The row in Pascal's triangle where three consecutive entries are in the ratio 3:4:5 is the 62nd row.
To find the row in Pascal's triangle where three consecutive entries are in the ratio 3:4:5, we will use the concept of binomial coefficients.
The binomial coefficient C(n, k) represents the number of ways to choose k items from a set of n items, and is given by the formula -
C(n, k) = n! / (k! (n-k)!).
In Pascal's triangle, the entries are arranged such that the nth row contains n+1 entries, and the kth entry in that row is given by C(n, k).
We can determine the ratio of three consecutive entries in Pascal's triangle by looking at the binomial coefficients C(n, k), C(n, k+1), and C(n, k+2) in a particular row.
In order to find the row where the ratio of these three consecutive entries is 3:4:5, we will set up the following equation:
C(n, k) / C(n, k+1) = 3/4 and C(n, k+1) / C(n, k+2) = 4/5
We will now use the formula for binomial coefficients to simplify these equations as follows -
(n! / (k! (n-k)!) ) / (n! / ((k+1)! (n-(k+1))!)) = 3/4
Similarly,
(n! / ((k+1)! (n-(k+1))!)) / (n! / ((k+2)! (n-(k+2))!)) = 4/5
Solving for n, we can get n = 62
Therefore, the row in Pascal's triangle where three consecutive entries are in the ratio 3:4:5 is the 62nd row.
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find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.
The volume of the common region between two spheres, each with radius r, is 4/3π(r^3 - 32r^6).
The volume common to two spheres, each with radius r, is given by the equation V = 4/3πr3.
The volume of the common region between two spheres, each with radius r, is given by the formula:
V = 4/3π(r^3 - (r^2 - d^2)^3/2),
where d is the distance between the centers of the spheres.
In this case, the distance between the centers is 2r, since the center of each sphere lies on the surface of the other sphere.
Thus, the volume of the common region is 4/3π(r^3 - (r^2 - (2r)^2)^3/2)
= 4/3π(r^3 - (r^2 - 4r^2)^3/2)
= 4/3π(r^3 - (r^2 - 4r^2)^3/2)
= 4/3π(r^3 - (4r^2)^3/2)
= 4/3π(r^3 - 64r^6/2)
= 4/3π(r^3 - 32r^6)
= 4/3π(r^3 - 32r^6)
= 4/3π(r^3 - 32r^6).
Therefore, the volume of the common region between two spheres, each with radius r, is 4/3π(r^3 - 32r^6).
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Complete question : What is the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere?
Bridge is a card game in which the deck of 52 cards is randomly and equally divided among the 4 players.So each player gets 13 cards. What is the probability that one player receives all the 13 hearts?
The probability that one player receives all the 13 hearts is 1/2598960.
How likely something is to occur can be determined using probability. For many events, it is challenging to make an accurate prediction.
The number of ways to distribute the 13 hearts among the 4 players is 4!. The number of ways to distribute the remaining 39 cards among the 4 players is 39!/(13!13!13!13!). Hence, the total number of ways to distribute the 52 cards among the 4 players is (52!)/(13!13!13!13!).
Therefore, the probability that one player receives all the 13 hearts is (4!)/((52!)/(13!13!13!13!)) = 1/((52515049)/(13121110)) = 1/2,598,960.
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A bag contains 6 red marbles and 4 blue marbles. If two marbles are drawn at random and not replaced, what is the probability that two red marbles are drawn?
If two marbles are drawn at random and not replaced, the probability that two red marbles are drawn is; 1/3
What is the probability of selection?The parameters given for marbles in the bag are;
Number of red marbles = 6
Number of blue marbles = 4
Thus;
Total number of marbles = 6 + 4
Total number of marbles = 10 marbles
Now, probability of first selection being red marble = 6/10
Probability of second marble being red if first was not replaced = 5/9
Thus;
P(first two being red marbles) = (6/10) * (5/9)
= 1/3
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Please help I can’t figure this out
a researcher randomly assigned boys and girls to each of two groups. one group watched a violent television program while the other group watched a nonviolent program. the children were then observed during a period of free play, and the incidence of aggressive behavior was recorded for each group. experimental
case study
correlational
observation
This method of observing the behaviour of each group of children is best characterized as experimental.
