Using the unitary method, we can estimate that $2000 will be spent on the purchase of 250,000 sheets of paper.
What is unitary method?The unitary method is a technique in mathematics for solving a problem by finding the value of a single unit, I.e.,1,(by dividing) and then finding the necessary value by multiplying the single unit value.A method of solving a problem that involves first determining the value of a single unit, And then multiplying that value to determine the required value.According to the question,
A company can buy packages of 500 sheets of paper for $4. At that rate, how much paper can be bought for $2000?let the number of paper be "X".$4=500$2000= 250,000Hence we can say that Using the unitary method, we can estimate that $2000 will be spent on the purchase of 250,000 sheets of paper.
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Find the X intercepts of the parabola with the vertex (1,-9) and y intercept of (0,-6)
The x-intercepts of the parabola with vertex (1,-9) and y-intercept of (0,-6) are of [tex]x = 1 \pm \sqrt{3}[/tex].
What is the equation of a parabola given it’s vertex?The equation of a quadratic function, of vertex (h,k), is given by:
y = a(x - h)² + k
In which a is the leading coefficient.
In this problem, the parabola has vertex (1,-9), hence h = 1, k = -9, and:
y = a(x - 1)^2 - 9.
The y-intercept is of (0,-6), hence when x = 0, y = -6, and this is used to find a.
-6 = a - 9
a = 3.
So the equation is:
y = 3(x - 1)^2 - 9.
y = 3x² - 6x - 6.
The x-intercepts are the values of x for which:
3x² - 6x - 6 = 0.
Then:
x² - 2x - 2 = 0.
Which has coefficients a = 1, b = -2, c = -2, hence:
[tex]\Delta = b^2 - 4ac = (-2)^2 - 4(1)(-2) = 12[/tex]
[tex]x_1 = \frac{2 + \sqrt{12}}{2} = 1 + \sqrt{3}[/tex]
[tex]x_2 = \frac{2 - \sqrt{12}}{2} = 1 - \sqrt{3}[/tex]
The x-intercepts of the parabola are [tex]x = 1 \pm \sqrt{3}[/tex].
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Chef Pierre can do something unique. Using a secret process, he can bake a nearly perfectly spherical pie consisting of a vegetable filling inside a thick crust. The radius of the whole pie is 12 cm, and the radius of the filling is 8 cm. What is the volume of the crust alone, to the nearest unit? Use p = 3.14.
The volume of Chef Pierre's pie crust is calculated by subtracting the volume of the vegetable filling from the volume of the entire pie, yielding approximately 5,091 cm³.
Explanation:To determine the volume of the crust of Chef Pierre's spherical pie, we need to calculate the volume of the entire pie and then subtract the volume of the vegetable filling. The formula for the volume of a sphere is V = (4/3)πr³. First, we calculate the volume of the whole pie (including crust) with a radius of 12 cm, and then the volume of the vegetable filling with a radius of 8 cm.
Volume of whole pie: Vwhole = (4/3)π(12 cm)³ = (4/3) * 3.14 * (12 cm)³ ≈ (4/3) * 3.14 * 1,728 cm³ ≈ 7,238.56 cm³
Volume of vegetable filling: Vfilling = (4/3)π(8 cm)³ = (4/3) * 3.14 * (8 cm)³ ≈ (4/3) * 3.14 * 512 cm³ ≈ 2,147.97 cm³
Subtract the volume of the filling from the volume of the whole pie to get the volume of the crust alone:
Volume of crust alone: Vcrust = Vwhole - Vfilling ≈ 7,238.56 cm³ - 2,147.97 cm³ ≈ 5,090.59 cm³
To the nearest unit, the volume of the crust is approximately 5,091 cm³.
$35,485.00 to $50,606.00 per year is equivalent to how much an hour
assuming it is based on a 40 hour work week, working 52 weeks per year:
35485/52 = 682.40 per week
682.40/40 = 17.06 per hour
50606/52 = 973.19 per week
973.19/40 = 24.33 per hour
so between 17.06 & 24.33 per hour
what ithe distance from (3 1/2,5) to (3 1/2,-12)
Hey!
Hope this helps...
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Questions like these are really simple to answer, for one, all you have to know is Rise over Run (or Rise/Run)...
This being:
Rise: the distance from one y value to the other...
Run: the distance from one x value to the other...
Naturally graph points are represented as (x, y).
So, all we need to is do the math....
