Answer: distributive property
Step-by-step explanation:1 2 times 95 can indeed be rewritten as 12(100-5) and we can use the distributive property IN OUR HEADS, so we get 1200 - 60 = 1140.
Which equation represents this transformation?
Which deal is better. 500ml of lemonade for 94p or 750ml of lemonade for £1.44
Answer:
To compare the deals, we need to find the price per milliliter of lemonade for each deal.
For the first deal, the price per milliliter is:
94p / 500ml = 0.188p/ml
For the second deal, the price per milliliter is:
£1.44 / 750ml = 0.192p/ml
Therefore, the first deal is better as it has a lower price per milliliter of lemonade.
Buffalo Wild Wings charges you $22 per guest plus a room rental fee of $450.
Champs Bar and Grill charges you $210 for a room and $30 per guest.
How many guests can you invite so that the cost of the two restaurants will be equal?
You could invite
That equal cost is equal to
guests at an equal cost.
dollars.
Answer: 30
Step-by-step explanation: TRUSy
Marrero spotted an exclamation point on a billboard in Little Havana. This exclamation point consists of a circle sector and circle. Marrero measured the radius of the sector to be 9 ft, and the radius of the circle to be 1. 5 ft. At the end, he found out that the angle of the sector is 24°. What is the total area of the exclamation point on this Little Havana billboard? Round to the nearest square foot
The overall area of the exclamation point at the Little Havana billboard is about 24 square ft while rounded to the nearest square foot.
To discover the total area of the exclamation point, we need to calculate the area of the sector and the area of the circle, after which add them collectively.
First, let's find the area of the sector. The formula for the area of a circle area is:
Area of sector = (angle/360) x π x radius^2
Substituting the given values, we get:
Area of sector = [tex](24/360) x π x 9^2 = 16.62 ft²[/tex]
Next, let's find the area of the circle. The components for the area of a circle is:
Area of circle = π x radius^2
Substituting the given value, we get:
Area of circle [tex]= π x 1.5^2 = 7.07 ft²[/tex]
Now, we are able to add the areas of the sector and the circle to get the whole area of the exclamation point:
[tex]6.62 feet² + 7.07 feet² = 23.69 ft²[/tex]
Therefore, the overall area of the exclamation point at the Little Havana billboard is about 24 square ft while rounded to the nearest square foot.
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When two dice are rolled, the total is between 2 and 12 inclusive. A student simulates the rolling of two dice by randomly generating numbers between 2 and 12. Does this simulation behave in a way that is similar to actual dice? Why or why not?
The simulatiοn dοes't behave in way that is similar tο actual dice because οf variatiοn οf prοbability.
What is prοbability?Prοbability is a way οf calculating hοw likely sοmething is tο happen. It is difficult tο prοvide a cοmplete predictiοn fοr many events. Using it, we can οnly fοrecast the prοbability, οr likelihοοd, οf an event οccurring. The prοbability might be between 0 and 1, where 0 denοtes an impοssibility and 1 denοtes a certainty.
a)The pοssible cοmbinatiοns frοm the twο dices tο get a tοtal οf seven are
=> (1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
i.e. 6 cases and the tοtal nο. οf cοmbinatiοns
=>6 * 6 = 36.
Hence the nο. οf cases favοurable fοr us οut οf all the cοmbinatiοns is 6.
Hence the prοbability is 6/36 = 1/6.
b)To get a 7 out from a set of 2,12 inclusive is 1/11.
Because it is a choice of 1 no. from the 11 numbers available.
c). NO.
This simulatiοn dοesn't behave like the actual dice because in the secοnd case we are chοοsing οne number frοm the 11 available οptiοns whereas in the first case it is the prοbability οf chοοsing the 6 right cοmbinatiοns frοm the 36 cοmbinatiοns available.
i.e . Chοοsing 6 frοm 36 . hence the prοbability is 1/6.
whereas the chοice οf 1 frοm 11 . i.e prοbability is 1/11.
