The within-treatments sum of squares is sometimes referred to as the "error sum of squares" because it represents the amount of variability in the data that is not accounted for by the treatment effect. In other words, it represents the variability that is due to random error or chance factors that are not related to the treatment. The error sum of squares is used to estimate the variance of the population from which the samples were drawn. The F-ratio is then calculated by comparing the treatment mean square to the error mean square. If the treatment effect is significant, then the treatment mean square will be larger than the error mean square, resulting in a larger F-ratio and a rejection of the null hypothesis. Therefore, the error sum of squares is an important component of the ANOVA table as it helps to determine whether the treatment effect is significant or not.
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How many times larger is 7 x 10^5 than 2 x 10^3?
A) 3.5
B) 35
C) 3,500
D) 350
[tex]\cfrac{7\times 10^{5}}{2\times 10^{3}}\implies \cfrac{7}{2}\times\cfrac{10^5}{10^3}\implies \cfrac{7}{2}\times 10^5\cdot 10^{-3}\implies \cfrac{7}{2}\times 10^{5-3} \\\\\\ \cfrac{7}{2}\times 10^2\implies 3.5\times 100\implies 350[/tex]
The legislature in a state has 103 seats. Apportion these seats to the four counties below using Jefferson's method.
The final apportionment of the 103 seats using Jefferson's method is: County A = 84 seats, County B = 119 seats, County C = 136 seats, and County D = 64 seats.
Here are the steps you can follow to apportion the 103 seats to the four counties;
Determine the population of each county; Let's assume the populations of the four counties are as follows;
County A; 250,000
County B; 350,000
County C; 400,000
County D; 200,000
Calculate the geometric mean of the populations; To calculate the geometric mean, we need to multiply the populations of all the counties together and then take the fourth root of the result.
Geometric mean = (250,000 x 350,000 x 400,000 x 200,000)^(1/4)
Geometric mean = 305,541.27
Calculate the ratio of each county's population to the geometric mean; To calculate the ratio for each county, we divide its population by the geometric mean.
County A; 250,000 / 305,541.27 = 0.8171
County B; 350,000 / 305,541.27 = 1.1458
County C; 400,000 / 305,541.27 = 1.3085
County D; 200,000 / 305,541.27 = 0.6542
Calculate the initial apportionment of seats; To calculate the initial apportionment of seats, we assign each county the number of seats equal to its ratio multiplied by the total number of seats.
County A; 0.8171 x 103 = 84.18, rounded down to 84 seats
County B; 1.1458 x 103 = 118.04, rounded up to 118 seats
County C; 1.3085 x 103 = 134.75, rounded up to 135 seats
County D; 0.6542 x 103 = 67.03, rounded down to 67 seats
Check if the initial apportionment is equal to the total number of seats; The initial apportionment gives a total of 404 seats, which is greater than the total number of seats available (103).
Adjust the apportionment to reach the total number of seats; To adjust the apportionment, we assign additional seats to each county based on their fractional remainders.
County C has the largest fractional remainder (0.75), so it gets the first additional seat.
The new apportionment is; A = 84, B = 118, C = 136, D = 65
The total number of seats is 403, so one seat is left unassigned.
We assign the last seat to the county with the largest fractional remainder, which is County B.
Therefore, the final apportionment of the 103 seats will be; County A = 84 seats, County B = 119 seats, County C = 136 seats, and County D = 64 seats.
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In which situation would it be most appropriate to use the mean of the set of numbers to describe the data set instead of the median?
О 152, 157, 159, 163, 164, 165, 976
О 0, 899, 906, 912, 915, 917, 919
O 165, 165, 177, 178, 186, 199, 211
O 0, 57, 847, 859, 866, 866, 872
The data set for which the mean is most appropriate to describe the set instead of the median is; 0, 57, 847, 859, 866, 866, 872.
