Answer:
To verify that [n+3]! / n! = (n+1), we can use the definition of factorial:
[n+3]! = (n+3) * (n+2) * (n+1) * n!
And we can cancel out n! from both sides of the equation:
(n+3) * (n+2) * (n+1) = n+1
Which is true, and thus [n+3]! / n! = (n+1) holds true.
Another way to verify this is by simplifying the RHS and LHS
(n+1) = (n+1)
And the LHS,
(n+3)! = (n+3)(n+2)(n+1)n! = (n+1)(n+2)(n+3)n! = (n+1)(n+2)(n+3)!/n!
which is also equal to (n+1)
So, [n+3]! / n! = (n+1) is true.
What is the slope of the line that passes through (-1, 4) and (3,- 8)?
Answer:
Slope =y2−y1/2−x1 =4−81+3=−1
Consider the quadratic function y = ax2. As |a| increases, the graph . As |a| decreases, the graph .
As the value of a decreases the graph will expand.
What is a quadratic function?In mathematics, a polynomial of degree two in one or more variables is referred to as a quadratic polynomial. The polynomial function that a quadratic polynomial defines is known as a quadratic function.
Given that consider the quadratic function y = ax². When the function is plotted on the for the value of an increases the graph contracts and if the value a decreases the graph expands.
The graph is attached with the answer below. In the graph, the quadratic function is shown at the two values of a.
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Juan's Cafe has regular coffee and decaffeinated coffee. This morning, the cafe served 70 coffees in all, 70% of which were regular. How many regular coffees did the cafe serve?
The cafe served 49 regular coffees.
How to calculate thisYou can use the proportion of regular coffees to the total number of coffees to find out how many regular coffees were served.
If 70% of the coffees served were regular, that means that 100-70 = 30% of the coffees served were decaffeinated.
To find out how many regular coffees were served, you can use the following proportion:
(Number of regular coffees) / (Total number of coffees) = 70% / 100%
You can then cross-multiply and solve for the number of regular coffees:
Number of regular coffees = (70/100) * (Total number of coffees)
Given the Total number of coffees = 70, you can substitute that and get:
Number of regular coffees = (70/100) * 70 = 49
Therefore, the number of regular coffees which the cafe served was 49.
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Let f be the function with derivative given by f′(x)=x^2−(a+b)x+ab=(x−a)(x−b), where a and b are constants such that a
Based on the given derivative function, the function f is decreasing for a < x < b, because f'(x) < 0 for a < x < b (A).
The given derivative function is the first derivative of function f(x), where:
f'(x) = x² - (a+b) x + ab = (x - a) (x - b)
a < b
Since the first derivative function f'(x) is still in the form of x² function, we can deduce that the original function f(x) has 2 extreme points, the maximum and the minimum. Those extreme points are (x = a) and (x = b).
However we cannot determine which one is the maximum point and which is the minimum one. We need to do another derivative work and check the second derivative function for each extreme points relevant to zero (0).
f'(x) = x² - (a+b) x + ab
f''(x) = 2x - (a+b)
(x = a)
f''(a) = 2(a) - (a+b)
f''(a) = 2a - a - b
f''(a) = a - b < 0 --> because a < b
(x = b)
f''(b) = 2(b) - (a+b)
f''(b) = 2b - a - b
f''(b) = b - a > 0 --> because a < b
Based on the second derivative function, we can define that (x = a) is the maximum point because f''(a) < 0 and (x = b) is the minimum point because f''(b) > 0. Then the value of f(x) between a < x < b should be on a downward slope.
However wee need to reconfirm this statement by choosing one set of random number that meet the rules of a < x < b. For this time, we use:
a = 1
b = 3
x = 2
Then:
f'(x) = x² - (1 + 3) x + (1) (3)
f'(x) = x² - 4x + 4
(x = 2)
f'(2) = (2)² - 4(2) + 3
f'(2) = 4 - 8 + 3
f'(2) = -1 < 0 --> supported statement A
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A researcher uses technology to estimate the effect of a car crash on different car models.a. experiment b. observational study C. survey d. simulation
The debriefing is an essential part of the informed consent process and is mandatory when the research study involves use of deception.
