3 has a mean of 120 and a standard deviation of 3.
Describe standard deviation using an example?
The dispersion of the data around the mean value is measured by the standard deviation. It might be helpful when comparing data sets that may have the same mean but a different range.
The means of the two numbers 15, 15, 15, 14, 16, and 2, 7, 14, 22, 30 are the same, for instance. The second is, however, obviously more dispersed.
mean of sample means = true mean = 120
std dev of sample mean = true std / sqrt(n)
= 15/5 = 3
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the admission fee at a small fair is $1.50 per child and $4.00 per adult. on a certain day, 2200 people enter the fair and $5050 is collected. how many children and how many adults attended?
Answer: 1,500+7=2,200 Y=700 2.
Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.x=1+(y−2)2,x=2
The volume of the solid obtained by rotating the region bounded by the given curves about the x-axis is 16π/3.
What is the bounded region?
A territory is said to be bounded if it has a boundary or other restrictions applied to it. In other words, a bounded shape is limited by a set of measurements or characteristics and cannot be an indefinitely huge area.
Here, we have
Given: x = 1+(y−2)², x=2
We have to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.
Rotating with the cylinder method about the x-axis means it must be a dy integral. The given parabola equation is already solved for x. Find the intersections:
2 = 1 + (y−2)²
0 = (y -1)(y-3)
y = 1, 3
We will integrate from
y = 1 to 3
The radius of each cylinder will be r=y and the height will be (greater minus lesser, i.e. right minus left).
h = 2 - (1 + (y-2)²)
h = 1 - y² + 4y -4
h = - y² + 4y - 3
Volume = [tex]\int\limits^a_b {2πrh} \, dy[/tex]
= 2π∫₁³ y.( - y² + 4y - 3)dy
= 16π/3
Hence, the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis is 16π/3.
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a and b are friends. a's father d is twice as old as a and b is twice as old as his sister c. the age of d and c differ by 40 years. write a linear equation representing the given situation, assuming the ages of a and b as x and y years respectively.
The A's age is 26 years and B's age is 24 years
What is meant by linear equation? A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables. Ax+By=C is the typical form for linear equations involving two variables. A linear equation in standard form is, for instance, 2x+3y=5. Finding both intercepts of an equation in this format is rather simple (x and y).Let the ages of A and B be x and y years respectively. Then, [tex]$x-y=\pm 2[/tex]Given
D's age [tex]$=2 x$[/tex]years and, C's age[tex]$=\frac{y}{2}$[/tex] years.
Clearly, D is older than C
[tex]$2 \mathrm{x}-\frac{\mathrm{y}}{2}=40 \Rightarrow 4 \mathrm{x}-\mathrm{y}=80$[/tex]
Thus, we have the following two systems of linear equations[tex]$x-y=2$[/tex].(i)
and, [tex]$4 x-y=80$[/tex](ii)
or
[tex]$x-y=-2$[/tex] (iii)
and, [tex]$4 x-y=80$[/tex]..(iv)
Subtracting equation (i) from equation (ii), we get [tex]$3 \mathrm{x}=78 \Rightarrow \mathrm{x}=26$[/tex]Putting [tex]$\mathrm{x}=26$[/tex] in equation (i), we get [tex]$\mathrm{y}=24$[/tex]or
Subtracting equation (iv) from equation (iii), we get [tex]$-3 \mathrm{x}=-82 \Rightarrow \mathrm{x}=\frac{82}{3}=27 \frac{1}{3}$[/tex]
Putting [tex]$\mathrm{x}=\frac{82}{3}$[/tex]in equation (iii), we get
[tex]$\mathrm{y}=\frac{82}{3}+2=\frac{88}{3}=29 \frac{1}{3}$[/tex]
Hence, A's age =26 years and B's age =24 years or
A's age =[tex]27 \frac{1}{3}$[/tex] years and B's age[tex]$=29 \frac{1}{3}$[/tex] years.
The complete question is,
A and B are friends and their ages differ by 2 years. A's father D is twice as old as A and B is twice as old as his sister C. The age of D and C differ by 40 years. Find the ages of A and B.
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Let x and y be functions of time t such that the sum of x and twice y is constant. Which of the following equations describes the relationship between the rate of change of with respect to time and the rate of
change of with respect to time?