What answer choices is best described by the mean instead of the median?It follows from the task content that the answer choice for which the dataset is better described by the mean instead of the median is to be identified.
When outliers exist in a dataset, the mean is a better measure of center for the data set.
On this note, it follows that the answer choice for which the mean is a better description of the dataset is; 0, 57, 847, 859, 866, 866, 872.
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Solve the equation 25* = 625" by equating bases.
30kg of clay divided to portions of 5/8kg each and enough to distribute to all students. how many students are there
There are 48 students.
To find the number of students, we need to divide the total amount of clay by the weight of each portion.
We are given that there are 30kg of clay. We are also told that the clay is divided into portions of 5/8kg each.
To divide one quantity by another, we can use the following formula:
quantity ÷ weight per portion = number of portions
In this case, we can write:
30kg ÷ (5/8)kg per portion = number of portions
30kg × (8/5) portions per kg = number of portions
Simplifying, we get:
48 portions = number of portions
Therefore, there are 48 portions of clay in total. To find the number of students, we need to divide the number of portions by the number of portions per student.
Since each portion weighs 5/8kg, we can write:
1 portion = 5/8kg
number of portions per student = 1 portion/student
So, the number of students would be:
48 portions ÷ 1 portion per student = 48 students
Therefore, there are 48 students.
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A family has two cars. The first car has a fuel efficiency of 30 miles per gallon of gas and the second has a fuel efficiency of 20 miles per gallon of gas. During one particular week, the two cars went a combined total of 1700 miles, for a total gas consumption of 65 gallons. How many gallons were consumed by each of the two cars that week?
Let x be the number of gallons consumed by the first car.
Then, the number of gallons consumed by the second car is 65-x (since the total gas consumption is 65 gallons).
Using the formula distance = fuel efficiency x gas consumption, we can set up two equations:
First car: distance = 30x
Second car: distance = 20(65-x)
Since the total distance is 1700 miles, we can set up another equation:
Total distance: 30x + 20(65-x) = 1700
Simplifying this equation, we get:
30x + 1300 - 20x = 1700
10x = 400
x = 40
Therefore, the first car consumed 40 gallons and the second car consumed 65-40 = 25 gallons.
Let's assume that the first car traveled x miles and the second car traveled y miles during that week. We can set up two equations based on the given information:
x + y = 1700 (the combined total of miles traveled by both cars)
x/30 + y/20 = 65 (the total gas consumption)
To solve for x and y, we can use the first equation to express one variable in terms of the other. For example, we can solve for y as follows:
y = 1700 - x
Substituting this into the second equation, we get:
x/30 + (1700 - x)/20 = 65
Multiplying both sides by the common denominator 60, we can simplify the equation:
2x + 3(1700 - x) = 3900
2x + 5100 - 3x = 3900
-x = -1200
x = 1200
So the first car traveled 1200 miles during the week. We can use this value to find the second car's mileage:
y = 1700 - x = 1700 - 1200 = 500
Therefore, the second car traveled 500 miles during the week. To find the gallons of gas consumed by each car, we can use the fuel efficiency rates:
Gallons used by first car = 1200 miles / 30 mpg = 40 gallons
Gallons used by second car = 500 miles / 20 mpg = 25 gallons
So the first car consumed 40 gallons of gas and the second car consumed 25 gallons of gas during that week.
Ava started reading at 3:30 P.M. She stopped for a snack at 4:15 P.M.
During this time, through what fraction of a circle did the minute hand turn?
How many degrees did the minute hand turn?
Answer:
270 degrees
Step-by-step explanation:
There are 60 minutes on a clock face, and the minute hand of a clock makes one complete revolution around the clock face in 60 minutes. This means that the minute hand turns through the angle of 360 degrees in 60 minutes.
The time from 3:30 P.M. to 4:15 P.M. is 45 minutes.