What is the researcher use to technology?The debriefing is an essential part of the informed consent process and is mandatory when the research study involves use of deception. The debriefing provides participants with a full explanation of the hypothesis being tested, procedures to deceive participants and the reason(s) why it was necessary to deceive them.
Research in science and technology is important because it contributes to knowledge and understanding of the world. It also contributes to the development of new technologies which are important in science and technology.
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Compute the p-value in order to test the following hypotheses
The p-value of the test hypothesis for each case is given as follows:
b) n = 9: 0.1151.
i. n = 25: 0.0228.
ii. n = 100: 0.
iii. When the sample size increases, the test statistic also increases, and hence the p-value decreases.
How to obtain the p-value?Before obtaining the p-value, the first step is obtaining the test statistic for the test.
The z-distribution is used, as the standard deviation for the population is known, and the equation is given as follows:
[tex]z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which the parameters are given as follows:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.[tex]\sigma[/tex] is the standard deviation of the population.n is the sample size.The fixed parameters for this problem are given as follows:
[tex]\overline{x} = 52, \mu = 50, \sigma = 5[/tex]
Hence, for n = 9, the test statistic is given as follows:
z = (52 - 50)/(5/3)
z = 1.2.
We have a right-tailed test, as we are testing if the mean is greater than a value, hence the p-value is of 1 subtracted by the p-value of z = 1.2, that is:
1 - 0.8849 = 0.1151.
With n = 25, we have that:
z = (52 - 50)/(5/5)
z = 2.
z = 2 has a p-value of 0.9772.
1 - 0.9772 = 0.0228.
With n = 100, we have that:
z = (52 - 50)/(5/10)
z = 4.
z = 4 has a p-value of 1.
1 - 1 = 0.
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Whats the answer for the radius the image is attached please help and provide work :)
The radius of the cylinder is 1.59 centimeters.
What is the radius of the cylinder?Remember that for a cylinder of radius R and height H, the volume is given by the formula:
V = pi*R^2*H
Where pi = 3.14
Here we know that the height is 76cm, and the volume is 600cm³, then we can replace these two in the equation to get:
600cm³ = 3.14*(76cm)*R^2
Solving for R, we will get:
R = √( 600cm³/3.14*(76cm)) = 1.59 cm
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The radius of the cylinder is r=50.129
What is the radius of a cylinder?The radius of the cylinder is the radius of a base. The altitude of the cylinder is a perpendicular segment from the plane of one base to the plane of the other and the height of the cylinder is the length of the altitude. The axis of a cylinder is the segment containing the centers of the two bases
The volume of a cylinder is 600,000.
The height is 76cm and the radius is unknown. How do you solve it?
The volume of a right cylinder can be determined by πr²h,
where
r is the radius,
h is the height, and
pi is a constant.
So:
76πr²=600000
πr²=600000/76=7894.736
r²=2512.972
r=50.129 cm^2
Hence, the radius is 50.129 cm^2
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What is the coefficient of the variable in the expression 2 − 5x − 4 + 8? (3 points) a −5 b −4 c 2 d 8
In the expression 2 − 5x − 4 + 8, the coefficient of x is -5. Thus, option a is correct.
What is a coefficient?A number or quantity combined with a variable is referred to as a coefficient. The variable is usually multiplied by an integer, which is then written next to it. It is assumed that the variables with no associated value have a coefficient of 1. For instance, in the expression 3x, the coefficient of x is 3, whereas in the expression x2 + 3, the coefficient of x2 is 1.
A variable's coefficient is the number or letter that the variable contains. As an illustration, the coefficient of variable x in the formula 2x + 3y is 2, and the coefficient of variable y is 3 in the same formula.
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Identify any vertical or horizontal asymptotes of the function: f(x) = x-2/ 5x^2 + 5x
if solid line is the parent function, f(x) Which of the following dotted line graphs shows the translation f(x)=f(x)+1? the translation f(x)=f(x)+1
Answer:
The graph that shows the translation f(x) = f(x) + 1 is the graph that is shifted upward by one unit on the y-axis. All the points on the graph have been moved upward by one unit, but the shape of the graph remains the same as the parent function.
The dotted line graph that shows the translation f(x) = f(x) + 1 is the one where the graph is shifted upward by one unit on the y-axis.
