Answer:$52.60
Step-by-step explanation: You have a initial cost of $42 and you have to add 30 %. When you add the 30 % you get 12.60 more dollars added to the amount. This will then give you an answer of $52.16.
For the function f(x) x 9, find (f f 1)(5)
the value of [tex]f o f^{-1}[/tex](5) of the function f(x) = x + 9 is 5
What is the inverse of a function?An inverse function or an anti function is defined as a function, which can reverse into another function. In simple words, if any function “f” takes x to y then, the inverse of “f” will take y to x. If the function is denoted by ‘f’ or ‘F’, then the inverse function is denoted by f^-1 or F^-1. One should not confuse (-1) with exponent or reciprocal here.
[tex]f^{-1}[/tex](x) =
if f(x) = x + 9
setting x + 9 = y and making x the subject we have
y = x + 9
x = y - 9
replacing x with [tex]f^{-1}[/tex](x) and y with x we have
[tex]f^{-1}[/tex](x) = x - 9
so the f o [tex]f^{-1}[/tex](x) can be gotten by substituting x in f(x) with x - 9 so that we have
f o [tex]f^{-1}[/tex](x) = (x - 9) + 9
f o [tex]f^{-1}[/tex](x) = x
so f o [tex]f^{-1}[/tex](5) will be x = 5
which becomes
f o [tex]f^{-1}[/tex](5) = 5
In conclusion f o [tex]f^{-1}[/tex](x) is 5
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pls whats the anwer im stuck 4+5(x-1)=34
Answer: 7
Step-by-step explanation:
To solve this equation, we first need to use the order of operations to simplify the expression on the left-hand side of the equation. The order of operations tells us that we should perform the operations within parentheses first, followed by any exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).
In this case, we have a set of parentheses with the term 5(x-1) inside. This means we need to calculate the value of 5(x-1) first. We can do this by multiplying 5 and (x-1), which gives us 5x-5.
Next, we need to add 4 to 5x-5 to get the final expression on the left-hand side of the equation. This gives us 4+5x-5, which simplifies to 5x-1.
Now we have the equation 5x-1=34. To solve this equation for the value of x, we need to isolate the x term on one side of the equation by moving all the other terms to the other side. We can do this by adding 1 to both sides of the equation, which gives us 5x = 35.
Finally, we can divide both sides of the equation by 5 to find the value of x. This gives us x = 35/5 = 7. Therefore, the value of x that makes the equation true is 7.
Answer:7
Step-by-step explanation:
4+5(x-1)=34
Distribute: 4+5(x - 1) = 34
4+5x - 5=34
Subtract the numbers: 4 + 5x - 5=34
-1 + 5x =34
Rearrange terms: -1 + 5x =34
5x - 1 = 34
Solution: x=7
Hope this helps!!
Question 3 (2 points)
For the following, find the arc length:
(Use π=3.14 and round your final answer to the hundredths.)
The length of the arc is 10.47 units
How to determine the length of the arc?From the question, we have the following parameters that can be used in our computation:
Radius = 10
Angle = π/3
The length of the arc can be calculated using
Arc length = Radius * Angle
Substitute the known values in the above equation, so, we have the following representation
Arc length = 10 * π/3
Evaluate
Arc length = 10.47
Hence, the arc length is 10.47
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find the maximum area of the rectangle inscribed in the triangle formed by the x x-axis, the y y-axis and the line y
A shape's highest possible area is called its maximum area.The size of the rectangle that will result in the greatest area is 4 by 2 units.
What is meant by rectangle?Four sides, four corners, and four 90° right angles make up the closed, 2-D shape of a rectangle. A rectangle has parallel, equal sides on either side. Rectangles are two-dimensional shapes, and as such, they have length and breadth as their defining characteristics. The rectangle's shorter side is its width, while its longer side is its length.How to calculate a rectangle's area. The size of a rectangle. A = l × b. Once the length and breadth are known for any rectangle, the area may be determined. The area of the rectangle is calculated as a square-unit dimension by multiplying length and width.The equation is given as:
x + 2y - 8 = 0
Rewrite as:
2y = 8 - x
Divide both sides by 2,
y = [tex]\frac{8 - x}{2}[/tex]
y = 4 - [tex]\frac{x}{2}[/tex]
The area (A) of the rectangle is:
A = xy
Substitute
y = 4 - [tex]\frac{x}{2}[/tex]
A = x[tex](4 - \frac{x}{2})[/tex]
Expand
A = 4x - [tex]\frac{x^{2} }{2}[/tex]
Differentiate
A' = 4 - x
Set the derivative to 0,
4 - x = 0
Collect like terms
x = 4
Recall that:
y = 4 - [tex]\frac{x}{2}[/tex]
Substitute 4 for x,
y = 4 - [tex]\frac{4}{2}[/tex]
y = 4 - 2
y = 2
So, we have:
x = 4 and y = 2
Hence, the dimension of the rectangle that will produce the maximum area is 4 units by 2 units.
