Answer: Part A:
If Isabella's mother has a garden with an area of 10 2/3 square feet, and she plants peas in 1/4 of the garden, we can calculate the area used for planting peas by multiplying the garden area by 1/4:
Area used for planting peas = (10 2/3 square feet) x (1/4) = (32/3) x (1/4) = 8/3 square feet
Therefore, the area used for planting peas is 8/3 square feet.
Part B:
If Isabella's mother sells the peas and gets 3 dollars for each square foot, we can calculate how much money she made by multiplying the area used for planting peas by 3:
Money made = (8/3 square feet) x (3 dollars/square foot) = 8 dollars
Therefore, Isabella's mother made 8 dollars by selling the peas she planted in her garden.
Your friend estimates that a bookcase is 2 1/2 feet wide. The actual width is 2/3 foot longer. What is the width of the bookcase?
I need the answers quick please !!
The number of triangles that can be formed from a common vertex on the polygon shown is three triangles.
How to find the number of triangles ?To form triangles on polygons, you can draw diagonals connecting the vertices of the polygon. A diagonal is a line segment that connects two non-adjacent vertices of a polygon. By drawing diagonals, you can create triangles within the polygon.
From a hexagon, we can draw three diagonals from each vertex. So, from a common vertex, we can draw three diagonals and form three triangles. Since there are six vertices in a hexagon, we can choose any one of these vertices as the common vertex and form three triangles.
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(Select all correct answers)
Consider the expression ([tex]x^{2}[/tex]−9) (x+4).
Select all the values of x for which ([tex]x^{2}[/tex]−9) (x+4)=0
3
9
-9
-3
-4
4
The only value of x for which (-9)(x+4) = 0 is x = -4.
Define the term expression?Calculations that include changeable altering multipliers, joining, removal, and random subdivision must be done. They could accomplish the following if they united: An algorithm, some data, and a mathematical problem.
The expression (-9)(x+4) can be simplified by using the distributive property of multiplication:
(-9)(x+4) = -9x - 36
To find the values of x for which (-9)(x+4) = 0, we can set the expression equal to zero and solve for x:
-9x - 36 = 0
Adding 36 to both sides gives:
-9x = 36
Dividing both sides by -9 gives:
x = -4
Therefore, the only value of x for which (-9)(x+4) = 0 is x = -4.
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Rowan is taking his siblings to get ice cream. they can't decide whether to get a cone or a cup because they want to get the most ice cream for their money. if w = 4 in, x =6 in, y = 6 in, z = 2 in, and the cone and cup are filled evenly to the top with no overlap, which container will hold the most ice cream? use 3.14 for π, and round your answer to the nearest tenth. a cone with a height of x and a radius of w, a cylinder with a diameter of y and a height of z the cup holds 56.52 in3 more ice cream than the cone. the cone holds 56.52 in3 more ice cream than the cup. the cup holds 43.96 in3 more ice cream than the cone. the cone holds 43.96 in3 more ice cream than the cup.
Therefοre , the sοlutiοn οf the given prοblem οf area cοmes οut tο be The cοne hοlds 43.96 in³mοre ice cream than the cup, sο that is the cοrrect respοnse.
What precisely is an area?Calculating hοw much space wοuld be needed tο fully cοver the οutside will reveal its οverall size. When determining the surface οf such a trapezοidal fοrm, the surrοundings are additiοnally taken intο accοunt. The surface area οf sοmething determines its οverall dimensiοns. The number οf edges here between cubοid's fοur trapezοidal extremities determines hοw much water it can hοld inside.
Here,
We must determine the vοlumes οf the cοne and the cup in οrder tο cοmpare hοw much ice cream each cοntainer can carry.
=> Cοne vοlume = (1/3)r²h.
=> Cup vοlume = (1/4)d²h
where r is the cοne's radius (w/2), h is the cοne's height (x), d is the cup's width (y), and z is the cup's height.