William put the tip of his pencil on the outer edge of a graph of the unit circle at He moved his pencil tip through a angle of % radians in the the point (0,"1)_ counterclockwise direction along the edge of the circle At what angle of the unit circle did William's pencil tip stop?
Therefore , the solution of the given problem of circle comes out to be
angle is 5π /6 .
Circle – what is it?A circle is created when every point as in plane that is specific distance apart from another point does so (center). As a result, it is a curve made up of points that are spaced apart from one another in the plane. It is rotationally symmetrical about the center at all angles. Every pair of endpoints in the plane that make up a circle are evenly spaced away from the "center" and form a closed, two-dimensional object. A specular symmetries line is produced by a line that passes through the circle. It is rotationally symmetrical about the center at all angles.
Here,
Given : 4π/3
=> 4π/3 * 180/π
=> 240°
Thus , Angle measured from positive x - axis is anticlockwise direction.
=> 90° + 60° = 150°
=> 150° * π/180°
=> 5π /6
Therefore , the solution of the given problem of circle comes out to be
angle is 5π /6 .
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answer for 10 points
Answer:
30 = 2·2·3
48 = 2·2·2·2·3
GCF = 6 (2·3)
LCM = 240 (2·3·5·8)
Step-by-step explanation:
A family buys an apartment for 200000. They pay a down payment of 20000. Their down payment is what percent of the purchase price
Answer:
the down payment of 20000 is 10% of the purchase price of 200000.
Step-by-step explanation:
To calculate the percentage of the down payment to the purchase price, you can use the following formula:
(down payment / purchase price) x 100
So in this case:
(20000 / 200000) x 100 = 0.1 x 100 = 10
Therefore, the down payment of 20000 is 10% of the purchase price of 200000.
Sally bought a soft drink for $3 and seven candy bars. she spent a total of $31. how much did each candy bar cost? Write an equation to represent the problem, and show the steps to solve it.
Answer:
31=3+x
31=-3+x
28=x
28/7
$4
Each Candy bar cost $4 each
Step-by-step explanation:
An education researcher claims that at most 5% of working college students are employed as teachers or teaching assistant. in random sample of 400 working college students, 6% are employed as teachers or teaching assistants. alpha=0.05,
1. Is there enough evidence to reject the researcher claim?
2. find the critical values and identify the rejection.
3. find the standardized test statistic z
4. decide whether to rejector fail to reject null hypothesis and interpret the decision.
Answer:
Step-by-step explanation:
Is there enough evidence to reject the researcher claim?
To determine if there is enough evidence to reject the researcher's claim, we can perform a hypothesis test with a null hypothesis of p <= 0.05 (at most 5% of working college students are employed as teachers or teaching assistants) and an alternative hypothesis of p > 0.05 (more than 5% of working college students are employed as teachers or teaching assistants).
Find the critical values and identify the rejection.
For a one-tailed test with a significance level of alpha = 0.05, the critical value for a z-test would be 1.645. This means that if the calculated z-score is greater than 1.645, we would reject the null hypothesis.
Find the standardized test statistic z
To find the standardized test statistic z, we can use the formula:
z = (p-hat - p0) / (sqrt(p0 * (1 - p0) / n))
where p-hat is the sample proportion of working college students employed as teachers or teaching assistants (6%), p0 is the proportion specified in the null hypothesis (5%), and n is the sample size (400).
z = (0.06 - 0.05) / sqrt(0.05 * (1 - 0.05) / 400)
z = 0.1 / 0.015811388300841898
z = 6.317
Decide whether to reject or fail to reject the null hypothesis and interpret the decision.
Since the calculated z-score (6.317) is greater than the critical value (1.645), we can reject the null hypothesis. This means that there is enough evidence
given f (x) = x^2 + 7x and g(x) = 9 - x^2 find f/g (x)
Answer:
(f/g)(x) = -(x²+7x)/(x²-9)
Step-by-step explanation:
Given f(x) = x^2 +7x and g(x) = 9 -x^2, you want f/g(x).
SubstitutionThe rational function (f/g)(x) is ...