A) dx/dt=2dy/dt
B) dx/dt=-2dy/dt
C) 2dx/dt+dy/dt=0
D) dx/dt + 2dy/dt=K, where K is a function of t
B) dx/dt=-2dy/dt is the relation of the rate of change for the conditon given in the question.
what is rate of change?Rate of change is a measure of how quickly a variable is changing. It is often represented by the symbol "d/dt" or "∆y/∆x" and is calculated by finding the ratio of the change in one variable to the change in another variable.
Given :
x + 2y =k
where k is a constant
differentiating above equation with respect to time we get
=> dx/dt + 2dy/dt =dk/dt
=> dx/dt + 2dy/dt =0
=> dx/dt = -2dy/dt
so Option B) dx/dt=-2dy/dt is correct option for the above problem
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In trapezoid ABCD we have AB parallel to DC, E as the midpoint of BC, and F as the midpoint of DA. The area of ABEF is twice the area of FECD. What is AB/DC ?
(A) 2
(B) 3
(C) 5
(D) 6
(E) 8
In trapezoid ABCD we have AB parallel to DC, E as the midpoint of BC
AB/DC = 5
given that
In American and Canadian English, a quadrilateral having at least one pair of parallel sides is referred to as a trapezoid (/traepzd/). It is referred to as a trapezium in British and other varieties of English (/trpizim/).
In Euclidean geometry, a trapezoid is a convex quadrilateral by definition. The bases of the trapezoid are the parallel sides. If the remaining two sides are parallel, they are referred to as the legs (or lateral sides); if not, the trapezoid is a parallelogram, and there are two pairs of bases. A scalene trapezoid is one that lacks equal-sized sides.
In trapezoid ABCD we have AB parallel to DC
E as the midpoint of BC
F as the midpoint of DA.
The area of ABEF is twice the area of FECD.
AB+EF/2 = 2(DC+EF/2)
AB+EF/2 = DC+EF
AB+EF = 2DC+2EF
EF is the midline between AB and DC
AB + (AB+DC/2) = 2DC+2(AB+DC/2)
[tex]\frac{3}{2}[/tex]AB +DC/2 = 2DC +AB + DC
[tex]\frac{3}{2}[/tex]AB +DC/2 = 3DC+AB
[tex]\frac{3}{2}[/tex]AB - AB = 3DC - DC/2
AB/2 = 5DC/2
AB = 5DC
AB/DC = 5
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The depth, d, of a lake is 500 m, rounded to the nearest integer.
Write the error interval for d in the form a ≤ d < b.
Answer: The depth, d, of a lake is 500 m, rounded to the nearest integer. The error interval for d is 499.5 ≤ d < 500.5
When a value is rounded to the nearest integer, the possible error is half of the smallest unit of measurement. In this case, the smallest unit of measurement is 1 meter, so the possible error is 0.5 meters. Therefore, the error interval for d is from -0.5 meters to +0.5 meters around the rounded value of 500 meters.
This can be expressed in the form a ≤ d < b as follows:
a = 500 - 0.5 = 499.5
b = 500 + 0.5 = 500.5
So, 499.5 ≤ d < 500.5
Step-by-step explanation:
For each of the following expression, write the shape of the resulting tensor. Use commas to separate thedimensions. For example, if the resulting tensor is a 2 by 3 matrix then write 2,3. If the expression is invalid,write None.ones (100) + ones (100)ones (100,1)+ ones (100,1)ones (100,1)+ones (1,100)ones (2,3) + 1
The result of this addition is a single tensor with 100 elements, which has the shape 100. Therefore, the shape of the resulting tensor is 100.
The expression is ones(100) + ones(100). This expression is valid, and the shape of the resulting tensor is a 100-dimensional vector.
We can calculate this by starting with the shape of each of the tensors. The first tensor, ones(100), is a 1-dimensional tensor with 100 elements. The second tensor, ones(100), is also a 1-dimensional tensor with 100 elements.
When we add these two tensors together, we are combining the elements of each tensor into a single tensor. In this case, we are combining the 100 elements from the first tensor with the 100 elements from the second tensor, resulting in a single tensor with 100 elements.
The result of this addition is a single tensor with 100 elements, which has the shape 100. Therefore, the shape of the resulting tensor is 100.
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Ask a question about your assignment
Answer: 146
Step-by-step explanation:
HK = GK x 2 because a rectangles diagonals are equal.
73 x 2 = 146
Answer:
146 that should be right
How do I check
89,235
-66,456
______
22,779
he triangles on the grid below represent a translation. On a coordinate plane, triangle A B C is shifted 5 units to the right and 3 units up to form triangle A prime B prime C prime. To form the image, the pre-image moved five units right and three units up. two units right and three units up. five units left and three units down. two units left and three units down.