For Rise: the distance from 5 to -12 is 17...
For Run: the distance from 3.5 (or 3 1/2) to 3.5 is 0, but because the denominator of ANY fraction can never be 0, we will change it to 1...
So, our equation looks like: 17/1 (or 17 over 1)...
And our answer is: The 2 points are EXACTLY 17 units apart...
A shipment of racquetballs with a mean diameter of 60 mm and a standard deviation of 0.9 mm is normally distributed. By how many standard deviations does a ball bearing with a diameter of 58.2 mm differ from the mean?
Answer:
By 2 standard deviations a ball does bearing with a diameter of 58.2 mm differ from the mean.
Step-by-step explanation:
It is given that a shipment of racquetballs with a mean diameter of 60 mm and a standard deviation of 0.9 mm is normally distributed.
[tex]Mean=60[/tex]
[tex]\text{Standard deviation}=0.9[/tex]
Absolute difference written diameter of 58.2 mm and average diameter is
[tex]|58.2-60|=1.8[/tex]
Divide the difference by standard deviation (i.e.,0.9), to find the by how many standard deviations does a ball bearing with a diameter of 58.2 mm differ from the mean.
[tex]\frac{1.8}{0.9}=2[/tex]
Therefore 2 standard deviations a ball does bearing with a diameter of 58.2 mm differ from the mean.
When a pair of six sided dice is rolled , each with faces numbered 1 to 6,is rolled once,what is the probability that the result is either 3 and 4 or a 5 and a prime number?
Hans deposits $300 into an account that pays simple interest at a rate of 2% per year. How much interest will he be paid in the first 5 years?
What is the answer to 40-2a squared when a=4?
Transform (5 square root x^7)^3 into an expression with a rational exponent
In order to transform (5 square root x^7)^3 into an expression with a rational exponent, first transform square root x^7 into x^(7/2), then raise entire expression to the power of 3. So, final expression is 125x^(21/2).
Explanation:To transform (5 square root x^7)^3 into an expression with a rational exponent, firstly simplify the expression inside the bracket, then apply the exponent of 3 to the simplified expression.
Inside the brackets, square root of x^7 can be written as x^(7/2). So, the first parenthesis can be transformed into 5x^(7/2). Now, raise this to the power of 3. The rule for powers of powers is to multiply the powers. So, 5 cubed is 125 and (x^(7/2))^3 is x^(21/2).
So, the transformed expression with a rational exponent is 125x^(21/2).
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which one is it? need help please
In mathematics, the nth harmonic number is defined to be 1 + 1/2 + 1/3 + 1/4 + ... + 1/n. so, the first harmonic number is 1, the second is 1.5, the third is 1.83333... and so on. write an expression whose value is the 8th harmonic number.
The nth harmonic number is defined to be 1 + 1/2 + 1/3 + 1/4 + ... + 1/n. the first harmonic number is 1.5 , the second is 1.5 the third is 1.83333... and soon, the harmonic expression will be written as follows
Given:
a1 = first term = 1
a2 = second term = 1.5
a3 = third term = 1.83333...
We will write expression in Harmonic term, as
= [tex]\rm 1.0 + \dfrac{1.0}{2.0} + \dfrac{1.0}{3.0} + \dfrac{1.0}{4.0} + \dfrac{1.0}{5.0} + \dfrac{1.0}{6.0} + \dfrac{1.0}{7.0} +\dfrac{ 1.0}{8.0}[/tex]
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8x-2y over 10xy if x=4 and y=-7
The value of the expression 8x-2y over 10xy when x=4 and y=-7 is -0.1643 (rounded to four decimal places).
Explanation:To evaluate the expression 8x-2y over 10xy when x=4 and y=-7, we substitute these values into the expression:
8(4)-2(-7) over 10(4)(-7)
Simplifying further,
32+14 over -280
46 over -280
Therefore, the value of the expression 8x-2y over 10xy when x=4 and y=-7 is -0.1643 (rounded to four decimal places).
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In a batch of 280 water purifiers, 12 were found to be defective. What is the probability that a water purifier chosen at random will be defective? Write the probability as a percent. Round to the nearest tenth of a percent if necessary
Answer:
[tex]\text{Probability}=4.3\%[/tex]
Step-by-step explanation:
Given : In a batch of 280 water purifiers, 12 were found to be defective.
To find : What is the probability that a water purifier chosen at random will be defective? Write the probability as a percent.