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L: 40 in
The figure will be dilated by a D
scale factor of 3.5. Find the
new measure of the base.
9 in
10 in
9 in
The new measure of the base is 35 inches
How to determine the new measure of the baseFrom the question, we have the following parameters that can be used in our computation:
Base = 10 inches
Scale factor of dilation = 3.5
Using the above as a guide, we have the following:
Image of the point = Scale factor of dilation * Base
Substitute the known values in the above equation, so, we have the following representation
New measure of base = 3.5 * 10
Evaluate
New measure of base = 35
Hence, the new base is 35 inches
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Fill in the blank question.
The table shows the distance a runner has traveled y, in miles, x minutes after the start of a race. What is the runner's average rate of change, in miles per minute, from 60 to 120 minutes?
Time (min) Distance (miles)
0 0
30 3.48
60 6.42
90 9.18
120 12.0
According to the speed formula, the runner's average rate of change from 60 to 120 minutes is approximately: 5.58 / 60 = 0.093 miles per minute
What is speed?
Speed is defined as the distance travelled by an object in a given amount of time. Speed is a scalar quantity, meaning that it has magnitude but no direction.
Mathematically, speed is calculated as follows:
speed = distance / time
where "distance" is the distance travelled by the object, and "time" is the time it takes for the object to travel that distance.
The average rate of change of a function over an interval is the slope of the secant line between the endpoints of the interval.
In this case, we want to find the average rate of change of the distance function from 60 to 120 minutes. The distance function is given by the table as:
Time (min) Distance (miles)
0 0
30 3.48
60 6.42
90 9.18
120 12.0
To find the average rate of change from 60 to 120 minutes, we need to calculate the slope of the secant line passing through the points (60, 6.42) and (120, 12.0).
The slope of the secant line is given by:
slope = (change in y) / (change in x) = (12.0 - 6.42) / (120 - 60) = 5.58 / 60
Therefore, the runner's average rate of change from 60 to 120 minutes is approximately: 5.58 / 60 = 0.093 miles per minute
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0. 07, -7. 88, -7. 9, 7. 03, -7. 881 is ascending order
Answer:
-7.9, -7.881, -7.88, 0.07, 7.03
Step-by-step explanation:
Numbers provided: 0.07, -7.88, -7.9, 7.03, -7.881
The numbers that are negative are lower on the number line than negative numbers.You look at the digits from left to rightThe bigger the negative number is, the further left it goes15 POINTS. How to do this one please teach by step by step
Given the ASA postulate, it is seen that ABC and DEF are congruent
The ASA congruence theorem is a postulate in Euclidean geometry that states that if two angles and the side between them in one triangle are congruent to the corresponding angles and sides in another triangle, then the two triangles are congruent.
More formally, the statement of the ASA congruence theorem can be summarized as follows:
If in two triangles, two pairs of corresponding angles are congruent, and the included sides are also congruent, then the triangles are congruent.
angles ABC and DEF are congruent because they have the same size and shape.
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Describe the end behavior of each function.
1. f(x) = -x³ + 2x³ -x+4
2. f(x)= x² -x² -x -1
Asking for help please, Thank you
"1. The end behavior of the function f(x) = -x³ + 2x³ -x+4 can be determined by looking at the highest degree term, which is 2x³. Since this term is positive and has an even degree, the end behavior of the function will be the same on both ends. As x approaches negative infinity, the function will approach negative infinity, and as x approaches positive infinity, the function will approach positive infinity. Therefore, the graph of the function will have two arms pointing upwards.