Inputting the numbers prοvided yields:
=> Cοne vοlume = (1/3)π(2²)(6) = 16π cubic inches
=> Cup vοlume = (1/4)π(6²)(2) = 9π cubic inches
As a result, the cοne has a larger capacity than the cup.
The cοne hοlds 43.96 in³mοre ice cream than the cup, sο that is the cοrrect respοnse.
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brainlest if correct write an equation
Answer:
y = x - 2
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (0, - 2) and (x₂, y₂ ) = (2, 0) ← 2 points on the line
m = [tex]\frac{0-(-2)}{2-0}[/tex] = [tex]\frac{0+2}{2}[/tex] = [tex]\frac{2}{2}[/tex] = 1
the line crosses the y- axis at (0, - 2 ) ⇒ c = - 2
y = x - 2 ← equation of line
Tell whether the ordered pair is a solution of the linear system.
(-2,-3).
y=3x+3.
y=-x-5
The given ordered pair is not a solution of the linear system.
What is linear equations?Linear equations are mathematical equations that describe the relationship between two or more variables in a straight line. They can be written in the form of:
y = mx + b
According to question:To check if the ordered pair (-2, -3) is a solution to the system of linear equations, we need to substitute x = -2 and y = -3 into both equations and see if they are satisfied.
For the first equation y = 3x + 3, we have:
-3 = 3(-2) + 3
-3 = -3
This is true, so (-2, -3) satisfies the first equation.
For the second equation y = -x - 5, we have:
-3 = -(-2) - 5
-3 = 3
This is false, so (-2, -3) does not satisfy the second equation.
Since (-2, -3) satisfies the first equation but not the second equation, it is not a solution to the system of linear equations. Therefore, the answer is no.
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Is this right?Help please
Answer: They are not similar.
Step-by-step explanation: The angle measures are not the same for both triangles which makes them not similar to each other.
solve the system of equations -3x-5y=14 and 7x+7y=0 by combining the equations.
Answer:
x=7 & y=-7
Step-by-step explanation:
from the 2nd equation 7x+7y=0 we can get that
x=-y
substitute that into the first equation to get
3y-5y=14
-2y=14
y=-7
then
x=7
Find the slope show your work you don't have to do them all I just need some of them done please
Answer: 1. Slope equals 3/4 or 0.75, 2. Slope equals -5/2 or -2.5
Step-by-step explanation:
1. Subtract y coordinates 4 and -2 to get 6. Then subtract x coordinates 4 and -4 to get 8, so that is equal to 6/8. Then simplify to get 3/4 or 0.75
2. Subtract y coordinates 10 and -5 to get -15. Then subtract x coordinates 0 and -6 to get 6, so that is equal to-15/6. Then simplify to get -5/2 or -2.5.
I need some help on these 4 questions
Therefore , the solution of the given problem of function comes out to be f(N) = 360/N the cost per individual for any number of contributors to the gift is provided by this function.
What does function actually mean?The math lesson includes a wide range of topics, such as mathematics, integers, as well as one's divisions, architecture, and both actual mostly and fictitious geographic locations. The relationships between various components that all cooperate to create the same outcome are covered by a work. A utility is composed of numerous unique elements that work together to produce unique outcomes for each input.
Here,
To figure out how much each individual would spend, we need to divide the cost of the gift by the number of people contributing. Here are the estimates for various populations:
Each person would spend $180 ($360 / 2) if there were two people splitting the expense.
The expense would be divided among three people, who would each pay
=> $120 = ($360 / 3).
If the cost were divided among five individuals, each person would pay => $72 = ($360 / 5).
Ten people would split the expense equally, each spending
=> $36 = ($360 / 10).
Each person would spend
=> $3.60 = ($360 / 100)
if there were 100 people splitting the expense.
II. The algorithm is as follows:
Cost per individual equals total expense / number of individuals
=> f(N) = C/N
where f(N) is the price per individual.
The function for the cost per individual, for instance, is:
=> f(N) = 360/N
The cost per individual for any number of contributors to the gift is provided by this function.