The complete question is:
A rectangle is inscribed in the region bounded by the x-axis, the y-axis, and the graph of x+2y-8=0. Approximate the dimensions of the rectangle that will produce the maximum area.
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I need help with the word problem
Answer:
BAG 1 IS BETTER. IT IS BETTER TO PAY $18 FOR 12 POUNDS OF CAT FOOD.
Step-by-step explanation:
To find the unit cost, we need to know how much one pound of cat food cost for each bag
Bag 1
It said you get 12 pounds for $18
Divide $18 by 12 to find how much ONE pound is.
18 ÷ 12 = 1.5
ONE POUND COST $1.5
This is the unit cost
Let's do the same for the other bag of cat food!
Bag 2
Divide $24 by 15
24 ÷ 15 = 1.6
ONE POUND COST $1.6
This is the unit cost
It is better to pay for $1.5 per pound because it is LESS than $1.6
You will save money this way.
BAG 1 IS BETTER. IT IS BETTER TO PAY $18 FOR 12 POUNDS OF CAT FOOD.
In an adjacency ____ representation, corresponding to each vertex v is a linked list such that each node of the linked list contains the vertex u, and there is an edge from v to u.
In an adjacency list representation, corresponding to each vertex v is a linked list such that each node of the linked list contains the vertex u, and there is an edge from v to u.
This is a common way to represent a graph in computer memory, where each vertex is represented by an object that stores its adjacent vertices in a list. This representation is useful when the graph is sparse, meaning it has relatively few edges compared to the number of vertices it has. It is not as efficient for dense graphs, where the number of edges is close to the maximum possible number of edges.
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Which points are included in the graph of the ceiling function f(x) = ⌈x⌉? Select all that apply.
A. (−0.5, −1)
B. (−1.5, −2)
C. (−3.5, −3)
D. (−3.5, −4)
E. (3.001, 4)
F. (−3.001, −3)
Answer:
C, E, F
Step-by-step explanation:
You want to know which points match the ceiling function among ...
A. (−0.5, −1)B. (−1.5, −2)C. (−3.5, −3)D. (−3.5, −4)E. (3.001, 4)F. (−3.001, −3)CeilingThe ceiling function is also known as the least integer function. It returns the smallest integer greater than or equal to its argument value. It is essentially a "round to next higher" function, with any integer input remaining unchanged.
The attachment shows the input and output of the ceiling function for the values on the answer choice list. C, E, F are points on the function's graph.
If the slope of the regression line is calculated to be 2.5 and the intercept 16 then the value of Y when X is 4 is:
Select one or more:
a.2.5
b.66.5
c.26
d.16
In order to calculate linear regression, we apply the formula Y = B + X. The current access level for 16 plus is 2.5. As a result, Option A is the right response. We get 26 from 16 + 10.
How do you interpret the slope and intercept of a regression line?The slope and intercept of a line show how steep it is, and the intercept shows where the line crosses an axis. It is possible to calculate an average rate of change using the slope and intercept, which describe the linear relationship between two variables.
An equation of the form Y = a + bX, where X is the explanatory variable and Y is the dependent variable, represents a linear regression line. A line's intercept (the value of y when x = 0) is equal to the slope of the line, which is b.
The regression slope intercept equation, [tex]y' = b_{0} + b_{1} x[/tex], where "[tex]b_{0}[/tex]" is the y-intercept and "[tex]b_{1} x[/tex]" is the slope, is actually just an algebraic version of the regression equation. Once the equation for linear regression has been discovered,
Therefore the correct answer is option a ) 2.5
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Business Organization
1. James invests $10,000 in a partnership with 3 other people. One of those people also invested
$10,000 and the other two invested $90,000 each. What percent of the business does James
own?
(please help me if I don't get help I won't graduate)
James owns just 5% of the company after the investment.
What is a percentage?Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100.
Percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%".
Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. It is given by:
Percentage = (value / total value) * 100%
How to find the Percentage owned by James;
James' investment: $10,000
The other 3 people invested;
x = $10,000
y = $10,000
z = $10,000
Total investment made was = $200,000
James' Percentage = [tex]\frac{10000}{200000\\}[/tex] x 100
James' Percentage = 5%
So, we can say that James owns 5% of the total investment in the business.
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I need help with my math homework
Answer: Work shown below. Have a nice day! :)
Caroline went to the grocery store and bought bottles of soda and bottles of juice.
Each bottle of soda has 30 grams of sugar and each bottle of juice has 15 grams of sugar. Caroline purchased a total of 6 bottles of juice and soda which collectively contain 150 grams of sugar. Graphically solve a system of equations in order to determine the number of bottles of soda purchased, ∞, and the number of bottles of juice purchased, y.
The solution of the system of equations is the point of intersection (4, 2), as shown in the graph below.
Caroline bought 4 soda bottles.
Caroline bought 2 juice bottles.
How to Solve a System of Equations?In order to determine the solution to this problem, use the information given to create two equations.
Number of soda bottle = x.
Number of juice bottle = y.
First Equation for the total number of bottle that Kelvin bought
x + y = 6 --> eqn. 1
Second Equation 2 for the total sugar in the soda bottles and total sugar in juice bottles that equals 150 grams of sugar.
30x + 15y = 150
Simplify by dividing by 5
6x + 3y = 30 --> eqn. 2
Rewrite eqn. 1:
y = 6 - x
Solve simultaneously for the variables of x and y, by substituting the value of y into equation 2:
6x + 3(6 - x) = 30
6x + 18 - 3x = 30
3x + 18 = 30
3x = 30 - 18
3x = 12
3x/3 = 12/3
x = 4
Caroline bought 4 soda bottles.
Substitute x = 4 into equation 1 to find the value of y:
x + y = 6
4 + y = 6
y = 6 - 4
y = 2
Caroline bought 2 juice bottles.
The graph below shows the point of intersection, (4, 2), which is the solution.
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Which of the following statements regarding the expansion of (x + y)^n are
correct?
A. The coefficients of x-1 and y- 1 both equal 1.
B. For any term xay in the expansion, a - b = n.
C. The coefficients of x and y both equal 1.
D. For any term xay in the expansion, a + b = n
The true statements are,
A. The coefficients of xⁿ and yⁿ both equal 1.
D. For any term x and y in the expansion, a + b = n
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ (x + y)ⁿ
Now,
We know that;
A binomial expansion is,
⇒ [tex](x + y)^n = x^n + nx^n^-^1 y + ^{n} C_{2} x^n^-^2y^2 + ..... + nxy^n^-^1 + y^n[/tex]
Clearly, For Option A:
The coefficients of xⁿ and yⁿ both equal 1.
If we see first and last term of the expansion, This statement is true.
For Option D:
For any term xᵃyᵇ in the expansion, a + b = n.
Let we take 3rd term of expansion
Here, a = n-2 and b=2
So, a + b = n-2+2 = n
Thus, a + b = n is true statement.
For Option B:
For any term xᵃyᵇ in the expansion, a - b = n.
Let we take 3rd term of expansion
Here, a=n-2 and b=2
If we do a-b = n-2-2=n-4≠n
Thus, a-b=n is false statement.
Thus, The true statements are given below,
A. The coefficients of xⁿ and yⁿ both equal 1.
D. For any term x and y in the expansion, a + b = n
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Maya has 120 caramel apples to sell. Each caramel apple is covered with one topping.
• of the caramel apples are covered with peanuts.
• are covered with chocolate chips.
• are covered with coconut.
• The rest are covered with sprinkles.
How many caramel apples are covered with sprinkles?
The number of caramel apples that are covered with sprinkles is; 20
How to solve algebra word problems?We are given;
Total number of caramel apples to sell = 120
1/5 of the caramel apples are covered with peanuts.
1/3 of the caramel apples are covered with chocolate chips.
3/10 of the caramel apples are covered with coconut.
The rest are covered with sprinkles.
Let 'n' be the number of caramel apples are covered with sprinkles. Thus, we can develop the equation as;
120 = (1/5 * 120) + (1/3 * 120) + (3/10 * 120) + n
120 = 24 + 40 + 36 + n
n = 120 - 100
n = 20
Which means that 20 are those covered with sprinkles
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Complete question is;
Maya has 120 caramel apples to sell. Each caramel apple is covered with one topping.