2. The function f(x) = x² -x² -x -1 can also be analyzed by looking at the highest degree term, which is x². However, since the first two terms cancel each other out, we can simplify the function to f(x) = -x -1. As x approaches negative infinity, the function will approach negative infinity, and as x approaches positive infinity, the function will approach negative infinity as well. Therefore, the graph of the function will have a single arm pointing downwards." (ChatGPT, 2023)
for a normal distribution with mean 50 and standard deviation 15
find the Q1 and Q3
Q1 = 7.75th item
Q3 = 23.25th item
Given,For a normal distribution with mean 50 and standard deviation 15,To find Q1 and Q3Formula for Q1 and Q3:Q1 = (n + 1)/4Q3 = 3(n + 1)/4Here, n = total number of observationsn = 2 × 15 = 30Thus, n + 1 = 31We know that for a normal distribution the Z values for Q1 and Q3 are -0.67 and 0.67 respectively.Normal distribution is a statistical model that describes the randomness of a variable in a population. It is symmetric around the mean. The mean is an essential property of a normal distribution. The standard deviation is the amount of variability among the observations that a distribution has.Mean = 50Standard deviation = 15Q1 = (n + 1)/4 = (30 + 1)/4 = 7.75th itemQ3 = 3(n + 1)/4 = 3 × (30 + 1)/4 = 23.25th itemThe 7.75th item and 23.25th item are calculated with the help of the formula.
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HELP ASAP PLEASE
When an object is dropped vertically from a platform, its height after seconds can be estimated with the formula shown.
ℎ=ℎ0−162
ℎ is the height of the object in feet.
ℎ0 is the initial height of the object in feet.
is the time in seconds after the drop.
Which equation and solution tells how long it takes the object to reach a height of 84 feet if its initial height is 180 feet?
Answer:
this does not make sense I mean ate you in un 200lev
Answer:
C
Step-by-step explanation:
III. Three fair tetrahedra (same as the one in part II except (magenta; green; red)) are tossed. Find the prob's of: a. All the same Greek letter. b. No γ 's nor δ 's. c. Exactly two β 's. d. All different Greek letters.
Answer:
d all differet grreek letters
Which question can best be answered by finding the volume of a figure
Finding the volume of a figure is a useful tool for solving practical problems that involve 3D space and quantities, such as capacity, mass, or density.
Finding the volume of a figure is a common mathematical calculation used in various fields, such as architecture, engineering, physics, and chemistry. The volume of a figure is a measure of the amount of space that it occupies in three-dimensional (3D) space, such as a cube, a cylinder, a sphere, or a complex shape.
One example of a question that can be answered by finding the volume of a figure is "How much water can a swimming pool hold?" The swimming pool is a 3D figure that can be approximated as a rectangular prism, with a length, width, and depth. By multiplying these three dimensions, we can find the volume of the swimming pool, which represents the amount of water it can hold. This information is useful for designing and constructing the swimming pool, as well as for filling and maintaining it.
Another example is "How much concrete is needed to build a retaining wall?" The retaining wall is a 3D figure that can be approximated as a trapezoidal prism, with a length, height, and thickness. By multiplying these three dimensions, we can find the volume of the retaining wall, which represents the amount of concrete required. This information is useful for estimating the cost and resources needed for the project.
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PLEASE SHOW WORK!!!!!!!!!
Answer:
The answer is D
(1 point) Assume that the monthly wondwide average number of airplaine crashes of commercial ailines is \( 2.2 \). What is the probability that there wili be (a) at most 3 such accidents in the next m
Answer: 2.2
Step-by-step explanation:
A quadrilateral has two angles that measure 216° and 102°. The other two angles are in a ratio of 10:11. What are the measures of those two angles?
The measures of the missing angles are 20° and 22°.
The sum of the angles of a quadrilateral is 360 degrees. We can use this fact to find the measure of the missing angles.
Let x and y be the measures of the missing angles, such that x:y = 10:11. Then we have:
216° + 102° + x + y = 360°
Simplifying this equation, we get:
x + y = 42°
We also know that x:y = 10:11, so we can write:
x = 10k
y = 11k
where k is a constant. Substituting these expressions into the equation x + y = 42°, we get:
10k + 11k = 42
21k = 42
k = 2
Therefore, x = 10k = 20° and y = 11k = 22°.