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This is the same information from the prior question.
Skidding distance can be determined using the formula d space equals space 2 square root of 5 x end root where d is the distance (in feet) and x is the speed (in miles per hour).
Calculate the approximate speed of the car when the skid marks are 20 feet long.
The approximate speed of the car when the skid marks are 20 feet long is 8.944 miles per hour.
What is importance of a formula?A formula is an expression in mathematics that denotes a connection or rule. In mathematics, science, and engineering, numbers and variables are described and computed using formulas. They can be stated verbally or symbolically, such as in verbal descriptions, or mathematically or symbolically, such in algebraic equations. Formulas can be applied to evaluate data, solve issues, and make predictions. They are utilised in a variety of disciplines, including physics, chemistry, biology, finance, and statistics, and are crucial for comprehending and explaining the physical world.
The skidding distance is given by the formula:
d = 2 * √(5x)
Isolating the value of x we have:
x = (d / (2 * √(5)))
Substituting the value of d = 20 we have:
x = (20 / (2 * √(5))) ≈ 8.944 miles per hour
Hence, the approximate speed of the car when the skid marks are 20 feet long is 8.944 miles per hour.
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27. If T and P are positive numbers,which of the following is always false?
A. P-T>O
B. P+T=0
C. P + T = 2P
D. P+T> P
Answer:
B
Step-by-step explanation:
The only way for two numbers to = zero when they're added together is if one is negative. There are no two positive numbers that when added together will equal 0, therefore if p and t are positive, their sum cannot be 0
Calculate the average daily balance, finance charge, and new balance using the average daily balance method.
The account balance on June 1st is $104. 0. On June 19th a payment of $80. 25 is made. The monthly rate is 1. 25%
Date
6/1 - 6/18
6/19
Payments Purchases Balance Number of Days Product/Sum
$104. 00 18
1,872. 00
$80. 25
$23. 75
23. 75
$23. 75
261. 25
2,157. 00
-
6/20 - 6/30
=
Total
The average daily balance = 2,157. 00 = 30 = $ 71. 90
Finance charge = monthly rate x average daily balance = $
New balance = previous balance - payment/credits + finance charge + new purchases = $
The average daily balance is $1,000, the finance charge is $14.7, and the new balance is $3,029.4.
To calculate the average daily balance, finance charge, and new balance using the average daily balance method, follow the below steps:
Step 1: Obtain the balance owed for each day. To do so, determine the balance at the beginning of each day and the amount charged or credited during the day.
Step 2: Add up the balances for each day during the billing cycle, then divide by the number of days in the billing cycle. This calculation results in the average daily balance.
Step 3: Multiply the average daily balance by the daily interest rate to calculate the finance charge.Step 4: Add the finance charge to the previous balance to get the new balance. Example: Suppose an account has the following daily balances and APR, calculate the average daily balance, finance charge, and new balance using the average daily balance method:
Day 1 balance: $1,000Day 2 balance: $1,200Day 3 balance: $800Day 4 balance: $1,000APR: 18%The number of days in the billing cycle is 30.Average daily balance:($1,000 + $1,200 + $800 + $1,000) ÷ 4 = $1,000Finance charge:$1,000 × 0.18 ÷ 365 = $0.49 per day.$0.49 × 30 = $14.7New balance: Previous balance + finance charge = $3,014.7 + $14.7 = $3,029.4
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A market researcher for a consumer electronics company wants to study the media viewing behavior of residents of a particular area. A random sample of 80 respondents is selected, and each respondent is instructed to keep a detailed record of time spent engaged viewing content across all screens in a particular week. The results are as follows: Content viewing time per week: X = 44 hours, S = 5.7 hours. 28 respondents have high definition (HD) on at least one television set. Complete parts (a) through (e) below. a. Construct a 90% confidence interval estimate for the mean content viewing time per week in this area. Isusa (Round to two decimal places as needed.) b. Construct a 90% confidence interval estimate for the population proportion of residents who have HD on at least one television set. Osas (Round to four decimal places as needed.) c. Suppose that the market researcher wants to take another survey in a different location. What sample size is required to be 90% confident of estimating the population mean content viewing time to within + 2 hours assuming that the population standard deviation is equal to 6 hours? (Type a whole number.) d. How many respondents need to be selected to be 90% confident of being within 10.05 of the population proportion who have HD on at least one television set if no previous estimate is available? (Type a whole number.) Enter your answer in each of the answer boxes. e. Based on (C) and (d), how many respondents should the market researcher select if a single survey is being conducted? (Type a whole number.) Enter your answer in each of the answer boxes.