1/5 of the caramel apples are covered with peanuts.
1/3 of the caramel apples are covered with chocolate chips.
3/10 of the caramel apples are covered with coconut.
The rest are covered with sprinkles.
How many caramel apples are covered with sprinkles?
A student is to be selected randomly from a group of students. For each classification of freshman and sophomore, there is a math major, an art major, and a biology major. The probability of each individual being selected is given in the following table:
Find the probability that
(a) a freshman is selected.
(b) an art major is chosen.
(c) a freshman math major or a sophomore biology major is chosen.
A freshman has a 0.35 chance of being chosen. A freshman has a 0.38 chance of being chosen. The likelihood of a freshman being chosen is 0.23.
What is probability?A probability is a number that represents the possibility or chance that a specific event will occur. Probabilities can be stated as proportions ranging from 0 to 1, as well as percentages ranging from 0% to 100%. It is predicated on the likelihood of something occurring. The rationale underlying probability is the foundation of theoretical probability. If a coin is tossed, the theoretical probability of receiving a head is 12.
Here,
.................. Math .. Art . Biology Total
Freshman . 0.10 . 0.08 . 0.17 .... 0.35
Sophomore 0.22 . 0.30 . 0.13 ... 0.65
Total ......... 0.32 . 0.38 . 0.30 ...... 1
(a) 0.35
(b) 0.38
(c) 0.10 + 0.13 = 0.23
The probability that a freshman is selected is 0.35. The probability that a freshman is selected is 0.38. The probability that a freshman is selected is 0.23.
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1. The graph of a quadratic function is called a(an)
Copy and complete the table of values for the function.
2.
y=-1/3 x²
The graph of a quadratic function is called a parabola
The complete table is
x = -6, -3, 0, 3 6
y = -12 -3 0 -3 -12
How to complete the table of values?From the question, we have the following equation that can be used in our computation:
y=-1/3 x²
From the table of values , we have the following x values
x = -6, -3, 0, 3 and 6
Substitute x = -6, -3, 0, 3 and 6 in y=-1/3 x²
So, we have the following representation
y = -1/3 (-6)² = -12
y = -1/3 (-3)² = -3
y = -1/3 (0)² = 0
y = -1/3 (3)² = -3
y = -1/3 (6)² = -12
So, the complete table of values is
x = -6, -3, 0, 3 6
y = -12 -3 0 -3 -12
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For Christmas you want to get your parents a framed family picture. At the framing store, there are 4 different styles each available in 5 different colors. You decide to use a blue mat board and there are 3 different shades of blue to choose from. How many different frames can you create?
The number of different frames you can create, if There are 4 different styles, each available in 5 different colors, there are 3 different shades of blue to choose from, is 12.
What is the combination?A combination is a choice of items from a group of different items, where the order of the choices is irrelevant.
Given:
There are 4 different styles, each available in 5 different colors, there are 3 different shades of blue to choose from,
Calculate the total number of frames, we can create as shown below,
The number of frames = different styles available for blue color × shades of blue
The number of frames = 4 × 3
The number of frames = 12
Thus, the number of frames you can create is 12.
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what are the coordinates of the point 3/5 of the way from A(3,-4) to B(13, 11)?
American children watch an average of 25 hours of television per week with a standard deviation of 8 hours. A random sample of 40 children is selected. What is the probability the mean number of hours of television they watch per week is less than 22? Question 15 options: 1) 0.3520 2) 0.0089 3) 0.9911 4) 0.6480
If American children watch an average of 25 hours of television per week with a standard deviation of 8 hours. the probability the mean number of hours of television they watch per week is less than 22 is: 2) 0.0089.
How to find the probability?First step is to find the standard deviation
σ = 8/√40
σ = 1.2649
Using this formula to find the Z-score
Z = x - μ/σ
Z =22-25/1.2649
Z = -3//1.2649
Z = -2.37
Hence,
P-value=P(Z < -2.37)
P-value =0.0089
Therefore the correct option is 2.
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Cómo se determina el condominio de la función cuadrática.
The domain is the set of all values in the set (- ∞, ∞).
What is a mathematical function, equation and expression?Function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function
Expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators
Equation : A mathematical equation is used to equate two expressions.