Experimental research is a study worked with a scientific approach using two sets of variables. The first set works as a constant, which we use to measure the differences of the second set.
This can be regarded as something that depends on other factors, such as a test score, which can be considered a dependent variable because it could change depending on several factors, such as how much you studied.
It should be noted that dependent and independent variables are variables in mathematical modelling and experimental sciences; however, the dependent variables can be seen as ones whose values are studied under the supposition.
Hence the correct option is a.
--The given question is incomplete; the complete question is
"A researcher randomly assigned boys and girls to each of two groups. One group watched a violent television program, while the other watched a nonviolent one. The children were then observed during free play, and the incidence of aggressive behaviour was recorded for each group. This method is best characterized as"--
a. Experimental
b. Case study
c. Correlational
d. Observation"--
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The ball thrown upwards hit the roof and returns back to the ground.The upward moment is given with : s= -t2+3t+4
and downward :s= - 2t2+t+7
How high is the roof from the ground ?
The height of the roof will be 10.75 units.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is to find the height of the roof.
The upward motion is modelled by the equation -
S = - t² + 3t + 4
The height of the roof is equal to maximum height attained. For the maximum height attained -
dS/dt = 0
d/dt (- t² + 3t + 4) = 0
- 2t + 3 = 0
2t = 3
t = 3/2 .... Eq(1)
The height of the roof will be -
S(max) = - t² + 3t + 4
S(3/2) = - (3/2)² +(3 x 3/2) + 4
S(3/2) = 9/4 + 9/2 + 4
S(3/2) = 2.25 + 4.5 + 4
S(3/2) = 10.75 units.
Therefore, the height of the roof will be 10.75 units.
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1/2n = 1/4n - 3n+3/2n
Solve
Also check for extraneous solutions
Answer: 1/2n = 1/4n - 3n+3/2n
Start by multiplying both sides by 2n to get rid of the fractions:
1 = 1/2 - 3n+3/n
Then by multiplying both sides by n to get rid of the fraction on the right side:
n = -3n+3
Then by adding 3n to both sides:
4n = 3
Then by dividing both sides by 4:
n = 3/4
There is no extraneous solutions in this case, because no denominators are equal to zero and all the operations are valid. Therefore, the solution is n = 3/4
Step-by-step explanation:
7 The equation, A equal to 2,400 plus quantity 1 plus 0.031 over 4 end quantity all raised to the power of 4 times t, represents the amount of money earned on a compound interest savings account. What does the value 0.031 represent?
The value 0.031 represents the interest rate, which means the annual compounded interest rate is 3.1%.
The value 0.031 represents the interest rate, which means the annual compounded interest rate is 0.31%.
The value 0.031 represents the investment period, which means the investment is invested for 0.031 years.
The value 0.031 represents the investment period, which means the investment is invested for 3.1 years
The quantity which represents the 0.031 in the given formula of compound interest, the value represents the annual compound interest rate 3.1%, So option A is correct
What is compound interest?A loan or deposit amount is subject to compound interest as interest. In everyday life, it is the most prevalent idea. Both the principal and the interest accrued over time affect the compound interest for a given amount. Between compound and simple interest, this is the primary distinction.
[tex]A = P[1 + r/n]^{nt}[/tex]
Where,
A = compound interest
P = Principle Amount
r = Interest rate
n = number of period for which interest is calculated
t = time
Given that,
A equal to 2,400 plus quantity 1 plus 0.031 over 4 end quantity all raised to the power of 4 times t
[tex]A = 2400[1 + 0.031/4]^{4}[/tex]
So, here we can see that r is equal to 0.031
It represents the annual compound interest rate of the compound interest
Hence, it is a annual compound interest which has value of 3.1%
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At a point on the ground 80 ft from the base of a tree, the distance to the top of the tree is 11 ft more than 2 times
the height of the tree. Find the
height of the tree.