a. Construct a 90% confidence interval estimate for the mean content viewing time per week in this area. Isusa
90% confidence interval estimate for the mean content viewing time per week: (41.75, 46.25)
b. Construct a 90% confidence interval estimate for the population proportion of residents who have HD on at least one television set. Osas
90% confidence interval estimate for the population proportion: (0.1867, 0.3833)
c. Suppose that the market researcher wants to take another survey in a different location. What sample size is required to be 90% confident of estimating the population mean content viewing time to within + 2 hours assuming that the population standard deviation is equal to 6 hours? (Type a whole number.)
Sample size required: 653
d. How many respondents need to be selected to be 90% confident of being within 10.05 of the population proportion who have HD on at least one television set if no previous estimate is available? (Type a whole number.) Enter your answer in each of the answer boxes.
Sample size required: 461
e. Based on (C) and (d), how many respondents should the market researcher select if a single survey is being conducted? (Type a whole number.) Enter your answer in each of the answer boxes.
Sample size required: 653
f(x)=-2x^2+6x+1 what is the rate of change
The rate of change of the function f(x) = -2x² + 6x + 1 at x = 1 is approximately 0.302.
What is the definition of a function?
In mathematics, a function is a rule that assigns to each element in a set (called the domain) exactly one element in another set (called the range). In other words, a function takes an input value and produces a unique output value. The input value is usually represented by the variable x, while the output value is represented by the variable y or f(x).
Now,
The rate of change of a function is the slope of the line that connects any two points on the graph of the function. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
slope = (y2 - y1)/(x2 - x1)
So to find the rate of change of the function f(x) = -2x² + 6x + 1, we need to choose two values of x and find the corresponding values of f(x), and then calculate the slope of the line passing through these two points.
Let's choose two values of x that are close to each other, say x1 = 1 and x2 = 1.01. Then, we have:
f(x1) = -2(1)² + 6(1) + 1 = 5
f(x2) = -2(1.01)² + 6(1.01) + 1 = 5.0302
Now, we can calculate the slope of the line passing through these two points:
slope = (f(x2) - f(x1))/(x2 - x1) = (5.0302 - 5)/(1.01 - 1) = 0.302
So,
the rate of change of the function f(x) = -2x² + 6x + 1 at x = 1 is approximately 0.302. Note that the rate of change of a function is not constant, but varies with the value of x.
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4) Two cars leave the same parking lot, with one heading north and the other heading east.
After several minutes, the northbound car has traveled 8 kilometers, and the eastbound car
has traveled 2 kilometers. Measured in a straight line, how far apart are the two cars? If
necessary, round to the nearest tenth.
Video
Answer:
8.2km
Step-by-step explanation:
North and East are perpendicular directions;
If you illustrate the distances covered in each direction and join the two lines, it will form a right-angle triangle;
This means pythagoras applies:
(8)² + (2)² = 64 + 4 = 68
The distance apart is, therefore:
[tex] \sqrt{68} = 8.2462112512 [/tex]
What is the surface area of the three-dimensional figure represented by this net?
The net pattern has a square base of side lengths 3 inches, 3 inches, 3 inches and 3 inches are attached with four congruent triangles of height 8 inches.