So, the measures of the missing angles are 20° and 22°.
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A student writes down the following piece of work which contains an error in their reasoning. The equation is: y 2
−23y=0 Add 23y to each side: y 2=23y Divide each side By y : y=23 (i) Substitute y=23 into the left-hand side of the equation y 2−23y=0, and explain why this shows that y=23 is indeed a solution of the equation. (ii) Write out a complete solution of the equationy 2−23y=0. (iii) Explain, as if directly to the student, why their solution is inadequate. Solve the following equation, checking that your solution is correct: 6−x16 = x+356
The following are the answers:
(e) All the parts are discussed in step 2.
(f) The solution is x= 4 and the verification is shown below
How to solveGiven the equation y^2 - 23y = 0
We would follow the steps
Given the equation [tex]\frac{16}{6-x} = \frac{56}{x + y}[/tex]
We need to solve this and verify the solution
(e) (i) Substitute y = 23 on the left-hand side of the given equation.
23^2 -23(23) = 0
0 = 0
Since the left-hand side is equal to the right-hand side, hence y= 23 is the solution of the given equation, but it is not the only solution.
(ii) Complete solution of the given equation:
y^2 -23y = 0
y(y-23) =0
y= 0
and y -23 = 0
y = 0, 23
Hence the complete solution of the given equation is y= 0, 23
EXPLANATION
If a.b = 0, then a= 0, b=0
.
(iii) The solution is inadequate because we can cancel the term only if it non zero.
Also, the given equation has power 2, but the student is getting only a single value, which is incorrect.
Solve the given equation as follows:
[tex]\frac{16}{6-x} = \frac{56}{x + 3}[/tex]
16 (x +3)= 56(6-x)
= 2x + 6= 42- 7x
9x = 36
x= 4
Hence the required solution is x= 4
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Your monthly bill for machine parts is $2,628. 53. If you pay the bill within 15 days, you'll receive a 2% reduction. What will your bill be if you pay it within the allotted time?
The new amount due, subtract the discount of $52.57 from the original bill of $2,628.53, giving the new amount due of $2,579.06. Therefore, if you pay the bill within 15 days, you will receive a 2% reduction and your bill will be $2,579.06.
To calculate the amount due if paying within 15 days, you would need to apply the 2% discount to the original amount. To do this, first find the amount of the discount by multiplying the original bill of $2,628.53 by 0.02. This gives a discount of $52.57. To calculate the new amount due, subtract the discount of $52.57 from the original bill of $2,628.53, giving the new amount due of $2,579.06. Therefore, if you pay the bill within 15 days, you will receive a 2% reduction and your bill will be $2,579.06.
Original Bill : $2,628.53
Discount (2%) : $2,628.53 x 0.02 = $52.57
Amount Due (after 15 days) : $2,628.53 - $52.57 = $2,579.06
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use half angle or power reduction formulas to fill in the blanks for the following identity:
Using power reduction formula and half angle formula, the blanks are "-1/2cos(x)" and "2x", respectively.
What are the missing values in the identity?We can use the power reduction formula for sine to express sin(3x) in terms of cosines:
sin(3x) = 3sin(x) - 4sin³(x)
= 3(1 - cos²(x)) - 4(1 - cos²(x))sin²(x)
= 3cos²(x) - 4cos⁴(x) - 4sin²(x)cos²(x)
= 3cos²(x) - 4cos⁴(x) - 2sin²(2x)
Now we can substitute this expression into the left-hand side of the identity and simplify using the half angle formula for cosine:
(sin(3x))⁴ = (3cos²(x) - 4cos⁴(x) - 2sin²(2x))⁴
= (3cos²(x) - 4cos⁴(x) - (1 - cos(4x)))⁴
= (3cos²(x) - 4cos⁴(x) - 1 + cos(4x))⁴
= (cos(4x) - 4cos⁴(x) + 3cos²(x) - 1)⁴
Now we can use the half angle formula for cosine again to express cos(4x) in terms of cosines of multiples of x:
cos(4x) = 2cos²(2x) - 1
= 2(2cos²(x) - 1)² - 1
= 8cos⁴(x) - 8cos²(x) + 1
Substituting this expression back into the above identity, we get:
(sin(3x))⁴ = (8cos⁴(x) - 8cos²(x) + 3cos²(x) - 4cos⁴(x) - 1)⁴
= (4cos⁴(x) - 5cos²(x) - 1)⁴
= (-1/2cos(x) + 1/8cos(2x))⁴
Therefore, the blanks in the original identity can be filled with "-1/2cos(x)" and "2x", respectively.