Solution :
Total number of batch of purifiers = 280
Number of defective purifiers = 12
The probability that a water purifier chosen at random will be defective is given by,
[tex]\text{Probability}=\frac{\text{Favorable outcome}}{\text{Total outcome}}[/tex]
[tex]\text{Probability}=\frac{12}{280}[/tex]
[tex]\text{Probability}=\frac{3}{70}[/tex]
Converting into percentage,
[tex]\text{Probability}=\frac{3}{70}\times 100[/tex]
[tex]\text{Probability}=4.28\%[/tex]
Round to nearest tenths,
[tex]\text{Probability}=4.3\%[/tex]
Why do we state restrictions for rational expression and when do we state the restrictions?
We state restrictions for rational expressions to ensure that the denominator does not equal zero, as division by zero is undefined in mathematics. The restrictions are the values of the variable that make the denominator equal to zero. We state the restrictions whenever we are simplifying, performing operations with, or solving rational expressions.
A rational expression is an expression that can be written in the form of a fraction, where the numerator and the denominator are polynomials. The denominator of a rational expression cannot be zero because division by zero is not defined in mathematics. Therefore, when working with rational expressions, it is crucial to identify the values of the variable that would make the denominator equal to zero. These values are the restrictions, or domain restrictions, for the rational expression.
For example, consider the rational expression [tex]\(\frac{1}{x-3}\)[/tex]. The denominator is[tex]\(x-3\)[/tex]. To find the restriction, we set the denominator equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[x - 3 = 0\][/tex]
[tex]\[x = 3\][/tex]
Therefore, the restriction for this rational expression is [tex]\(x \neq 3\)[/tex], meaning that [tex]\(x\)[/tex] can be any real number except 3.
We must state these restrictions whenever we perform operations such as simplifying, adding, subtracting, multiplying, or dividing rational expressions, as well as when we are solving rational equations. This ensures that the operations are valid and that the solutions to the equations do not include any undefined expressions.
In summary, stating restrictions for rational expressions is a critical step in avoiding mathematical errors and ensuring that the expressions and equations we work with are well-defined.
(05.03 MC)
Eva has borrowed 200 songs from her friend. She plans to download an equal number of songs on her music player each week for 5 weeks. The graph shows the number of songs left to download, y, for a certain number of weeks, x:
A graph titled Song Downloading shows Number of Weeks on x-axis and Number of Songs Left to Download on y-axis. The x-axis scale is shown from 0 to 5 at increments of 1, and the y-axis scale is shown from 0 to 280 at increments of 40. A straight line joins the ordered pairs 0, 200 and 1, 160 and 2, 120 and 3, 80 and 4, 40 and 5, 0.
Part A: What is the rate of change and initial value of the function represented by the graph, and what do they represent in this scenario? Show your work to find the rate of change and initial value. (6 points)
Part B: Write an equation in slope-intercept form to model the relationship between x and y. (4 points)
A) The rate of change is -40 songs each week, that is because the amount of songs left to be downloaded decrease by 40 each week
B) lets use 2 points of the graph p1(0, 200), p2(1, 160)
calculate the slope:
m = (y2- y1)/(x2 - x1) = (160 - 200)/(1 - 0)
m = -40
now use line equation in form point-slope:
y - y1 = m(x - x1)
y - 200 = -40(x - 0)
y = -40x + 200
Part A:
Each week the amount of songs that need to download decreases by 40. So The rate of change is -40 songs each week. The initial value is 200 because that is the number of songs left to download.
Part B:
y = -40x + 200
You have 4500 cubic centimeters of wax. how many cylindrical candles can you make from the wax if each candle is 15 centimeters tall and has a diameter of 10 centimeters?
The number of cylindrical candles of 15cm height and 10cm diameter to be made from 4500[tex]cm^{3}[/tex] of wax is : 3.81 approximately 4
What is a cylinder?A cylinder is a solid geometrical shape with two parallel sides and two oval or circular cross-sections.
Analysis:
Given data:
Volume of wax = 4500[tex]cm^{3}[/tex]
Diameter of candle = 10cm
Radius of candle = diameter/2 = 10/2 = 5cm
Height of candle = 15cm
Volume of each cylindrical candle = π[tex]r^{2}[/tex]h
Volume of each cylindrical candle = [tex]\frac{22}{7}[/tex] x [tex](5)^{2}[/tex] x 15 = [tex]\frac{8250}{7}[/tex][tex]cm^{3}[/tex]
Volume of wax = n x volume of each cylindrical candle
n = number of candles
n = [tex]\frac{volume of wax}{volume of each cylindrical candle}[/tex]
n = [tex]\frac{4500}{\frac{8250}{7} }[/tex] = 3.81 approximately 4
In conclusion, the number of cylindrical candles to be made from 4500 cubic centimeters wax is 4.