So, during this time, the minute hand of the clock turns through a fraction of a circle equal to:
Fraction of a circle = (angle turned by the minute hand) / (angle for a complete circle)
= (45 minutes) / (60 minutes)
= 3/4
Therefore, the minute hand of the clock turned through 3/4 of a circle during this time.
To find the number of degrees turned by the minute hand, we can multiply the fraction of a circle turned by the total number of degrees in a complete circle:
Degrees turned by the minute hand = (Fraction of a circle turned) x (360 degrees)
= (3/4) x (360 degrees)
= 270 degrees
Therefore, the minute hand of the clock turned through 270 degrees during this time.
Side lengths 8 13 and x
A cylindrical swimming pool has a diameter of 14 feet and a height of 5 feet. How many gallons of water can the pool contain? Round your answer to the nearest whole number.(1 ft3 ≈ 7.5 gal)
Step-by-step explanation:
The volume of a cylinder can be calculated using the formula:
V = πr^2h
where r is the radius of the cylinder, and h is the height of the cylinder.
The diameter of the pool is 14 feet, so the radius is half of that, or 7 feet. The height is 5 feet. Therefore, the volume of the pool is:
V = π(7)^2(5) ≈ 769.39 cubic feet
We can convert this volume to gallons using the conversion factor given:
1 cubic foot ≈ 7.5 gallons
Therefore:
769.39 cubic feet * 7.5 gallons/cubic foot ≈ 5,770 gallons
So the pool can contain approximately 5,770 gallons of water (rounded to the nearest whole number).
A pottery class, 30kg of clay are divided into portion of 5/8kg each, and it just enough to distributes to all the students. how many students are there in the class?
There are 48 students in the pottery class
30kg of clay is divided into 5/8kg chunks, and there is just enough for all of the kids. To find the number of students in the class, we need to divide the total amount of clay by the amount of clay per student.
The amount of clay per student is 5/8kg.
So, number of students = (total amount of clay) / (amount of clay per student)
Dividing the total amount of clay by the amount of clay per student, we get:
30 / (5/8) = 30 x 8/5
= 48
Therefore, there are 48 students in the pottery class
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Help plsss!!!
Quadrilateral A' B' C' D' is the image of quadrilateral A B C D under a translation.
Quadrilateral A'B'C'D' is the image of quadrilateral ABCD under a translation of 2 units to the right and 5 units to left.
Given information:
Quadrilateral A'B'C'D' is the image of quadrilateral ABCD.
In Euclidean geometry, a translation is a geometric transformation that entails shifting each point in a figure, shape, or space by the same amount in one direction.
A translation may also be conceived of as changing the coordinate system's origin or as adding a constant vector to each point.
As per the diagram:
A(-5, 3) translates to A'(-3, -2).
As per the translation rule:
A translates to A' as per the rule of 2 units to the right and 5 units to left.
Similarly, all the remaining coordinates follow the same translation rule.
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Find the equation of the parabola with focus (-3,2) and directrix x-y + 1 = 0.?
Step-by-step explanation:
To find the equation of the parabola with focus (-3,2) and directrix x-y+1=0, we can use the definition of a parabola: the set of all points that are equidistant to the focus and directrix.
Let P(x, y) be an arbitrary point on the parabola, and let d(P, directrix) be the distance from P to the directrix. The distance from P to the focus is given by the distance formula:
d(P, focus) = √[(x - (-3))^2 + (y - 2)^2] = √[(x + 3)^2 + (y - 2)^2]
Since P is equidistant from the focus and directrix, we have:
d(P, directrix) = |x - y + 1| / √(1^2 + (-1)^2) = |x - y + 1| / √2
Therefore, the equation of the parabola is given by:
d(P, focus) = d(P, directrix)
√[(x + 3)^2 + (y - 2)^2] = |x - y + 1| / √2
Squaring both sides and simplifying, we get:
(x + 3)^2 + (y - 2)^2 = (x - y + 1)^2 / 2
Expanding the right-hand side and simplifying, we get:
2(x + 3)^2 + 2(y - 2)^2 = (x - y + 1)^2
Expanding the right-hand side again and simplifying, we get:
2x^2 + 8xy + 2y^2 - 8x - 12y + 20 = 0
Therefore, the equation of the parabola is:
2x^2 + 8xy + 2y^2 - 8x - 12y + 20 = 0
which is in general form.