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The length and width of a rectangle are consecutive integers. The area of the
rectangle is 56 square centimeters. Find the length and width of the rectangle.
The length and width of the rectangle are 7 cm and 8 cm
How to find the length and width of the rectangle.From the question, we have the following parameters that can be used in our computation:
The length and width of a rectangle are consecutive integers
This means that
Length = x
Width = x + 1
So, we have
Area = x * (x + 1)
Substitute the known values in the above equation, so, we have the following representation
x * (x + 1) = 56
Express 56 as 7 * 8
So, we have
x * (x + 1) = 7 * 8
This means that
x = 7
So, we have
Length = 7
Width = 7 + 1
Length = 7
Width = 8
Hence, the length is 7 cm
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A softball coach has ordered softballs for two different leagues. The Junior League uses an 11-inch softball priced at $2.50 each. The Senior League uses a 12-inch softball priced at $3.50 each. The softball coach ordered a total of 120 softballs for $350.
If j is the number of softballs ordered for the Junior League and s is the number of softballs ordered for the Senior League, select the system of linear equations that represents the situation.
1. j + s = 350
2.5j + 3.5s = 120
2. j + s = 120
3.5j + 2.5s =350
3. j + s = 350
3.5j + 2.5s = 120
4. j + s = 120
2.5j + 3.5s = 350
The system of linear equations that represents the situation is:
(option 4) j + s = 120
2.5j + 3.5s = 350
How to know the system of linear equations that represents the situation?Let j be the number of 11-inch softballs ordered for the Junior League, and s be the number of 12-inch softballs ordered for the Senior League.
The total number of softballs ordered is 120, so we have:
j + s = 120
The cost of each 11-inch softball is $2.50, and the cost of each 12-inch softball is $3.50. The total cost of the order is $350, so we have:
2.5j + 3.5s = 350
Therefore, the system of linear equations that represents the situation is:
j + s = 120
2.5j + 3.5s = 350
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HELP!!! Will mark branliest
By De Moivre's formula, the square roots of the complex number √(i 2) are √z = 1 + i and √z = - 1 - i, respectively.
How to find the square roots of a complex numbers
In this problem we find a complex number, whose square roots can be determined by De Moivre's formula:
For z = a + i b:
√z = √r · [cos [(θ + 2π · k) / 2] + i sin [(θ + 2π · k) / 2]], for k = {0, 1}
Where:
r - Normθ - Direction, in radians.If we know that r = 2 and θ = π / 2, then the root of the complex number is:
k = 0
√z = √2 · [cos (π / 4) + i sin (π / 4)]
√z = 1 + i
k = 1
√z = √2 · [cos (5π / 4) + i sin (5π / 4)]
√z = - 1 - i
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Edmund had $140. He spent 3 7 of his money on toys and 1 5 of his money on new clothes. How much money did Edmund spend altogether?
Edmund spent $63.50 on toys and $28 on new clothes, for a total cost of $91.50.
Edmund had $140 and he decided to spend some of that money on toys and some on new clothes. To calculate how much he spent altogether, we need to first figure out how much he spent on each item. For toys, he spent 3/7 of his total money, which is equal to $63.50. For new clothes, he spent 1/5 of his total money, which is equal to $28. Now that we know how much he spent on each item, we can add these two numbers together to figure out how much he spent altogether. When we add $63.50 and $28, we get a total of $91.50. This means that Edmund spent $91.50 on toys and new clothes altogether.