[tex](f/g)(x)=\dfrac{f(x)}{g(x)}=\dfrac{x^2+7x}{9-x^2}\\\\\\\boxed{(f/g)(x)=-\dfrac{x^2+7x}{x^2-9}}[/tex]
__
Additional comment
We simply use the values of f(x) and g(x) in the desired ratio. No simplification is possible.
find the total number of relations on the set s={1,2,3}
The total number of relations in the set s = {1,2,3} is given as follows:
9.
How to obtain the total number of relations in a set?
The total number of relations in a set is given by the cardinality of the square of the set.
The square of a set is composed by all the possible pairs that can be formed with the elements in the set.
The set for this problem is defined as follows:
s = {1, 2, 3}.
The set has a cardinality of 3, as it has 3 elements, hence the cardinality of the square of the set is given as follows:
3² = 9.
Which is the total number of relations in the set.
These relations are all the ordered pairs with elements of s, hence:
Relations = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}
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HELP ASAP
a. Calculate the percentage change in nominal GDP between 2010 and 2019.
%
b. Calculate real GDP in 2019 at prices of 2010.
$
c. Calculate the percentage change in real GDP between 2010 and 2019.
%
d. What is the percentage change in nominal GDP between 2010 and 2019 attributed to changing prices?
%
e. What is the percentage change in nominal GDP between 2010 and 2019 attributed to a growing economy?
%
a) The percentage change in nominal GDP between 2010 and 2019 is;
47.54%
b) Real GDP in 2019 at prices of 2010 is; 20,597.113 billions
c) The percentage change in real GDP between 2010 and 2019; 72.4%
d) The percentage change in nominal GDP between 2010 and 2019 attributed to changing prices is; 24.45%
How to calculate the percentage change?The formula to calculate percentage change is;
Percentage change = (Final Value - Original Value)/Original Value * 100%
a) The percentage change in nominal GDP between 2010 and 2019 is;
(21433 - 14527)/14527 * 100%
= 47.54%
b) Real GDP in 2019 at prices of 2010 is;
21433 * 0.961
= 20,597.113 billions
c) The percentage change in real GDP between 2010 and 2019;
real GDP in 2010 = 14527 * 0.961
= 13,960.447 billion
Real GDP in 2019 = 21433 * 1.123
= 24,069.259 billion
Percentage change = (24069.259 - 13960.447)/13960.447 * 100%
= 72.4%
d) The percentage change in nominal GDP between 2010 and 2019 attributed to changing prices is;
Percentage change = [(21433/1.123) - (14527/0.961)]/(14527/0.961) * 100%
= 24.45%
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Answer ASAP PLEASE!!!
The fraction form of the given decimal number [tex]8.3 \overline3[/tex] is 75/90.
Decimal numbers into fractions:Any number with a decimal point in between the full amount and the fractional portion is said to be decimal. These two components of the decimal are separated by the point. It is known as a decimal point as a result the figures following the decimal point are always less than one.
We can covert every decimal number into fractions as follows
Here we have
[tex]8.3 \overline3 = 10F[/tex] ---- (1)
[tex]- 0.8\overline3 = - F[/tex] ---- (2)
Subtract (2) from (1)
=> [tex]8.3 \overline3 - 0.8\overline3 = 10F - F[/tex]
=> 7.5 = 9F
=> F = 7.5/9
=> F = 75/90 [ Multiplied by 10 on both sides ]
Therefore,
The fraction form of the given decimal number [tex]8.3 \overline3[/tex] is 75/90.
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Some students collected the data below. They will graph the data. What is the y-intercept? Your answer should be an ordered pair ( , )
Answer:
(0, 28)
Step-by-step explanation:
Overview
To answer the question given, we have to find the point on the line that goes through the points given in the table when the x-axis represents the month of the year (1 is Jan, 2 is Feb, etc.) and the y-axis represents the average temperature for that month.
Method 1
I've attached a graph of the line for reference.
When we look at the graph of the line that goes through the points given in the table, we can determine that its y-intercept is (0, 28).
Method 2
To determine the y-intercept of the line without actually graphing it, we can construct an equation using the given points.
We know that the line's slope is defined as [tex]\dfrac{\textrm{rise}}{\textrm{run}}[/tex].
Using that information, we can solve for the line's slope by using any two of the given points. I will use the points for January (1, 42) and February (2, 56).