Find the volume, in cubic inches, of the composite solid below, which consists of a rectangular prism box that has a cube shape cut out of the center of the box. All measurements shown are in inches. Enter only the number. An image shows a rectangular box that has length of 8 inches, width of 6 inches and height of 6 inches. The cube is 4 inches on each sides. The solution is
The composite solid has a volume of 224 inches³.
Define the term volume?Volume is a mathematical notion that indicates how much space in three dimensions a solid, liquid, or gas occupy.
It is commonly measured in cubic length units like cubic metres, cubic feet, or cubic centimetres.
The volume of the cube must be subtracted from the volume of the rectangular prism in order to determine the volume of the composite solid.
Volume of the rectangular prism = length x width x height = 8 x 6 x 6 = 288 cubic inches
Volume of the cube = side [tex]length^3[/tex] = [tex]4^3[/tex] = 64 cubic inches
The composite solid's volume is thus:
Volume of rectangular prism - Volume of cube = 288 - 64 = 224 cubic inches.
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For any spin of the wheel, what is the probability of NOT getting a number 1 through 18? a.) 9 over 19 b.) 10 over 19 c.) 1 over 19 d.) 18 over 19
The probability of not getting a number 1 through 18 is b) 10 over 19.
There are a total of 38 slots on the roulette wheel, and 18 of them are numbered 1 through 18. Therefore, there are 20 slots that are not numbered 1 through 18. The probability of not getting a number 1 through 18 on a single spin of the roulette wheel is the same as the probability of getting one of the 20 non-numbered slots.
Thus, the probability of not getting a number 1 through 18 is 20/38 or 10/19.
Therefore, the answer is option b) 10 over 19.
Correct Question :
A roulette wheel has 38 slots total, 36 of which are numbered 1 through 36, and 2 green slots labeled "O" and "00." For any spin of the wheel, what is the probability of NOT getting a number 1 through 18?
a.) 9 over 19
b.) 10 over 19
c.) 1 over 19
d.) 18 over 19
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What is the domain of the equation?
The domain of the equation include the following: D. all real numbers.
What is a domain?In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.
How to identify the domain any graph?In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.
By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:
Domain = [-∞, ∞] or all real numbers.
Range = [-4, ∞}
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I’m completely lost pls help
The probability that exactly 2 of the pencils in the package are painted green is given as follows:
p = 9/30.
How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
In this problem, we have a set of 30 outcomes, in which the number of outcomes in which the letter G appears exactly twice is given as follows:
9.
Hence the probability that exactly 2 of the pencils in the package are painted green is given as follows:
p = 9/30.
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Exercise 2: We are costing out our drink menu, and wish to compare the costs and contribution margin
of our "house" vodka Czarina, vs. a "premium" brand Grey Goose. Include 1 olive garnish and cocktail
napkin.
Equivalents: 750 ml = 25.4 oz 1 Liter = 33.8 oz
DRINK RECIPE:
"House Vodka Martini"
2 oz Czarina Vodka
½ oz Dry Vermouth
Selling Price $4.50
"Grey Goose Vodka Martini"
2 oz Grey Goose Vodka
½ oz Dry Vermouth
Selling Price $5.50
How do the contribution margins of the two drinks compare? Can our less expensive drinks make more
$$ than expensive brands? What would be your recommendations to upper Mgmt. on the pricing of
these two drinks in the menu?
The "House Vodka Martini" may sell less when its price is raised since some consumers could prefer the premium brand.