Given is a quadratic function as follows -
ax² + bx + c
For any quadratic equation (with vertex at origin), the value of [y] or the range is either the set of all positive values from 0 to infinity when the quadratic function open upwards and the set of all negative values from 0 to -infinity when the quadratic function open downwards. The domain is the set of all values in the set (- ∞, ∞).
Therefore, the domain is the set of all values in the set (- ∞, ∞).
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{Question in english language is as follows -
How the domain of the quadratic function is determined.}
Weekly wages at a certain factory are
normally distributed with a mean of
$400 and a standard deviation of $50.
Find the probability that a worker
selected at random makes between
$500 and $550.
Answer:
The probability that a worker selected at random makes between $500 and $550 is approximately 0.24.
Step-by-step explanation:
To calculate this probability, we can use the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. We can convert the weekly wage amounts to standard units by subtracting the mean and dividing by the standard deviation.
For wages between $500 and $550, the corresponding standard units are:
$500 - $400 = $100 / $50 = 2
$550 - $400 = $150 / $50 = 3
We can use a standard normal table or a calculator to find the probability that a value chosen at random from the standard normal distribution falls within the range of 2 to 3 standard units. This probability is approximately 0.24.
Since the weekly wages at the factory are normally distributed with a mean of $400 and a standard deviation of $50, this probability is also the probability that a worker selected at random makes between $500 and $550.
You are playing a new video game. The table shows the proportional relationship between the number of levels completed and the time it took you to complete them.
Number of Levels 3 5
Time (hours) ? 2.5
How many minutes does it take you to complete 3 levels?
105 minutes
90 minutes
60 minutes
50 minutes
PLS HELP
Answer:
the answer is 90 mins option b
Step-by-step explanation:
Answer:
the answer is 90 minutes or B
hope this helps
mark me brainliest
Step-by-step explanation:
On May 26, your balance is $897.96. On May 27, you write check #117 to your landlord, Mrs. Wall, for $366.00. On May 28, you use receive a rebate check for $62.52. On May 29, you write a check for $137.32 to Bank of America for your car payment. On May 30, you receive your monthly check from your employee of $658.50 On May 31, you write a check for $685 to your parents for supporting you the previous month. What is the ending balance for the month of May?
Bank reconciliation is an accounting process to bring the balance as per the bank statement to agree with the balance as per the checkbook.
What is bank reconciliation?The journal entry to adjust the checkbook balance is as follows:
May 1 Debit Bank Charges $20
Credit Checkbook $20
The preparation of the Bank Reconciliation Statement is as follows:
Balance as per bank statement $16,833.19
Add Uncleared Deposits $1,650.00
Less Outstanding Checks $2,040.84
Balance as per adjusted checkbook $16,442.35
Checkbook balance on May 31 = $16,462.35
Bank Statement balance on May 31 = $16,833.19
Uncleared Deposits:
May 29 $575
May 30 $1,075 $1,650
Outstanding Checks:
Check 189 - $127.50
Check 191 - $462.19
Check 192 - $569.42
Check 194 - $881.73 $2,040.84
Bank charges = $20.00
Checkbook Adjustment:
Balance as per checkbook $16,462.35
Less Bank charges 20.00
Adjusted checkbook balance $16,442.35
Transaction Analysis for Journal:
Bank Charges $20 Checkbook $20
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coordinate plane notes pratice
help pleaseeee tyyyvm :)
The graph in which point are plotted is given below and Point D and F lie on Y axis and X axis respectively other lie on the different quadrant
What does a math quadrant mean?
The term "quadrant" refers to each of the four divisions of the coordinate plane. Furthermore, when we speak of the sections, we also mean the sections as the coordinate axes split them. This is where the x-axis is, and this is where the y-axis is on the up-down axis. On a coordinate plane, it divides the space into four parts, as you can see.
Point A (-5,4)
Lies in 2 quadrant of the graph
Point B (-3,-2)
Lies in 3 quadrant of the graph
Point C (4,6)
Lies in 1 quadrant of the graph
Point D (0,-4)
Lies on negative y axis of the graph
Point E (2,-1)
Lies in 4 quadrant of the graph
Point F (1,0)
Lies on positive x axis of the graph
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pls help
A. G
B. H
C. The functions have the same y-intercept.
Answer:
A.) G
Step-by-step explanation:
I'd say A.) G because when you look at the graph the y-intercept is 4. When you look at the equation the y-intercept is 6. 6 is bigger than 4 so then the answer should be A.) G
The fare charged for a rideshare service is a function of
the distance traveled. However, the fare differs
according to the time of day, availability, and other
variables. The distance and fares for 10 rides are
shown in the table. The equation of the least-squares
regression line is ý = 5.20 +2.33x, where y is the
predicted fare and x is the distance.