[tex]\textrm{slope} = \dfrac{56-42}{2-1} = \dfrac{14}{1} = 14[/tex]
Then, we can plug this value into the point-slope form equation
[tex]y - b = m(x - a)[/tex]
along with one of the given points [tex](a,b)[/tex] -- I'll use January (1, 42):
[tex]y-42=14(x-1)[/tex]
Next, to solve for the y-coordinate of the line's y-intercept, we can plug 0 into the equation for x and solve for y.
[tex]y - 42 = 14(0 - 1)[/tex]
[tex]y - 42 = 14(-1)[/tex]
[tex]y - 42 = -14[/tex]
[tex]y = -14 + 42[/tex]
[tex]y = 28[/tex]
Finally, we can plug the solved y-coordinate and 0 for the x-value (because we solved for when y when x = 0) into the format of a Cartesian coordinate:
(0, 28)
What is the work done on the box from x = 0m to 4.0m?
The work done on the given box from x = 0 m to 4.0 m will be 100 Nm.
What is Work Done?Work is performed when an item moves as a result of a force, causing energy to flow from one store to another. Work is measured in Joules, and one joule equals one newton of force on an object traveling one meter.
The graph is given in the question, as shown
We have to determine the work done on the given box from x = 0 m to 4.0 m
As per the given graph, the required solution would be as:
Work done = Force × covered area between the graph and displacement axis
Work done = 1/2 × (30 × 4) + 10 × 4
Work done = 1/2 × 120 + 40
Work done = 100 Nm
The work done on the given box will be 100 Nm.
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Match the following function operations
The function operations are given as follows:
(f + g)(x) = -3x + 6.hf(x) = 2x³ - x² - 2x + 1.(f/h)(x) = (x² - 1)/(2x - 1)(g - f)(x) = -2x² - 3x + 8.(h - g)(x) = x² + 5x - 8.How to do the function operations?The addition of functions is done combining the like terms, hence:
(f + g)(x) = f(x) + g(x)
(f + g)(x) = x² - 1 + 7 - 3x - x²
(f + g)(x) = -3x + 6.
The multiplication of the functions is done multiplying the terms and then combining the like terms, as follows:
hf(x) = (2x - 1)(x² - 1)
hf(x) = 2x³ - x² - 2x + 1.
The quotient of the functions is quite straightforward, we just replace their definitions, as follows:
(f/h)(x) = (x² - 1)/(2x - 1)
The subtraction of the functions is similar to the addition, combining the like terms, as follows:
(g - f)(x) = 7 - 3x - x² - (x² - 1)
(g - f)(x) = 7 - 3x - x² - x² + 1
(g - f)(x) = -2x² - 3x + 8.
(h - g)(x) = 2x - 1 - (7 - 3x - x²)
(h - g)(x) = 2x - 1 - 7 + 3x + x²
(h - g)(x) = x² + 5x - 8.
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pls help with this question asap
Therefore , the solution of the given problem of slope comes out to be
1/10 , 2/15 , 3/20 and 4/25 slopes are respectively.
Explaining slope.steep a line is defined by its slope. The phenomenon of gradient overflow occurs in equations based on gradients. By dividing the median of the run (width difference) at two points even by rise (vertical distinction) between these very same two points, one may calculate the slope. Modeling the fixed path problem involves using an equation of the hill variety, denoted by the formula y = mx + b. The location of the line's y-intercept, when the slope is m, a = b, and also the equation are all true (0, b).
Here,
Given:
To find the slope
we have to find dx/dy is the formula for slope,
Thus ,
=> x/y = 1/10
=> x/y = 2/15
=> x/y = 3/20
=> x/y = 4/25
Therefore , the solution of the given problem of slope comes out to be
1/10 , 2/15 , 3/20 and 4/25 slopes are respectively.
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y+1 = 4/5(x+5)
solve in standard form
Answer:
y = 4/5x + 3
Step-by-step explanation:
The equation y + 1 = 4/5(x + 5) can be solved by first isolating the y variable on one side of the equation. To do this, we'll first distribute the 4/5 across the parenthesis:
y + 1 = 4/5x + 4
Next, we'll subtract 1 from both sides of the equation:
y = 4/5x + 3
This is the equation in standard form, with the y variable on the left side and the x variable on the right side, and the constants added and subtracted.