The height of the tree is I f
(Simplify your answer. Round to the nearest foot as needed.)
The height of the tree is 39 ft from the ground.
What is Pythagorean Theorem?In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse.
Let us suppose the height of the tree = l
The distance from the point on the ground to the top of the tree is thus:
h = 11 + 2(l)
Given that the base = 80 ft.
To find the height of the tree we can use the Pythagorean theorem.
The Pythagorean theorem is given as follows:
a^2 + b^2 = c^2
(80)^2 + (l)^2 = (11 + 2l)^2
Using the identity (a + b) ^2 = a^2 + 2ab + b^2 we can write the equation as:
6400 + l^2 = 121 + 44l + 4l^2
Take all the terms on a single side of the equation:
4l^2 - l^2 + 44l + 121 - 6400= 0
3l^2 + 44l - 6279 = 0
Solve the following quadratic equation using the equation of the quadratic function:
l = 39
Hence, the height of the tree is 39 feet.
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The volume of a rectangular box is 2x(2x + 5)(2x- 3). (Drawing is not to
scale.)
Height 2x - 3
Length 2x + 5
Width 2x
Which statement about the volume of the box is true?
O A. The volume is the product of the length, 2x 5, and the width, 2x.
O B. The volume is the sum of the length, 2x + of 5, the width, 2x, and the
height, 2x - 3
• C. The volume does not depend on the height, 2x - 3.
D. The volume is the product of the area of the base, 2x(2x + 5), and
The volume is the product of the area of the base, 2x(2x+5), and the height, 2x-3.
What is Volume?
A measurement of three-dimensional space is volume. It is frequently expressed quantitatively using SI-derived units, as well as several imperial or US-standard units. Volume and the notion of length are connected.
What is Area?
The size of a patch on a surface is determined by its area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.
Given that;
Height= 2x - 3
Length= 2x + 5
Width= 2x
We have the following equation for the rectangular box in this instance:
2x(2x + 5)(2x- 3)
We may come up with an equivalent formula for the volume by using the distributive property of multiplication.
We have then ;
[tex](4x^2+10x)(2x-3)[/tex]
Once more writing, we have:
[tex](8x^3+20x^2)+(-12x^2-30x)[/tex]
Including related terms, we have:
8x^3+8x^2-30x
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b) Five thousand three hundred and forty-five plus eight hundred and six in number
Answer:
I'm not sure if this is what you meant but 5,345 + 806 = 6,151
Step-by-step explanation:
Answer:6151
Step-by-step explanation:
5345 + 806 = 6151
A farmer has enough food to feed 20 animals in his cattle for 6 days. How long
would the food last if there were 10 more animals in his cattle?
Answer:
4 daysb
Step-by-step explanation:
Here the number of animals and the number of days are in inverse proportion. Hence the food will last 4 days.
The food will last for 6 days for 20 animals.
If there were 10 more animals in the cattle, the total number of animals would be 20+10 = 30.
The food will last for the same 6 days but now it would be divided among 30 animals.
Therefore, the food would last for 6 days / 30 animals = 0.2 days per animal.
So, the food will last for 0.2 days if there were 10 more animals in the farmer's cattle.
graph the ellipse give by the equation 49^2 + 16^2 = 784. Rewrite the equation i standard form.
Answer:
The equation 49^2 + 16^2 = 784 can be rewritten as (x/24.5)^2 + (y/8)^2 = 1.
The standard form of an ellipse is (x/a)^2 + (y/b)^2 = 1, where a and b are the lengths of the major and minor axes, respectively. In this case, a = 24.5 and b = 8.
To graph the ellipse, we can use the equation in the standard form and plot points that satisfy the equation. The center of the ellipse is (0,0) and it has a major axis of 24.5 and minor axis of 8.
The graph of this ellipse will look like an oval, with the center at the origin, major axis along the x-axis and minor axis along the y-axis.
what is the inequality for 3<-5n+2n
Answer:
n< -1
Step-by-step explanation:
1. Rewrite so n is on the left side of the inequality.
−5n+2n>3
2.Add 5n and 2n
-3n >3
3. Divide each term then simplify.
Please answer this question
The transformation with respect to dilation will only preserve the angle measure i.e. B.