Distance
(Miles)
1
3
5
8
10
12
15
16
56
Mark this and return
Fare
(Dollars)
1.91
13.68
16.52
24.15
24.79
39.87
40.24
53.84
What is the residual for the rideshare cost with a
distance of 16 miles?
2.33
5.21
11.35
42.49
Answer: 11.35
Step-by-step explanation: The residual for the rideshare cost with a distance of 16 miles can be calculated using the equation of the least-squares regression line and the observed fare for that distance. The residual is defined as the difference between the observed value and the predicted value.
To find the predicted fare for a distance of 16 miles, we can substitute 16 for x in the equation of the least-squares regression line: ý = 5.20 + 2.33x. This gives us ý = 5.20 + 2.33 * 16 = 42.49.
The observed fare for a distance of 16 miles is 53.84, so the residual is 53.84 - 42.49 = 11.35. Therefore, the answer is 11.35.
2^3 + (-16) / 4^2 x 5 - (-3)
which angle corresponds with 3
∠8 is the corresponds with ∠3.
Hence, option B is the correct answer.
What is Corresponding pairs ?Congruent angles are those that correspond. Corresponding pairs are all angles that are positioned in relation to the parallel and transversal lines in the same way. When two parallel lines are intersected by another line, comparable angles are the angles that are created in matching corners or corresponding corners with the transversal (i.e. the transversal).
we know that
m∠3=m∠8 --------> by corresponding angles
m∠7=m∠8 -------> by vertical angles
so
m∠3=m∠7 -------> by alternate interior angles
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NOTE: The given question is incomplete on the portal. Here is the complete question.
QUESTION: Which angle corresponds to ∠3?
A) ∠6
B) ∠8
C) ∠1
D) ∠7
The graph of linear function k passes through the points (-7, 0) and (1, 8).
Domain
2
Range
Slope
Y-intercept
X-intercept
The linear function is y = x + 7 and it has it domain and range at (-∞, ∞) respectively while it x and y - intercepts are (-7, 0) and (0, 7) respectively.
Linear FunctionA linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y.
The linear function always model the form of y = mx + c
m = slopec = y-interceptTaking the points, we can find the slope of the line
m = y₂ - y₁ / x₂ - x₁
m = 8 - 0 / 1 - (-7)
m = 8 / 8
m = 1
The slope of the line is 1
Using the slope to find the y - intercept;
y = mx + c
8 = 1(1) + c
8 = 1 + c
c = 8 - 1
c = 7
The linear function is y = x + 7
The range of this function is (-∞, ∞) and the domain of the function is
(-∞, ∞).
The x and y -intercept of the function are
x - intercept = (-7, 0)
y - intercept = (0, 7)
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a small bookshelf is 8/12 yard long. How many books can fit on the shelf if the width of each book is 1/24 yard? Explain
Answer:
16 books
Step-by-step explanation:
8/12 is equivalent to 16/24. If each book is 1/24 width, 16/1 = 16
suppose 2n c 1 numbers are selected from f1; 2; 3; : : : ; 4ng. using the pigeonhole principle, show that there must be two selected numbers whose difference is 2. clearly indicate what are the pigeons, holes, and rules for assigning a pigeon to a hole.
Yes, there must be two selected numbers whose difference is 2. Any assignment of n+1 pigeons to n holes. Then there exists at least one hole with at least 2 pigeons in it.
The Pigeon-Hole Principle :A general rule, called the pigeon -hole principle, is that if there are more pigeons than holes, there should be at least one hole with at least two pigeons. If we put n + 1 objects into n boxes, we can say that at least one box contains 2 or more objects. Example: If there are Kn+ 1 (where k is a positive integer) pigeons distributed in n holes, then the holes contain at least k + 1 pigeons. Consider 2n pair of pigeon holes (1,3), (2,4), (5,7),(6,8), .. ..,(4n-3,4n-1),(4n-2,4n). The 2 numbers in each pair have the difference 2.
If we choose 2n+1 numbers(pigeons), then they cannot all belong to different pairs.
Thus by pigeon hole principle, there must be two selected numbers whose difference is 2.
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