Use the on-screen keyboard to type the correct number of square inches in the box below.
$$
The surface area of the three-dimensional figure represented by this net is 54 square inches, which is calculated by multiplying the area of the base (9 square inches) by the height of the four congruent triangles (8 inches).
The surface area of the three-dimensional figure represented by this net is 54 square inches. To calculate this, first we need to find the area of the base. The base is a square, with side lengths of 3 inches each. Therefore, the area of the base is 3 x 3 = 9 square inches. Next, we need to find the height of the four congruent triangles. The height is 8 inches. Finally, we can calculate the surface area by multiplying the area of the base (9 square inches) by the height of the four congruent triangles (8 inches). This gives us a total surface area of 54 square inches.
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1 Divide the given polynomial by the given monomial. (1) (5x² - 6x) ÷ 3x C C 31 2-2-2 (V)
The solution to the expression (5x² - 6x) / 3x is (5/3)x - 2
What is an equation?An equation is an expression that shows the relationship between numbers and variables using mathematical operators such as addition, subtraction, exponent, multiplication and division.
Equations can be classified based on degree as linear, quadratic, cubic and so on.
Given the polynomial:
5x² - 6x
Dividing the polynomial by the monomial 3x:
= (5x² - 6x) / 3x
= [x(5x - 6)] / 3x
= (5x - 6) / 3
= (5/3)x - 2
The solution to the equation is (5/3)x - 2
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HELP ASAP is the relationship between the amount of dry dog food and the amount of wet dog food proportional the numbers are for dry god food (cups) (Y) there 4.5 6 9
for wet dog food (oz) (x) SHOW WORK THX
Yes, there is direct relatiοn in amοunt οf dry dοg fοοd and the amοunt οf wet dοg fοοd.
If amοunt οf dry dοg fοοd is x and the amοunt οf wet dοg fοοd is y then the relatiοn is given as: x = 3y
Hοw tο find the prοpοrtiοnality?Tο determine whether the relatiοnship between the amοunt οf dry dοg fοοd and the amοunt οf wet dοg fοοd is prοpοrtiοnal, we need tο check whether the ratiο οf the twο quantities remains cοnstant.
A) yes, there is direct relatiοn in amοunt οf dry dοg fοοd and the amοunt οf wet dοg fοοd. as amοunt οf dry dοg fοοd increases there alsο increase in amοunt οf wet gοοd.
if amοunt οf dry dοg fοοd is x and the amοunt οf wet dοg fοοd is y then the relatiοn is given as:
x = 3y
fοr example lets take values frοm first rοw
4.5 = 3 × 1.5
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Complete question:
4 points
A 400ft tall building cast a shadow that is 75ft long. What is the angle of
depression from the sun, that makes this shadow? (Round to the nearest
tenth) *
Your answer
The angle of depression from the sun that makes the shadow is 69.1°.
The angle of depression from the sun can be calculated using the formula, tan θ = h/l, where θ is the angle of depression, h is the height of the building, and l is the length of the shadow. In this case, h is 400ft, and l is 75ft. Plugging these numbers into the formula gives us tan θ = 400/75, which simplifies to [tex]θ = tan^-1(400/75)[/tex]. Using a calculator, this gives us an angle of depression of θ = 69.1°. Therefore, the angle of depression from the sun that makes the shadow is 69.1°.
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PLEASE HELP!! SPECIAL RIGHT TRIANGLES I"LL GIVE YOU BRAINLIEST OR WHATEVER
Using the trigonometry function, the value of y = 3√2.
What exactly does math trigonometry mean?The area οf mathematics knοwn as trigοnοmetry is cοncerned with analyzing the relatiοnships between the sides and angles οf a right-angled triangle. A trigοnοmetric ratiο called the sides' ratiο serves as a representatiοn οf the relatiοnship. Trigοnοmetric ratiοs cοme in six varieties: sine, cοsine, tangent, cοtangent, secant, and cοsecant.