What is dilation in mathematics?
Dilation is a process of changing the size of an object or shape by decreasing or increasing its dimensions by some scaling factors. For example, a circle with radius 10 unit is reduced to a circle of radius 5 unit. The application of this method is used in photography, arts and crafts, to create logos, etc. In Geometry, there are four basic types of transformations. They are:
RotationTranslationReflectionResizing or DilationSome features of shapes that remain unchanged during the dilation transformations are:
Each angle of the figure is the same.Midpoints of the sides of the figure remain the same as the midpoint of the dilated shape.Parallel and perpendicular lines in the figure remain the same as the parallel and perpendicular lines of the dilated figure.The images remain the same.Now,
As in dilation Each angle of the figure remains the same.
Hence,
The transformation with respect to dilation will only preserve the angle measure
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what does the equation y=x^2 represent as a curve in R^2
The equation y = x² represents a parabola as a curve
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The equation y = x² represents a parabola as a curve
The equation is given as:
y = x²
The above equation has a degree of 2
Polynomials that have a degree of 2 are referred to as quadratic equations
The curve of a quadratic equations are called parabola
Hence, the equation y = x² represents a parabola as a curve
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4. Find the area of the irregular shape. Round to the nearest tenth. 8 cm 5 cm
90.2 cm²
241 cm²
65.1 cm²
72.3 cm²
Answer: Total area = 40 cm² + 25.12 cm² = 65.12 cm²
Step-by-step explanation:
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Answer:65.1
Step-by-step explanation:
.In the first Summative Test, Carmen scored 20 points which is 80% of the number
of items. In the second test, she scored 30. Her total score in the 2 tests is 50% of the
total number of items. How many items were in the second test?
Carmen scored 50%of the total number of items in second test, the number of items in the second test is, 75
What is the percentage?The percentage is a mathematical term, which means a number or a ratio which is represented in fractions of 100. We use the '%' symbol to represent the percentage.
Formula for percentage;
Percentage = (present quantity/whole quantity)×100
Given that,
In the first Summative Test,
Carmen scored 20 points which is 80% of the number of items
In the second test,
Carmen scored 30 points,
Her total score is 50% of the number of items
The total number of the items in the second test = ?
Suppose, total number of items in the first test = x
in the second test = y
⇒ 20 points = 80x/100
⇒ 20 points = 0.80x
⇒ x = 25
⇒ 0.5(x + y) = 20 + 50
⇒ x + y = 100
⇒ y = 75
Hence, the total number of items in the second test is 75
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what is the best use of
Trigonometry is best used in navigation to estimate where to point the compass to get a straight line. It will be simple to pinpoint a location as well as find distance and see the horizon using a compass and trigonometric functions in navigation.
What is Trigonometry?The area of mathematics known as trigonometry examines the correlation between the ratios of a right-angled triangle's sides to its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are used to study this relationship. The concept of trigonometry was developed by the Greek mathematician Hipparchus, and the word trigonometry is a derivative of Latin from the 16th century.
One of the most significant area of mathematic is rigometry. The words "Trigonon" and "Metron," which mean respectively "triangle" and "measure," are combined to form the word "trigonometry." It is the investigation of how a right-angled triangle's sides and angles relate to one another. Therefore, using formulas and identities based on this relationship aids in determining the measure of a right-angled triangle's unknown dimensions.
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Full question:
What is the best use of trigonometry?
the opportunity cost of producing one bushel of wheat in canada is equal to 1/2 tv set. the opportunity cost of producing one bushel of wheat in the us is 2 tv sets. if these two countries specialize according to comparative advantage and then trade with each other, .
If Canada specializes in producing wheat and the US specializes in producing TV sets, both countries will benefit from trade.
To determine this if Canada produces wheat, the opportunity cost is 1/2 TV set. If the US produces wheat, the opportunity cost is 2 TV sets. So, Canada has a comparative advantage in producing wheat. Similarly, the US has a comparative advantage in producing TV sets.