To form the image, the pre-image should move five units right and three units up.
What are coordinates?A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane. The x-coordinate and the y-coordinate are two numbers that define a point's location on a 2D plane.
Given, The triangles on the grid below represent a translation.
On a coordinate plane, triangle A B C is shifted 5 units to the right and 3 units up to form triangle A prime B prime C prime.
In geometry, figures in a plane can be transformed in a variety of ways, including shifts and scaling, to produce new shapes. The new (transformed) shapes are called images and the original, unaltered shapes are called preimages.
Therefore, The pre-image should travel five units right and three units up to form the image.
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How do you find all six trigonometric function of θ if the point (-5,12) is on the terminal side of θ?
The six trigonometric functions of an angle θ in a right triangle are sine (sin θ), cosine (cos θ), tangent (tan θ), cotangent (cot θ), secant (sec θ) and cosecant (csc θ).
To find these functions for a given angle θ, we need to know the length of the sides of a right triangle with angle θ.
Given that the point (-5,12) is on the terminal side of θ, we can apply the Pythagorean Theorem to find the length of the sides of the triangle. The Pythagorean Theorem states that a² + b² = c² where a and b represent the lengths of the sides of the triangle and c is the length of the hypotenuse. Using the point (-5,12), we can solve for c, which is 13.
Therefore, the sides of the triangle have lengths 5, 12, and 13. We can now use these lengths to calculate the trigonometric functions of θ.
sin θ = 12/13
cos θ = 5/13
tan θ = 12/5
cot θ = 5/12
sec θ = 13/5
csc θ = 13/12
Therefore, the six trigonometric functions of θ are sin θ = 12/13, cos θ = 5/13, tan θ = 12/5, cot θ = 5/12, sec θ = 13/5 and csc θ = 13/12.
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Directions: solve for x round to the nearest tenth
Using the trigonometric ratios for the angle A-cosθ= Base/ Hypotenuse =11.8726
What is trigonometric?
Trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions in relation to a right triangle are displayed in the figure.
For example, the triangle contains an angle A, and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of A, or sin A; the other trigonometry functions are defined similarly. These functions are properties of the angle A independent of the size of the triangle, and calculated values were tabulated for many angles before computers made trigonometry tables obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures.
Here, if we consider the ∠A=θ, then AB is the base, BC is the perpendicular, and AC is the hypotenuse.
And it is given that, ∠A=θ=32°,AB=x, and C=14 units.
Now, using the trigonometric ratios for the angle A-cosθ= Base/Hypotenuse
cosA=AB/AC
cos30°=x/14
x=14×cos32°
=11.8726
≈11.9units
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Ana is 2 years older than Wanda. let w represent Wanda's age.
The expression that represents Ana's age is 2 + w.
What is an expression?A mathematical expression is made up of a statement, at least two integers or variables, and one or more arithmetic operations. This mathematical operation enables the multiplication, division, addition, or subtraction of numbers. The following is the structure of an expression:
Expression: (Number/Variable, Math Operator, Math Operator)
Given:
let w represent Wanda's age.
let a represent Ana's age.
Also given Ana is 2 years older then Wanda
Ana's age can be calculated as Wanda's age plus 2
Hence the expression which represents Ana's age is given below
Ana's age = 2 + Wanda's age
a = 2 + w
Hence Ana's age is 2 + w.
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An imported item costs 500$. The retailer's plans to sell with a 70% increase in import price. After selling for a while, the seller decides to discount 20% of the first sale price and continue to discount 10% of the second sale price. Find the final sale price?
Profit is determined using the equation Profit = Selling Price - Cost Price. The final price after discount is $612.
Following this, the formula for calculating profit percentage is: profit percentage = (profit/cost price) x 100.
What is the loss percent formula?
The firm is considered to be losing money if a product's cost price is higher than its selling price; conversely, if the product's cost price is lower than its selling price, a profit might be made. Loss% is equal to Loss/CP multiplied by 100.
Profit is calculated as cost less selling price. Similar to this, the cost price is higher than the targeted selling price when a loss occurs. A profit percentage is computed by dividing the cost price by 100. 100 divided by the cost price is the loss percentage.
Determine the net profit and loss from a company's operations over the course of an accounting period with the help of the profit and loss account, also referred to as the income statement. The trading account's gross profit or loss is transferred to this account.