How to solve
Contribution Margin = Selling Price - Total Cost
Let's start with the "House Vodka Martini":
Czarina Vodka: 2 oz * (Czarina cost per oz)
Dry Vermouth: 0.5 oz * (Dry Vermouth cost per oz)
Olive garnish: 1 * (Olive cost per unit)
Cocktail napkin: 1 * (Napkin cost per unit)
Total Cost (House Vodka Martini) = sum of the costs above
Contribution Margin (House Vodka Martini) = $4.50 - Total Cost (House Vodka Martini)
Let's solve for "Grey Goose Vodka Martini":
Grey Goose Vodka: 2 oz * (Grey Goose cost per oz)
Dry Vermouth: 0.5 oz * (Dry Vermouth cost per oz)
Olive garnish: 1 * (Olive cost per unit)
Cocktail napkin: 1 * (Napkin cost per unit)
Total Cost (Grey Goose Vodka Martini) = sum of the costs above
Contribution Margin (Grey Goose Vodka Martini) = $5.50 - Total Cost (Grey Goose Vodka Martini)
Other considerations, like client preferences and the drinks' perceived value, must also be taken into account. The "House Vodka Martini" may sell less when its price is raised since some consumers could prefer the premium brand.
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HELP PLEASEE
01 yards
01/15
04 yards
yards
yards
Question 6(Multiple Choice Worth 2 points)
(Factoring MC)
The requried total amount of metal needed for the project is 4 1/2 yards.
The first piece of metal measures 1 1/5 yards, which is equivalent to 6/5 yards. The second piece of metal measures 3/10 yards. To find the total amount of metal needed, we need to multiply the sum of the two pieces by 3:
Total amount of metal = 3 x (6/5 + 3/10) yards
Simplifying the expression inside the parentheses, we have:
Total amount of metal = 3 x (12/10 + 3/10) yards
Total amount of metal = 3 x 15/10 yards
The total amount of metal = 9/2 yards
Total amount of metal = 4 1/2 yards
Therefore, the total amount of metal needed for the project is 4 1/2 yards.
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The equation that represents ⨀A is (x+1)2+(y−1)2=16. Determine whether point B(3,1) is on the circle.
Answer:
Step-by-step explanation:
center=(-1,1)
radius=√16=4
distance between (-1,1) and (3,1)=√[(3+1)²+(1-1)²]=√[16+0]=√16=4=radius
Hence point lies on the circle.
write a verbal expression for 0.7x-11
The verbal expression for 0.7x-11 is "seven tenths of x minus eleven."
What is verbal expression in maths?A scenario expressed using mathematical terms is condensed into a conversational statement. It identifies the connections between and movements made by amounts. Example: The product of a number x and one multiplied by five.
Words are present in verbal utterances. For instance, the phrase "three more than a double a number" is a verb. The algebraic equivalent of the phrase "three more than a double a number" is the expression 2x + 3.
Verbal models are another name for verbal expressiveness. When we communicate ourselves verbally, we describe the mathematical data using language. An expression becomes a verbal expression when it is written in terms or spoken in words. However, compared to verbal expression, an algebraic expression is entirely different.
The verbal expression for 0.7x-11 is "seven tenths of x minus eleven."
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Please help with the attached problem
give question oh t is so easy
Sasha is bisecting a segment. First, she places the compass on one endpoint, opens it to a width larger than the compass the same width and places it on the other endpoint. What is her next step?
Swing arcs on both sides to intersect the first two arcs created.
Swing an arc that intersects the segment.
Swing arcs that intersect a point that is not on the segment.
Swing an arc that intersects the opposite endpoint.
Sasha's next step is to swing arcs on both sides of the segment to intersect the first two arcs created. Option A is correct.
By doing this, she will be creating a two points of intersection on the segment. She can then draw a straight line which is connecting these two points, which will bisect the segment into two equal parts. This method of bisecting a segment is called as the compass-and-straightedge construction, and it is a fundamental technique in Euclidean geometry.