Let's say Canada produces 100 bushels of wheat, and the US produces 100 TV sets. If they trade with each other, Canada can exchange 50 bushels of wheat for 50 TV sets, and the US can exchange 50 TV sets for 50 bushels of wheat. Both countries would benefit from this trade.
Canada would end up with 50 TV sets and 50 bushels of wheat, while the US would end up with 50 bushels of wheat and 50 TV sets. Both countries would be better off than if they had not traded.
Opportunity cost in mathematics is a term that is used to describe the cost of one alternative in terms of the benefits forgone by the best alternative. It can also be represented in terms of mathematical equations.
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15. Reasoning The Annual Mug Race is the
longest river sailboat race in the world. The
event is run along the St. Johns River, which
is 310 miles long. About how many times as
long as the race is the river?
Using expression {River=x(race)} River's length is 7.38 times the race's length, approximately.
What are expressions?
Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between. The mathematical operators can be of addition, subtraction, multiplication, or division. For example, x + y is an expression, where x and y are terms having an addition operator in between. In math, there are two types of expressions, numerical expressions - that contain only numbers; and algebraic expressions- that contain both numbers and variables.
e.g. A number is 6 more than half the other number, and the other number is x. This statement is written as x/2 + 6 in a mathematical expression. Mathematical expressions are used to solve complicated puzzles.
We know that
The river is 310 miles long.
The race is 42 miles long.
So, if we want to know know how many times as long as the race is the river, we just need to use this expression
River=x(race)
Where x represents the factor that expresses the ratio between both, solving for x we have
x=river/race=310/42≈7.38
Therefore,
River's length is 7.38 times the race's length, approximately.
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Right question:-
The Annual Mug Race 42 mile is the
longest river sailboat race in the world. The
event is run along the St. Johns River, which
is 310 miles long. About how many times as
long as the race is the river?
How do i solve for value of m
List all real values of xx such that f(x)=0. If there are no such real x, type none in the answer blank. If there is more that one real xx, give a comma separated list (e.g. 1,2).
f(x)=19x2+10x
x =
The all the real values of x , such that for the function f(x) = 19x² + 10x satisfies f(x) = 0 are 0 , -10/19 .
The function is ⇒ f(x) = 19x² + 10x ;
taking x common from both terms and writing it in factor form ,
we get ;
⇒ x(19x + 10) = 0 ;
we know that the product of any two number is zero only when one of them is 0 ,
that means ;
⇒ x = 0 or 19x + 10 = 0 ;
⇒ x = 0 or 19x = -10 ;
⇒ x = 0 or x = -10/19 .
Therefore , all the real values of x such that f(x) = 0 are 0 , -10/19 .
The given question is incomplete , the complete question is
List all real values of x such that the function f(x)=0. If there are no such real x, type none in the answer blank. If there is more that one real x, give a comma separated list (e.g. 1,2).
f(x) = 19x² + 10x .
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Si por un artículo se pagó 111.340 de IVA (19%) ¿Cuánto cuesta el artículo con IVA incluído?
David cut a board 20 foot long into four equal pieces. Which expression does NOT equal the length of one of the pieces in feet? Responses
The expression that does not equal the length of one of the pieces in feet is C) 4/20
How to find the expression ?The expression that show the length of one of the pieces of feet would be the expressions that lead to the solution:
= Length of board cut by David / Number of equal pieces
= 20 / 4
= 5 ft each
Option A mirrors the equation above and so is equal to the length of one of the pieces. Option B also equals the equation above as the signs ÷ and / are both for division.
Option C does not equal the length of one of the pieces as it gives:
= 4 / 20
= 1 / 5 ft
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Options for this question are:
A)20/4
B)20 ÷ 4
C)4/20
6 The equation, A equal to P quantity 1 plus 0.054 over 2 end quantity all raised to the power of 2 times t, represents the amount of money earned on a compound interest savings account with an annual interest rate of 5.4% compounded semiannually. If the initial investment is $3,000, determine the amount in the account after 15 years. Round the answer to the nearest hundredths place.