Given,
the cost price = $500
Selling price = 70% increase in import price
70 / 100 × $500 = $ 350
Selling price = $350 + $500
Selling price = $850
Selling price after 20% discount = 20% of $850
⇒ 0.2 × 850 = 170
⇒ Selling price after 20% discount = $850 - $170 = $680
Selling price after discount 10% = 0.10 × $680 = $68
Selling price after discount 10% = $680 - $68 = $612
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Select the correct answer from the drop-down menu.
June collected data showing the monthly expenses of some girls at her school. She wants to know the difference between the highest monthly expenses and the lowest monthly expenses.
To do that, she has to find the ________
To find the difference between the highest monthly expenses and the lowest monthly expenses, June has to find the range of the data.
help what's the answer. What is the value of 12 – 4 + 2h when h = 3?
Answer:
14
Step-by-step explanation:
To find the value of 12 – 4 + 2h when h = 3, we need to substitute the value of h into the expression and simplify.
When h = 3, we can substitute this value into the expression:
12 – 4 + 2h = 12 - 4 + 2*3 = 12 - 4 + 6 = 12 - 4 + 6 = 8 + 6 = 14
So the value of 12 – 4 + 2h when h = 3 is 14.
Answer:
12 - 4 + 2h = 14
Step-by-step explanation:
Given expression,
→ 12 - 4 + 2h
Now we have to use,
→ h = 3
Let's simplify the expression,
→ 12 - 4 + 2h
→ 12 - 4 + 2(3)
→ 12 - 4 + 6
→ 8 + 6
→ 14 => {final value}
Hence, required value is 14.
b. Mai says that the tiles have the same area as one-foot by one-foot tiles. Do you
agree with Mai? Explain or show your reasoning.
Answer:
yes most are 12x12 inch tiles so yes
Step-by-step explanation:
Sarah and Thomas each roll a die. Whoever rolls the higher number wins; if they both roll the same number, neither wins. a.What is the probability that Thomas wins?
b.If Sarah rolls a 3, what is the probability that she wins?
c.If Sarah rolls a 3, what is the probability that Thomas wins?
d.If Sarah wins, what is the probability that Thomas rolled a 3
e.If Sarah wins, what is the probability that Sarah rolled a 3?
Rolls the higher number wins; if they both roll the same number, neither wins.
The answer is-
a)They both start the game with an equal chance of winning, therefore 1/2
b) If she rolls a 3, Thomas must roll a 1 or 2 for her to win, making her chance of winning 1/3.
c) Thomas has a 50% chance of winning if she rolls a three because he has to roll a 4,5 or 6 to do so.
d) In order for Sarah to win, she had to roll a 2, 3, 4, or 6. If Thomas had rolled a 1, 2, 3, 4, or 5, he would have lost. Thus, there is a 1/5 chance that he will roll a 3.
e) If Sarah wins she must have rolled a 2,3,4,5 or 6. So the probability that she rolls a three is also 1/5.
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do the ordered pairs plotted in the graph below represents a function explain
Answer: it's not a function
Step-by-step explanation:
(4,3) (4,-3)
one to many
it isn't a function.
When looking for the domain of a graph, which values are you looking for?
x-values
y-values
Every one of a method's x-values, or entries, make up the domain, and all of a substring y-values, or outputs, make up the range. The domain of a vertex is every value in the structure, from left to right. The graph's range includes all integers from lower to higher.
Find the future value of a simple interest loan of $15,500 at 6.2% interest for 11 months
$
The amount with simple interest is A = $ 16,380.92
What is Simple Interest?Simple interest is a method of calculating interest that ignores the impact of compounding. While interest frequently compounds throughout the course of a loan's set periods, simple interest does not. Simple interest is calculated by multiplying the principal amount by the interest rate, times the number of periods.
Simple Interest = ( Principal Amount x Rate x Time Period ) / 100
Given data ,
Let the simple interest be represented as I
Now , the amount with simple interest = A
And , the principal P = $ 15,500
The rate of interest R = 6.2 %
The number of months n = 11 months
So , the number of years T = 11/12 years
Simple Interest I = ( Principal Amount x Rate x Time Period ) / 100
Substituting the values in the equation , we get
Simple Interest I = ( 15,500 x 6.2 x ( 11/12 ) ) / 100
On simplifying the equation , we get
Simple Interest I = $ 16,380.916987
Simple Interest I = $ 16,380.92
Hence , the future value if $ 16,380.92
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each side of a regular septagon has length 5 inches find the area of the region between the sptagon and its circumscribing circle?