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Final answer:
To find the number of cylindrical candles that can be made from a given volume of wax, one needs to calculate the volume of one candle with the formula for the volume of a cylinder and then divide the total wax volume by a single candle's volume.
Approximately 3 candles can be made from 4500 cm^3 of wax if each candle is 15 cm tall with a 10 cm diameter.
Explanation:
To calculate the number of cylindrical candles that can be made from 4500 cubic centimeters of wax, with each candle being 15 centimeters tall and with a diameter of 10 centimeters, we use the formula for the volume of a cylinder, V = πr^2h.
First, we need to calculate the radius of the cylinder by dividing the diameter by 2. The diameter is 10 cm, so the radius is 5 cm. Next, we apply the formula to find the volume V of one candle:
V = (π)(5 cm)^2(15 cm) = 3.14159 × 25 cm^2 × 15 cm = 1177.5 cm^3 approximately
To find out how many candles we can make, we divide the total volume of wax by the volume of one candle:
{4500 cm^3/}{1177.5 cm^3} approx 3.82
As it is not possible to make a fraction of a candle, you can make 3 complete candles with the given amount of wax.
Find the coordinates of point Q that lies along the directed line segment from R(-2, 4) to S(18, -6) and partitions the segment in the ratio of 3:7.
Answer:
(2 2/7, 5 1/3)
Step-by-step explanation:
The coordinates of point Q, lies along R(-2,4) and S(18,-6)
thus, QR and RS, that is in ratio of QR : RS = 3 : 7
Let point Q = (x,y)
Hence, QR = -2 - x; RS = -6 - 4
Thus, QR/RS = 3/7, which is: (-2 - x)/(-6 - 4) = 3/7
7(-2 - x) = -30
-14 - 7x = -30
7x = 16
∴ x = 16/7 = 2 2/7
If x : y = 3 : 7 ( where x = 2 2/7)
Hence, (2 2/7)/y = 3/7
3y = 16
∴ y = 16/3 = 5 1/3
The coordinates of point Q = (2 2/7, 5 1/3)
When graphed,a system shows the exact same lines. How many solutions will the system have?
The cost to produce a product is modeled by the function f(x) = 5x2 − 70x + 258 where x is the number of products produced. Complete the square to determine the minimum cost of producing this product.
The minimum cost of producing this product is:
13
Step-by-step explanation:The function which is used to represent the cost to produce x elements is given by:
[tex]f(x)=5x^2-70x+258[/tex]
Now, on simplifying this term we have:
[tex]f(x)=5(x^2-14x)+258\\\\i.e.\\\\f(x)=5(x^2+49-49-14x)+258\\\\i.e.\\\\f(x)=5((x-7)^2-49)+258\\\\i.e.\\\\f(x)=5(x-7)^2-5\times 49+258\\\\i.e.\\\\f(x)=5(x-7)^2-245+258\\\\i.e.\\\\f(x)=5(x-7)^2+13[/tex]
We know that:
[tex](x-7)^2\geq 0\\\\i.e.\\\\5(x-7)^2\geq 0\\\\i.e.\\\\5(x-7)^2+13\geq 13[/tex]
This means that:
[tex]f(x)\geq 13[/tex]
This means that the minimum cost of producing this product is: 13
What is the 5th term of an arithmetic sequence if t3 = 10 and t7 = 26?
18
20
22
24
Evaluate 4(a2 + 2b) - 2b when a = 2 and b = –2.
The given expression is [tex]4(a^2 + 2b) - 2b[/tex]. when a = 2 and b = –2 then the answer would be 4.
What is a simplification of an expression?Usually, simplification involves proceeding with the pending operations in the expression.
Like, 5 + 2 is an expression whose simplified form can be obtained by doing the pending addition, which results in 7 as its simplified form.
Simplification usually involves making the expression simple and easy to use later.
The given expression is
[tex]4(a^2 + 2b) - 2b[/tex]
when a = 2 and b = –2.