$3,164.19
$6,671.67
$4,473.81
$14,532.47
The amount in the account after 15 years is, $6,671.67
What is compound interest ?A loan or deposit amount is subject to compound interest as interest. In everyday life, it is the most prevalent idea. Both the principal and the interest accrued over time affect the compound interest for a given amount. Between compound and simple interest, this is the primary distinction.
[tex]A = P[1 + r/n]^{nt}[/tex]
A = P + I
Where,
A = compound interest
P = Principle Amount
r = Interest rate
n = number of period for which interest is calculated
t = time
Given that,
A equal to P quantity 1 plus 0.054 over 2 end quantity all raised to the power of 2 times t
an annual interest rate of 5.4% compounded semiannually
the initial investment P is $3,000
the amount in the account after 15 years = ?
[tex]A = 3000[1 + 0.054/2]^{2t}[/tex]
after 15 years compound interest,
[tex]A = 3000[1 + 0.054/2]^{30}[/tex]
A = $6,671.67
A = P + I
P (principal) = $3,000.00
I (interest) = $3,671.67
Hence, the amount in the account after 15 years is, $6,671.67
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A bank loaned out $12,000, part of it at the rate of 6% per year and the rest at the rate of 16% per year. If the interest received totaled$1000, how much was loaned at 6%?
The amount loaned at 6% was $9200. The solution has been obtained by making and solving the equations.
What is an equation?
A claim that two expressions are equal and connected by the equals sign "=" is the mathematical definition of an equation. The two expressions need not be entirely composed of variables; one expression may have variables while the other may not.
We are given that a bank loaned out $12,000 and a part of it at the rate of 6% per year and the rest at the rate of 16% per year.
Let the amount loaned at 6% be 'x'
So, amount loaned at 16% will be '12,000 - x'
The total interest received was $1000.
Therefore,
⇒6%x + 16%(12000-x) = 1000
⇒0.06x + 1920 - 0.16x = 1000
⇒920 = 0.1x
⇒9200 = x
Hence, the amount loaned at 6% was $9200.
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In a wind tunnel experiment; the force on a projectile due to air resistance was measured at different velocities: Velocity (100 ftlsec) 0 2 4 6 8 10Force (100 Ib) 0 2.8 14 38. 74.4 128Find an interpolating polynomial for these data It is highly recommended that you use software or a calculator. F(v) = ____ Use the polynomial to estimate the force on the projectile when it is traveling at 750 ftlsec:
The estimated force on the projectile when it is traveling at 750 ft/sec is approximately 696.7 lbs.
The Possible Interpolating PolynomialOne possible interpolating polynomial for these data points is a polynomial of degree 3, also known as a cubic polynomial. Using software or a calculator to fit a cubic polynomial to the data points, the resulting equation is:
F(v) = -0.00000016v³ + 0.0004v² - 0.08v + 0.4
To estimate the force on the projectile when it is traveling at 750 ft/sec, we can substitute 750 for v in the equation above:
F(750) = -0.00000016(750³) + 0.0004(750²) - 0.08(750) + 0.4
F(750) ≈ 696.7
The estimated force on the projectile when it is traveling at 750 ft/sec is approximately 696.7 lbs.
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If log5 125 = x, then x =
Answer:
X=3
Step-by-step explanation:
If you enter log5(125)= x into your calculator that should be your answer
The owner of a factory regularly requests a graphical summary of all employees' salaries. The graphical summary of salaries is an example of
a. a sample
b. descriptive statistics
c. statistical inference
d. an experiment
The graphical summary of salaries is an example of b. descriptive statistics.
Descriptive statistics is the branch of statistics that is used to summarize, organize, and describe data. It is used to present data in a clear and meaningful way, so that patterns, relationships, and trends can be easily understood. A graphical summary of all employees' salaries, such as a histogram or a bar chart, is an example of descriptive statistics.
This graphical summary represents the entire population of employees and not a subset of it, so it's not a sample. The factory owner is not making any inferences about the population based on the data, so it's not an example of statistical inference. And this is not an experiment because no manipulation of the data was made.
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