The area of the region is equal to the area of the septagon minus the area of the circumscribing circle. The area of the septagon = 7 x (5^2 x (1 + sqrt(3))/4). The area of the circumscribing circle = pi x (5^2/2).
The area of the region between the septagon and its circumscribing circle can be found by subtracting the area of the circumscribing circle from the area of the septagon. The side length of the septagon is 5 inches. The area of the septagon can be calculated using the formula A = 7 x (5^2 x (1 + sqrt(3))/4). This is equal to 58.82 inches squared. The area of the circumscribing circle can be calculated using the formula A = pi x (5^2/2). This is equal to 78.54 inches squared. Therefore, the area of the region between the septagon and its circumscribing circle is 58.82 inches squared minus 78.54 inches squared, which is equal to 19.72 inches squared.
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What fraction of 350 gallons is 135 litres
Answer:
10.19%
Step-by-step explanation:
1 gallon = 3.785 liters
350 gallons = 350 × 3.785 liters = 1324.75 liters
percent = part/whole × 100%
percent = 135 liters / 1324.75 liters × 100%
percent = 10.19%
Joel's town voted on a new law. 48 out of 50
votes were in favor of the law. What is the ratio of the number of votes against the law to the number of votes in favor of the law?
The total number of votes cast is 480
What is the ratio of the number of votes against the law to the number of votes in favor of the law?Ownership of more than 50% of voting shares generally gives the right of control and consolidation. In special cases, control is possible without having to own more than 50% of voting stock.
Let x represent total number of votes.
step1:
We have been given that Joel got 50% of the votes, which was 48 votes more than Sean.
50% of total votes:50/100[tex]x[/tex]=0.50[tex]x[/tex]
Since Joel got 50% of total votes, so Sean got 40% of total votes that is
40/100[tex]x[/tex]=0.40[tex]x[/tex]
Since Joel got 38 votes more than Sean, so we can represent this information in an equation as:
0.50[tex]x[/tex]=0.40[tex]x[/tex]+48
0.50[tex]x[/tex]-0.40[tex]x[/tex]=0.40[tex]x[/tex]-0.40[tex]x[/tex]+48
0.10[tex]x[/tex]=48
step2:
0.10[tex]x[/tex]/.10=48/.10
[tex]x[/tex]=480
Therefore, the total number of votes cast is 480
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The value of a baseball card in dollars has been found to be 0.35y + 0.15, where y is the number of years since it was released. By how much is the baseball card's value increasing per year?
35%
0.15%
.35%
15%
The value of baseball cards is increasing by 35% per year.
Option A is the correct answer.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 4 = 9 is an equation.
We have,
The value of a baseball card in dollars is given by,
0.35y + 0.15
Where y is the number of years since it was released.
This means,
The value of the baseball card is 0.15.
There is a yearly increase of 35%.
35% = 35/100 = 0.35
Thus,
The value of baseball card is increasing at 35% per year
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one unit of a is made of two units of b, three units of c, and two units of d. b is composed pf one unit of e and two units of f. c is made of two units of f and one unit of d. e is made of two units of d. items a, c, d, and f have one-week lead times; b and e have lead times of two weeks. lot-for-lot (l4l) lot sizing is used for items a, b, c, and d; lots of size 50 and 180 are used for items e and f, respectively. item c has on-hand (beginning) inventory of 15; d has an on-hand inventory of 50; all other items have zero beginning inventories. we are scheduled to receive 20 units of item e in week 2; there ae no other scheduled receipts. construct simple and low-level-coded bill-to-materials (product structure tree) and indented and summarized parts lists. if 20 units of a are required in week 8, use the low-level-coded bill-of-materials to find the necessary planned-order releases for all components.
Data handling is the process of ensuring that research data is stored, archived, or disposed of in a safe and secure manner during and after the conclusion of a research project.