Trigοnοmetric functiοns are used tο calculate the lengths and angles οf geοmetric fοrms frοm knοwn οr measured angles. As a result οf the need tο cοmpute angles and distances in numerοus fields, including astrοnοmy, mapmaking, surveying, and artillery range finding, trigοnοmetry was develοped.
Since it is a 90-60-30 triangle, y is s √2
s = 3
so, y = 3√2
y = 3√2
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PLEASE I NEED HELP BADLY.
We have drawn the number line and included the given points, see attached number line.
What is the fraction?
A fraction is used to represent the portion/part of the whole thing. It represents the equal parts of the whole. A fraction has two parts, namely the numerator, and denominator. The number on the top is called the numerator, and the number on the bottom is called the denominator.
On the number line above, the decimal values and fractions are labeled as follows:
0.63 is labeled between 0 and 1.
34/100 is labeled at the same point as 0.34, between 0 and 1.
0.02 is labeled between 0 and 1.
7/10 is labeled at the same point as 0.7, between 0 and 1.
Therefore, we have drawn the number line and included the given points, see attached number line.
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The pressure that a box exerts on a shelf is 200 N/m². The force that the box exerts on the shelf is 180 N. Work out the area of the base of the box. If your answer is a decimal, give it to 1 d.p. Question attached!
Step-by-step explanation:
To solve this problem, we can use the formula for pressure:
pressure = force / area
We are given the pressure and the force, so we can rearrange the formula to solve for the area:
area = force / pressure
Substituting the values we have:
area = 180 N / 200 N/m²
area = 0.9 m² (to 1 decimal place)
Therefore, the area of the base of the box is 0.9 square meters.
Pleas can someone help!
Answer:
99.6 cm²
Step-by-step explanation:
You want to know the area of a sector that has radius 6 cm, and subtends an arc of 317°.
Sector areaThe formula for the area of a sector is ...
A = 1/2r²θ . . . . . . where θ is the central angle in radians
ApplicationA = 1/2(6 cm)²(317·π/180) ≈ 99.6 cm²
The area of the sector is about 99.6 square centimeters.
a scientist recorded the duration of the eruptions of the old faithful geyser in yellowstone national park that occurred during a one-month time period. the histogram below shows the distribution of the duration, in seconds, of the eruptions.
A scientist recorded the duration of the eruptions of the Old Faithful geyser in Yellowstone National Park that occurred during a one-month time period. The histogram below shows the distribution of the duration, in seconds, of the eruptions. By analyzing this data, the scientist can gain a better understanding of the behavior of the Old Faithful geyser and potentially make predictions about future eruptions.
The histogram shows that the majority of the eruptions had a duration between 120 and 140 seconds. There were also a significant number of eruptions with a duration between 100 and 120 seconds, and between 140 and 160 seconds.
It can also help them identify any unusual eruptions that may have a significantly shorter or longer duration than the majority of the eruptions
This data can help the scientist understand the typical duration of eruptions at the Old Faithful geyser in Yellowstone National Park..
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40 is what percent of 80?
Answer:
50%
Step-by-step explanation:
it's just half 80 so it's 50%
What is the equation of the line?
Answer:
[tex]y = -\dfrac{1}{4}x + 1[/tex]
Step-by-step explanation:
Slope of the line can be found by taking two points on the line, (x1, y1) and (x2, y2)
Find the difference in y coordinates, y2 - y1 (called rise)
Find the difference in corresponding x coordinates x2 - x1 (called run)
rise/run gives the slope
Two distinguishable points on the line graph are (0, 1) and (4, 0)
rise = 0 - 1 = - 1
run = 4 - 0 = 4
Slope = -1/4
The equation of the line in general form is
y = mx + b
where
m = slope
b = y-intercept (value of y when x = 0)
Here m = -1/4
b = 1
So equation of the line is
[tex]y = -\dfrac{1}{4}x + 1[/tex]
Calculate the difference between the amount of commission for selling goods worth Rs.20 lakhs and Rs .30 lakes. Sales between 15-25 lakhs is 1% and sales between 25-54 is 1.5%.