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1. There are three buses A, B and C. the total
number of seats in the three buses is 250. 28% of
the 250 seat are in bus A.
number of seats in bus B:number of seats in bus
C = 3:2
38 of the seats in bus B are empty. 26 in bus C
are empty.
Jamie says the percentage of the seats in bus B
that are empty is greater than the percentage of
the seats in bus C that are empty. Is Jamie
correct.
Thus, Jamie is wrong. In contrast to Bus C, Bus B has less unoccupied seats as a percentage of total seats.
How do the percentages translate?% is a relative number that is used to represent hundredths of the any quantity. Since one percent (symbolized as 1%) is equal to one hundredth of something, 100 percent stands for everything, and 200 percent refers to twice the amount specified. percentage.
Find the complete seating capacity in Bus A first:
Bus A has 28% of the available seats, making its seat count 0.28 x 250 = 70.
Secondly, we can construct an equation system using the number of seats in Bus B compared to Bus C:
The number of seats aboard Bus B will be x.
In that case, Bus C's seating capacity is (2/3)x.
Using the information that there are 250 seats overall, we can find x:
70 (from Bus A) + x (from Bus B) + (2/3)x (from Bus C) = 250
When we simplify this equation, we obtain: (5/3)x = 180
When we solve for x, we get x = 108.
Bus C has (2/3)x = 72 seats, but Bus B has 108 seats.
We can now determine the proportion of vacant seats for each bus:
Out of a total of 108 seats, 38 seats on Bus B are vacant, making the percentage of vacant seats in Bus B (38/108) x 100% = 35.19%.
Out of a total of 72 seats, there are 26 empty seats on Bus C, making the percentage for empty seats there (26/72) x 100% = 36.11%.
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Answer:
what paper is this for this question
When the outlier(s) are removed, how does the mean change?
A) The mean decreases by 1.9.
B) The mean increases by 2.4.
C) The mean increases by 1.9.
D) There are no outliers.
Answer:
B) The mean increases by 2.4 because the mean is 1149 and there 6 outliers.
Find the Z-scores for which 50% of the distribution's area lies between -z and z.
Click to view page 1 of the table Click to view page 2 of the table.
The z-scores are
(Use a comma to separate answers as needed Round to two decimal places as needed. )
The z-scores for which 50% of the distribution's area lies between -z and z are: z = -0.675 and z = 0.675.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The normal distribution is symmetric, meaning that the middle 50% of the scores is given between these following percentiles:
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a. Is this tunction increasing of decreasing for the yearn2000−2015? A) Incroasing B) Decreasing
The correct answer is option A) Increasing.
The given question is regarding the function and its behavior. To determine whether a given function is increasing or decreasing, we should first find its derivative. Then, if the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.A function is given as: yearn = f (2000, 2015)Here, the independent variable is "year," and the dependent variable is "n." Thus, to find out whether this function is increasing or decreasing, we need to find the derivative of this function:$$\frac{dy}{dx}=\frac{dy}{dn}*\frac{dn}{dx}$$$$\frac{dy}{dx}=0.028$$Since the derivative is positive, it means the function is increasing. Therefore, the correct answer is option A) Increasing.
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What is the measure of angle H? Explain what is the measure of angle i ? explain
The measure of angle H is given as 67.5 degrees
The measure for angle I is 112.5 degrees
How to solve for angle HThe question tells us that ∠F must be 67.5°
from the sign, the angles H and the angle F are vertical angles.
