinate.
7(2) + 4(6) = -23
Simplify.
14 + 24 = -23
38 = -23
Since it is not equal, making it false, (2,6) is not a solution.
(-5,3):
Plug in coordinate.
7(-5) + 4(3) = -23
Simplify.
-35 + 12 = -23
-23 = -23
Since it is equal, (-5,3) is a solution.
(6,-7):
Plug in coordinate.
7(6) + 4(-7) = -23
Simplify.
42 - 28 = -23
14 = -23
Since it is not equal, making it false, (6,-7) is not a solution.
Answer:
The solutions to the equation are: (-1,-4) and (-5,3).
please answer and explain. show work 100 POINTS
In 2029, there will be an estimated A, 8.66 billion people in the world.
C, t = (ln(N/N₀))/k is the equation rewritten to solve for t.
How to determine exponential growth model?Part A:
Using the given exponential growth model, find the population in 2029 as follows:
N = N₀e^kt
N₀ = 7.95 billion (present population)
k = 1.08% = 0.0108 (rate of increase)
t = 2029 - 2022 = 7 (number of years)
N = 7.95 billion × e^(0.0108×7)
N ≈ 8.66 billion
Therefore, the world's population is expected to be 8.66 billion in 2029. Answer choice A is correct.
Part B:
To solve for t, isolate it on one side of the exponential growth model equation. Taking the natural logarithm of both sides:
ln(N/N₀) = kt
Divide both sides by k:
t = ln(N/N₀)/k
Therefore, the equation rewritten to solve for t is t = (ln(N/N₀))/k. Answer choice C is correct.
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(From Unit 7, Lesson 3. )
The Colorado state flag consists of three horizontal stripes of equal height. The
side lengths of the flag are in the ratio 2 : 3. The diameter of the gold-colored disk is
equal to the height of the center stripe. What percentage of the flag is gold?
The total area of the flag is the sum of the areas of the three stripes and the percentage of the flag that is gold is approximately 0.459%
Let's call this height "h".
We also know that the side lengths of the flag are in the ratio 2:3. This means that if the shortest side is 2x, then the longest side is 3x. Since the three stripes are of equal height, each one must be h/3 in height.
Now we can use this information to find the area of the gold-colored disk and the total area of the flag. The area of a circle is given by the formula A = πr^2, where r is the radius. Since the diameter is equal to the height of the center stripe (which is h/3), the radius of the disk is h/6. So the area of the disk is:
A_disk = π(h/6)^2 = πh^2/36
The total area of the flag is the sum of the areas of the three stripes. Since each stripe is h/3 in height and the shortest side is 2x, the area of each stripe is:
A_stripe = (2x)(h/3) = 2hx/3
So the total area of the flag is:
A_flag = 3(A_stripe) = 6hx
To find the percentage of the flag that is gold, we need to divide the area of the gold-colored disk by the total area of the flag and multiply by 100.
percentage = (A_disk / A_flag) x 100
Substituting our expressions for A_disk and A_flag, we get:
percentage = (πh^2/36) / (6hx) x 100
Simplifying, we can cancel out the factor of h and get:
percentage = (π/216)x100
So the percentage of the flag that is gold is approximately 0.459% (rounded to three decimal places).
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Does anyone have the answer
Answer:does anyone have the image
Step-by-step explanation:
F(x) and g(x) are polynomial functions.
determine whether each expression is always, sometimes, or never a polynomial.
f(x)+g(x)
and
f(x) / g(x)
F(x) and G(x) are polynomial functions in which the expression f(x) + g(x) is always a polynomial and f(x)/g(x) is sometimes a polynomial.
Let F(x) be a polynomial = 2x + 4
g(x) be a polynomial = 6x² + 12x
putting the value in the expression
f(x) + g(x) = 2x + 4 + 6x² + 12x
f(x) + g(x) = 6x² + 14x + 4
6x² + 14x + 4 is a polynomial
Now, putting the value in the equation
f(x)/g(x) = 2x + 4/6x² + 12x
Taking 3x common from 6x² + 12
We get 3x(3x+4)
f(x)/g(x) = 2x+4/3x(2x+4)
f(x)/g(x) = 1/3x
1/3x is not a polynomial
Hence, it sometimes a polynomial.
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Find the linearization L(x) of the function at a. T f(x) = 7cos(x), a = - (Consider a=3.14159265359 ) 9 L(x)"
To find the linearization L(x) of the function f(x) = 7cos(x) at a = 3.14159265359, we'll use the formula:
L(x) = f(a) + f'(a)(x - a)
where f'(x) is the derivative of f(x) with respect to x.
First, let's find the value of f(a) at a = 3.14159265359:
f(a) = 7cos(a)
f(3.14159265359) = 7cos(3.14159265359) ≈ -7
Next, let's find the value of f'(a) at a = 3.14159265359:
f'(x) = -7sin(x)
f'(a) = -7sin(a)
f'(3.14159265359) = -7sin(3.14159265359) ≈ 0
Now we have all the pieces we need to plug into the formula for L(x):
L(x) = f(a) + f'(a)(x - a)
L(x) = -7 + 0(x - 3.14159265359)
L(x) = -7
So the linearization of the function f(x) = 7cos(x) at a = 3.14159265359 is:
L(x) = -7
To find the linearization L(x) of the function f(x) = 7cos(x) at a specific point a, we'll use the formula:
L(x) = f(a) + f'(a)(x - a)
Given that a = 3.14159265359 (approximating π), first we need to find f(a) and f'(a).
1. f(a) = 7cos(a) = 7cos(3.14159265359) ≈ -7
2. To find f'(x), we take the derivative of f(x):
f'(x) = -7sin(x)
Now, we can find f'(a):
f'(a) = -7sin(3.14159265359) ≈ 0
Finally, we can plug these values into the linearization formula:
L(x) = -7 + 0(x - 3.14159265359)
Simplifying, we get:
L(x) = -7
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In a circle with radius 6 and angle intercepts an arc of length 3pi find the angle in radians in simplest form
In a circle with radius 6 and angle intercepts an arc of length 3π , the angle in radians in simplest form is π/2.
In a circle, the length of an arc is proportional to the angle that it intercepts. The ratio of the arc length to the circumference of the circle is equal to the ratio of the angle in radians to 2π. Thus, we can write:
(arc length) / (circumference) = (angle) / (2π)
In this problem, we are given that the circle has a radius of 6 and that the arc length is 3π. We can use the formula for the circumference of a circle, which is C = 2πr, to find the circumference of this circle:
C = 2πr = 2π(6) = 12π
Now we can use the formula above to find the angle in radians:
(3π) / (12π) = (angle) / (2π)
Simplifying this equation, we get:
angle = (3π * 2π) / 12π = 1/2 * π
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