The difference in commission for selling goods worth Rs. 20 lakhs and Rs. 30 lakhs is Rs. 7,500.
How to calculate the difference between the amount of commission for selling goods ?
The commission on sales worth Rs. 20 lakhs would be:
1% of (20 lakhs) = 0.01 x 20,00,000 = Rs. 20,000
The commission on sales worth Rs. 30 lakhs would be:
1% of (25 lakhs) + 1.5% of (5 lakhs) = (0.01 x 25,00,000) + (0.015 x 5,00,000) = Rs. 27,500
The difference in commission between the two sales would be:
Rs. 27,500 - Rs. 20,000 = Rs. 7,500
Therefore, the difference in commission for selling goods worth Rs. 20 lakhs and Rs. 30 lakhs is Rs. 7,500.
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what are the trig functions?
Answer:
sin
cos
tan
csc
sec
cot
Step-by-step explanation:
easy to remember that they are reciprocals of each other
Fifty pairs of individuals playing in a bridge tournament have been seeded 1,. , 50. In the first part of the tournament, the 50 are randomly divided into 25 east-west pairs and 25 north-south pairs.
(a) What is the probability that x of the top 25 pairs end up playing east-west?
(b) What is the probability that all of the top five pairs end up playing the same direction?
(c) If there are 2n pairs, what is the pmf of X = the number among the top n pairs who end up playing east-west? What are E(X) and V(X)?
a .The probability that x of the top 25 pairs play east-west is: P(X = x) = (25 choose x) / (50 choose 25) , b. The probability that the top 5 pairs all play in the same direction is: P(all 5 play the same direction) = (1/2) * (24/49) * (23/48) * (22/47) * (21/46), c. The pmf of X is given by P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)], and the expected value and variance of X are E(X) = n/2 and V(X) = (n² - 1)/12.
P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)] yields the pmf of X. , and X's expected value and variance are, respectively, E(X) = n/2 and V(X) = (n² - 1)/12.
(a) Let X be the number of top 25 pairs that play east-west. Since there are 25 east-west pairs and 25 north-south pairs, the total number of ways to choose 25 pairs to play east-west is (50 choose 25) = 50! / (25! * 25!). Similarly, the total number of ways to choose x top pairs to play east-west is (25 choose x) * (25 choose 25 - x) = (25 choose x), since the remaining 25 pairs automatically play east-west. Therefore, the probability that x of the top 25 pairs play east-west is:
P(X = x) = (25 choose x) / (50 choose 25)
(b) The probability that the top pair plays east-west is 25/50 = 1/2. Once the top pair has been assigned a direction, the probability that the second pair plays in the same direction is 24/49, the probability that the third pair plays in the same direction is 23/48, and so on. Therefore, the probability that the top 5 pairs all play in the same direction is:
P(all 5 play the same direction) = (1/2) * (24/49) * (23/48) * (22/47) * (21/46)
(c) Let X be the number of top n pairs that play east-west, where there are 2n pairs total. Then X can take on values from 0 to n. To find the pmf of X, we can use the same method as in part (a):
P(X = x) = [(2n - n) choose x] / [(2n) choose n]
= [(n choose x) * (n choose n - x)] / [(2n choose n)]
The expected value of X is:
E(X) = sum[x * P(X = x), {x, 0, n}]
= sum[x * (n choose x) * (n choose n - x) / (2n choose n), {x, 0, n}]
Using the identity sum[x * (n choose x) * (n choose k - x), {x, 0, k}] = k * (n choose k), we can simplify this to:
E(X) = n/2
The variance of X is:
V(X) = E(X²) - E(X)²
= sum[x² * P(X = x), {x, 0, n}] - (n/2)²
= sum[x * (n choose x) * (n choose n - x) / (2n choose n), {x, 0, n}] - (n/2)²
= (n² - 1)/12
Therefore, the pmf of X is given by P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)], and the expected value and variance of X are E(X) = n/2 and V(X) = (n² - 1)/12, respectively.
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