Jayla's history class is going to a museum. The museum charges $25 for an adult ticket and $12.50 for a student ticket. The class must spend $375 to rent buses for the trip. The total cost of the museum visit is modeled by the expression 25 x + 12.5y + 375. Which is the meaning of y in the expression?
the number of students going
$12.50 is the cost of the STUDENT ticket, so that knocks 2 answers down then, would you times the total by 12.5 to get the actual cost. no so it has to be the number of students going
A a triangular object is placed in front of a plane mirror. Use the law of reflection and principles of ray diagramming to construct the image that the subject will formed behind the mirror. After during the image, describe it in terms of 1) distance from the mirror 2) it's size compared to the image 3) orientation (inverted from right to left or inverted from left to right) and 4) type real or virtual
Please find attached the drawing, created with MS Word, of the image of the triangle following the reflection across the mirror surface.
1) The object and the image are equidistant from the mirror
2) The object and the image following the reflection across a plane mirror are have the same side lengths and are therefore congruent
3) The image of the triangle is upright, and virtual and inverted from left to right
What are the laws of reflection?The laws of reflection are;
1) The angle made by the incident ray and the normal line (the angle of incidence) is equal to the angle made by reflected ray and the normal line
2) The incident ray, the reflected ray and the normal are coplanar
3) The incident ray is on the other side of the normal with regards to the refracted ray
1) The distance of the points on the object from the mirror are the same as the distance of the corresponding point from on the image from the mirror
2) A reflection of an object across a plane mirror is a rigid transformation such that the size of the object and the size of the image are the same.
Therefore, ΔABC ≅ ΔA'B'C'
3) The coordinates of the image formed following the reflection of a point (x, y) across the y-axis is; (-x, y)
The y-coordinate remains the same, and the x-coordinates changes from positive to negative, and the location of the object on the left of the y-axis, produces an image on the right of the y-axis. Therefore
Therefore, the image is upright and inverted from right to left
The image of the triangle cannot be played on the screen, therefore, the image is virtual
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A number is no more than negative three
Based on the given figure, we refer to a as the length of the side _________ angle 0 , b as the length of the side __________ angle 0 , and c as the length of the __________.
Based on the given figure, we refer to angle a as the length of the side opposite to the angle θ, b as the length of the side adjacent to the angle θ and c as the length of the hypotenuse.
What are the features of a right triangle?A right triangle is composed by two sides, that form a right angle = angle of 90º between them, plus the hypotenuse, which is the segment connecting these two sides.
The angles are classified as adjacent and opposite as follows:
Adjacent side: same side of the angle.Opposite angle: different side of the angle.Missing InformationThe problem is given by the image presented at the end of the answer.
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State the domain and range for the following relationThen determine whether the relation represents a function \{(4, 7), (- 3, 7), (6, 9), (4, 12)\}
Answer:
Domain {-3, 6}
Range: 7, 12}
Not a function
Step-by-step explanation:
Domain
Domain is the set of all x values which result in real and defined values for y
Range
Range is the set of y values that are possible from the values of x in the domain
Both are expressed in the form {min-value, max-value}
For x we see the minimum is -3 and maximum is 6
So domain = {-3, 6}
For y values we see the minimum is 7 and maximum is 12
So range = {7, 12}
A function can have only one output(y) for a specific input(x)
Here we see that for different values of x, x = 4 and x = -3, we get the same y-value
Hence it is not a function
Suppose the following data show the introductory interest rates on a sample of 5 credit cards: 5.7, 1.4, 3.6, 4.9, 4.0. Calculate the mean interest rate of the sample.
The mean interest rate of the sample is 3. 92
What is mean?Mean of values is mathematically described as a quantity having a value that is intermediate between those of the extreme members of a given se of numbers.
It is referred to as the average of numbers or values. It is also described as the central value of a set of numbers.It is calculated by dividing the sum of the values or numbers by the total number of numbers in the set.From the information given, we have the set of values;
5.7, 1.4, 3.6, 4.9, 4.0
The formula for calculating mean is expressed as;
Mean = sum of numbers/ total number of numbers
Substitute the values
Mean = 5. 7 + 1. 4 + 3. 6 + 4. 9 + 4. 0/5
Add the values
Mean = 19. 6/5
divide through
Mean = 3. 92
Hence, the value is 3.92
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