The following are the given details:
Item Leadtime On hand Inventory Lot sizing criteria Schedulereceipts
A 2 0 L4L 10 in week 2
B 1 0 LAL 0
C 1 10 50 0
D 2 0 50 0
E 1 50 180 50 in week 1
F 1 180 L4L 50 in week 1
The Complete MRP schedule can be seen in the attached images below:
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The gross requirements for each component are as follows:
b = 2 x a = 44 units in week 7
c = 3 x a = 66 units in week 7
d = 2 x e + 1 x c + 2 x a
= 110 units in week 3, 56 units in week 6, 44 units in week 7
e = 1 x b = 44 units in week 5
f = 2 x b + 2 x c = 88 units in week 5 and 112 units in week 6
Net requirements = Gross requirements - Projected on hand
Projected on hand = Planned order receipts - net requirements
Planned order receipts = net requirements
Planned order releases = net requirements adjusted as per the lead time
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the present age of Neha is 4 times the age of her son 4 years ago .if her son is 14 years old, find the present age of Neha.
Answer: 4*4= 16
Step-by-step explanation:
if Neha is 4 years older than her son, so she will be sixteen.
Given an integer N where 1 ≤ N ≤ 105, the task is to find the number of positive integers less than or equal to N that have an odd number of digits without leading zeros.
if N < 10 then the count of numbers will be N , if N / 10 < 10 then 9 , if
N / 100 < 10 then 9 + N – 99 , if N / 1000 < 10 then 9 + 900 , if N / 10000 < 10 then 909 + N – 9999.
in all other case 90909.
what is leading zeros?
Leading zeros are zeros that appear at the beginning of a number or numerical value. These zeros are used to align digits or to indicate the number of digits in a number, but do not affect the value of the number itself. For example, 007 and 7 are the same number but the leading zeros indicate that there are 3 digits in the number.
The above contions specifies the count of number having odd number of digits and without leading zeros.
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1. Which are you trying to show is true in Step 3 of a proof by mathematical induction for the statement "For all the positive integers, n³ + 2n is divisible by 3"?
On³+2n=k³ +2k is divisible by 3
O (k+1)³ +2(k+1) is divisible by 3
On³ +2n is divisible by 3
Ok³+2k is divisible by 3
n³ +2n is divisible by 3 is the statement that we want to prove and A) n³+2n=k³ +2k is divisible by 3 is not a valid statement.
Option (A) is correct.
What is mathematical induction?
Mathematical induction is a method of proof used to establish the truth of a statement for all positive integers (or natural numbers).
In step 3 of a proof by mathematical induction, you are trying to show that if the statement is true for some positive integer k, then it is also true for the next positive integer (k+1). So you would be trying to show that:
B) (k+1)³ +2(k+1) is divisible by 3
is true, assuming that
D) k³+2k is divisible by 3
is true.
In other words, you are trying to show that if the statement is true for some positive integer k, then it is also true for the next positive integer (k+1).
Hence, n³ +2n is divisible by 3 is the statement that we want to prove and A) n³+2n=k³ +2k is divisible by 3 is not a valid statement.
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2. What is the third step in proving by mathematical induction that for every positive integer n, 11" - 6 is divisible by 5, is true?
O Show that 11k - 6 is divisible by 5.
O Assume that 11¹ - 6 is divisible by 5.
O Assume that 11*+1 - 6 is divisible by 5.
Show that 11*+1 - 6 is divisible by 5
To show that 11k+1 - 6 is divisible by 5 using mathematical induction, the first step is to assume that the statement is true for any given positive integer k.
What is the third step in proving by mathematical induction that for every positive integer n, 11" - 6 is divisible by 5, is true?This means that 11k - 6 is divisible by 5.Next, we need to show that this assumption implies that 11k+1 - 6 is also divisible by 5.We can do this by substituting 11k+1 - 6 for 11k + 5 - 11 + 6, which can be simplified to 11k + 5 - 11 + 6.Then, we can subtract 11k from both sides, leaving us with 5 - 11 + 6. Finally, we can add 11 to both sides, resulting in 16, which is divisible by 5.Therefore, 11k+1 - 6 is divisible by 5.To show that 11k+1 - 6 is divisible by 5, we will use mathematical induction.First, we assume that 11¹ - 6 is divisible by 5.Then, we assume that 11*+1 - 6 is divisible by 5.To prove that the statement is true, we must show that 11*+1 - 6 is divisible by 5.We can do this by multiplying both sides of the equation by 11: 11*+1 - 6 = 5(11k - 5).This equation shows that 11k+1 - 6 is equal to 5 times 11k - 5, which is divisible by 5.Therefore, 11k+1 - 6 is divisible by 5.This completes the proof by mathematical induction that for every positive integer n, 11" - 6 is divisible by 5 is true.To learn more about the mathematical induction refer to:
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