Alternatively, if you want to bisect a segment without using the compass-and-straightedge construction, you can measure the length of the segment and mark its midpoint. Then you can draw a line connecting the two endpoints through the midpoint to bisect the segment.
Hence, A. is the correct option.
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--The given question is incomplete, the complete question is
"Sasha is bisecting a segment. First, she places the compass on one endpoint, opens it to a width larger than the compass the same width and places it on the other endpoint. What is her next step? A) Swing arcs on both sides to intersect the first two arcs created. B) Swing an arc that intersects the segment. C) Swing arcs that intersect a point that is not on the segment. D) Swing an arc that intersects the opposite endpoint."--
Choose the inequality whose solution set is (-14, ♾️)
Choose the inequality whose solution set is [-9,♾️)
Write the solution set to x (greater than or equal too) 7 in interval notation
Write the solution set to x (greater than or equal too) -8 in interval notation
With the solution set using interval notation x < 3.14/3 and x< 3.14/ 6
The inequality with the set of solutions (-14, infinity ) can be expressed as follows: x > -14
What is an inequality?Inequalities are simply created through the connection of two expressions. In this case, it should be noted that the expressions in an inequality are not always equal. Inequalities implies that the expressions are not equal. Here, they are denoted by the symbols ≥ < > ≤.
Any value of x bigger than -14 will fulfill the inequality, according to this inequality, and the solution set contains all real numbers greater than -14 but excluding -14.
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A construction company is repaving 27 miles of a road. They can repave 2.4 miles per day.
How many days, to the nearest hundredth, will it take the company to repave the road?
Answer:
It would take the company 11.25 days to repave the road.
Step-by-step explanation:
27 miles to repave total
2.4 repavement per day
27 / 2.4 = 11.25
Therefore, it would take the company 11.25 days to repave the road.
Suppose the manufacturer of a board game will supply 10,000 games if the wholesale price is $1.50 each but will supply only 5000 if the price is $1.00 each. Assuming that the supply function is linear, write its equation. Use p for price and q for quantity.
The equation of the supply function is q = 20000p - 10000
How to determine the equation of the supply functionLet's use the point-slope form of a linear equation:
slope = (change in quantity) / (change in price) = (10000 - 5000) / (1.50 - 1.00) = 10000 / 0.50 = 20000
Using the point-slope form with the point (1.50, 10000):
q - 10000 = 20000(p - 1.50)
Simplifying:
q = 20000p - 20000 + 10000
q = 20000p - 10000
So the equation of the supply function is q = 20000p - 10000
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Solids A and B are similar. Use the given information and scale factor k from solid A to solid B to find the volume of solid B.
Round your answer to the nearest hundredth if necessary.
The volume of cone A is 167 cubic centimeters and k =
3
2
The volume of cone B is cubic centimeters.
4
The volume of cone B is of 2.37π cubic centimeters.
What is a dilation?A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.
The scale factor in this problem is given as follows:
k = 3/2.
Hence:
A/B = 3/2.B/A = 2/3.The scale factor measures the ratio of the side lengths, in units, while the volume is given in cubic units, hence the ratio of the volumes is given as follows:
Vb/Va = (2/3)³
Vb/Va = 4/27
Hence the volume of solid B is given as follows:
Vb = 4/27 x 16π
Vb = 2.37π.