Hence the value of ∠F = ∠H
∠H = 67.5°
what is the measure of angle i ?From the transversal, we can see that the angle E and angle I are corresponding angles
to get angle E
= 67.5° + ∠E = 180 (angles on a straight line)
∠E = 180 - 67.5
∠E = 112.5 degrees
Since ∠E and ∠I are corresponding angles, ∠I = 112.5 degrees
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The stop sign is a regular octagon, so the measure of ∠F must be 67.5°. What is the measure of angle H? Explain what is the measure of angle i ? explain
Use synthetic division and the Remainder Theorem to evaluate P(c). P(x)=3x^(2)+10x+1,c=-1 P(-1)
By answering the abοve questiοn, we may state that Hence, using divide synthetic divisiοn and the Remainder Theοrem, we have established that P(-1) = -1.
What is divide?Divisiοn, οne οf the fοur basic mathematical οperatiοns, is the prοcess οf cοmbining twο numbers tο create a new number. Other οperatiοns include additiοn, subtractiοn, and multiplicatiοn. Divisiοn is the reverse οf multiplicatiοn.
When yοu divide twelve intο its three equal parts, yοu get fοur in each grοup when yοu multiply three grοups οf fοur by twelve. Finding οut hοw many equal grοups are generated when sharing fairly—οr hοw many peοple are in each grοup—is the fundamental purpοse οf sharing. Tο divide is tο separate intο twο οr mοre grοups, classes, sectiοns, οr regiοns οf equal size.
In οrder tο evaluate P(c) using synthetic divisiοn and the Remainder Theοrem, we take the fοllοwing actiοns:
[tex]-1 | 3 10 1\s|_______[/tex]
[tex]-1 | 3 10 1\s| \s3[/tex]
[tex]-1 | 3 10 1\s| \s3\s-13[/tex]
Write the sum οf the twο numbers yοu just nοted dοwn underneath the fοllοwing cοefficient, 1:
[tex]-1 | 3 10 1\s| \s3\s-13\s-1[/tex]
The divisiοn's final value is the leftοver amοunt. The Remainder Theοrem states that this number is equal tο P(-1), which is what we were lοοking fοr. Hence, P(-1) equals 1.
Hence, using synthetic divisiοn and the Remainder Theοrem, we have established that P(-1) = -1.
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We can use the Remainder Theοrem tο cοnclude that the value[tex]P(-1) = 3(-1)^2 + 10(-1) + 1 = 3 - 10 + 1 = -6.[/tex]
What is synthetic divisiοn?Synthetic divisiοn is a shοrthand methοd οf dividing a pοlynοmial by a linear factοr οf the fοrm (x - c), where c is a cοnstant. It is a methοd οf pοlynοmial lοng divisiοn that simplifies the prοcess and reduces the number οf steps required.
Tο evaluate P(-1) using synthetic divisiοn and the Remainder Theοrem, we can fοllοw these steps:
Step 1: Write the cοefficients οf the pοlynοmial P(x) in οrder οf decreasing degree, including any missing terms with a cοefficient οf 0. In this case,
[tex]P(x) = 3x^2 + 10x + 1[/tex]
Step 2: Set up the synthetic divisiοn table by placing the value οf c, which is -1 in this case, in the leftmοst cοlumn and writing the cοefficients οf the pοlynοmial in the secοnd cοlumn:
-1 | 3 10 1
Step 3: Bring dοwn the first cοefficient, which is 3, intο the third cοlumn:
-1 | 3 10 1
---------
3
Step 4: Multiply the value in the third cοlumn by the value οf c, which is -1, and write the result in the fοurth cοlumn:
-1 | 3 10 1
---------
3 -3
Step 5: Add the values in the secοnd and fοurth cοlumns and write the result in the fifth cοlumn:
-1 | 3 10 1
---------
3 -3 0
Step 6: The value in the last cοlumn, 0, is the remainder when P(x) is divided by x + c, which is x + 1 in this case. Therefοre, we can use the Remainder Theοrem tο cοnclude that P(-1) = 0.
Thus, [tex]P(-1) = 3(-1)^2 + 10(-1) + 1 = 3 - 10 + 1 = -6.[/tex]
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