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Answer:
Answer:
169.65
Step-by-step explanation:
Solving ones like these, I use the formula: Volume b/Volume a=k^3. I will be using “•” to represent multiplication, “^” is used for exponent. 16•pie is the volume for shape A, and k is 3/2. The reason why k is exponent by 3 is because this is for volume. Then, plug in the values, Volume B/16•pi=3/2^3 is what we have. With the piwer of inverse operations, we can isolate for Volume B. By multiplying 16•pi by each side, Volume B=16•pi•3/2^3. Lets do 3/2^3 first since it is following the order of PEMDAS. To recall what PEMDAS is, it is a method my school uses for solving equations. Parentheses first, Exponent next, then Multiply, Divide, Add, and subtract. Basically the order of solving any equation. 3/2^3= 27/8. Now, back to multiplying! I use desmos scientific calculator. 27/8•16•pi= =169.6460033. The qeustion says to round to the nearest hundreth, so it would be 169.65. Have a great day everybody
Step-by-step explanation:
8. The expression
(x^(2a+1))³/(x^(a+3)) ²
can be written as x^n, where n depends on the value of a.
(a) If a = 5, then find the value of n. Show your
work.
(b) Find a binomial expression for n in general
terms of “a”
a. If a = 5, the value of n will be 17.
(b) A binomial expression for n in general
terms of “a” will be n = 4a - 3.
How to calculate the value(a) In the case that a is equal to 5, then the expression transforms into:
(x^(2a+1))³/(x^(a+3))² = (x^(2(5)+1))³/(x^(5+3))² = (x^11)³/(x^8)² = x^(33-16) = x^17
Therefore, for a = 5, n = 17.
(b) We can simplify this by making use of the rules of exponents:
(x^(2a+1))³/(x^(a+3))² = x^(6a+3)/(x^(2a+6)) = x^(6a+3-2a-6) = x^(4a-3)
Therefore, the binomial expression for n in terms of a equals: n = 4a - 3
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Triangle XYZ is an isosceles triangle.
XY = YZ
mY = 42°
What is mZZ?
Y
42°
N
If triangle XYZ is isosceles, then the two sides (XY and YZ) are congruent in addition to the two base angles.
Angle Y (the one at the top of the triangle) = 42 degrees
Angles X and Z are congruent because of the properties of an isosceles triangle. Let's call both angle measures "x" in our equation.
x + x + 42 = 180
2x + 42 = 180
2x = 138
x = 69
Answer: The measure of Angle Z is 69 degrees.
Hope this helps!
Find all solutions of the equation in the interval [0, 2).
sin x-1= cos x
Write your answer(s) in radians in terms of π.
If there is more than one solution, separate them with commas.
x=
Answer: The solutions to the equation sin x - 1 = cos x in the interval [0, 2) are x = -π/2 and x = 3π/4, in radians in terms of π.
Step-by-step explanation:sin x - 1 = cos x
Subtracting cos x from both sides:
sin x - cos x - 1 = 0
Using the identity sin(x - π/4) = sin x cos π/4 - cos x sin π/4, we get:
sin(x - π/4) = -1/√2
Taking the inverse sine of both sides, we get:
x - π/4 = -π/4 - π/2 = -3π/4
or
x - π/4 = π/2 + π/4 = π/2
Adding π/4 to both sides:
x = -3π/4 + π/4 = -π/2
or
x = π/2 + π/4 = 3π/4
Note that the interval [0, 2) contains 0, π/2, and π, but none of these values satisfy the equation. Therefore, the solutions in the given interval are:
x = -π/2 and x = 3π/4.
Hence, the solutions to the equation sin x - 1 = cos x in the interval [0, 2) are x = -π/2 and x = 3π/4, in radians in terms of π.
D is the midpoint of CE. E has coordinates (5,-10), and D has
coordinates (11,4). Find the coordinates of C.
[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ C(\stackrel{x_1}{x}~,~\stackrel{y_1}{y})\qquad E(\stackrel{x_2}{5}~,~\stackrel{y_2}{-10}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 5 +x}{2}~~~ ,~~~ \cfrac{ -10 +y}{2} \right) ~~ = ~~\stackrel{\textit{\LARGE D} }{(11~~,~~4)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ 5 +x }{2}=11\implies 5+x=22\implies \boxed{x=17} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ -10 +y }{2}=4\implies -10+y=8\implies \boxed{y=18}[/tex]