[tex]4(2^2 + 2(-2)) - 2(-2) \\\\ =4(4-4) + 4 \\\\= 4\times 0 + 4 = 4[/tex]
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Suppose you buy a CD for $500 that earns 2.5% APR and is compounded quarterly. The CD matures in 3 years. How much will the CD be worth at maturity?
Solve:The quantity 2 x minus 20 divided by 3= 2x
A Ferris wheel has a diameter of 42 feet. It rotates 3 times per minute. Approximately how far will a passenger travel during a 5-minute ride?
132 feet
659 feet
1,978 feet
3,956 feet
Approximately 1978 ft a passenger travel during a 5-minute ride and this can be determined by using the formula of the perimeter of a circle.
Given :
A Ferris wheel has a diameter of 42 feet. It rotates 3 times per minute.
The following steps can be used in order to determine the total distance travel by the passenger during a 5 minutes ride:
Step 1 - First determine the perimeter of the circle. The formula of the perimeter of the circle is given by:
[tex]\rm C = 2\pi r[/tex]
Step 2 - Now, substitute the value of known terms in the above formula.
[tex]\rm C=2\pi\times(21)[/tex]
[tex]\rm C= 131.94\;ft[/tex]
Step 3 - In one minute passenger travels:
[tex]\rm =131.94\times 3=395.82\; ft[/tex]
Step 4 - So, in three minutes passenger travels:
[tex]=395.82\times5[/tex]
= 1978 ft
So, approximately 1978 ft a passenger travel during a 5-minute ride.
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25 decreased by 1/5 of a number is 18
The equation for the student's question is 25 - (1/5)x = 18. Solving for x involves simple algebraic manipulation, resulting in x being equal to 35.
Explanation:The student's question '25 decreased by 1/5 of a number is 18' is a basic algebra problem. We could represent the unknown number as x. So the equation would be 25 - (1/5)x = 18.
To solve the equation 25 decreased by 1/5 of a number is equal to 18, we can set up the equation as 25 - (1/5)x = 18, where x is the unknown number.
To isolate x, we first subtract 25 from both sides of the equation:
- (1/5)x = -7.
Next, we can multiply both sides of the equation by -5 to eliminate the fraction:
x = (-7) * (-5) = 35.
To solve for x, first, add (1/5)x to both sides to get 25 = 18 + (1/5)x.
Then, subtract 18 from both sides to obtain 7 = (1/5)x. Finally, multiply both sides by 5 to find the value of x. Thus, x equals 35.
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Edgar started with 2 poems in his journal. Then he started writing 3 poems each day. Which of the following graphs represents Edgar's poem writing
Answer:
Let x represents the number of the days, and y represents total number of the poem.
According to the question,
Edgar started with 2 poems in his journal. Then he started writing 3 poems each day.
Thus, the line that describes the above situation is,
y = 2 + 3 x
The x -intercept of the line is [tex](-\frac{2}{3} , 0)[/tex] or (-0.667 , 0)
And, the y-intercept of the line is (0,2)
Also, if x = 1 y = 5
if x = 2 y = 8
If x = -1 y = - 1
And, if x = - 2 , y = -4
Thus, the points by which the line will pass are,
(1,5), (2,8) , (-1,-1) and (-2,-4)
Therefore, with the help of the above information we can plot the graph of the line. ( shown below)
A rectangular shipping container has a volume of 2500 cubic cm. The container is 4 times as wide as it is deep, and 5cm taller than it is wide. What are the dimensions of the contaner?
The question asks for the dimensions of a rectangular container with given volume and specific proportional relationships between its dimensions. Setting up and solving the equation 2500 = d × (4d) × (4d + 5) leads us to find the distinct depth, width, and height of the container.
Explanation:The subject matter of the student's question pertains to the mathematics concepts of volume and dimensional relationships of rectangular prisms. Let's represent the depth of the shipping container as d, the width as 4d (since it is four times the depth), and the height as 4d + 5 (since it is 5cm taller than the width). The volume of a rectangular prism (such as our shipping container) is given by the formula Volume = length × width × height. Given the volume is 2500 cubic cm, or 2500 cm³, we can set up the equation 2500 = d × (4d) × (4d + 5).
Solving this equation leads us to find the dimensions of the container, wherein the depth, width, and height are represented by the variables d, 4d, and 4d + 5
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The Partnership for 21st Century Learning lists core subject areas that all employees need to know about. What are two of those core subjects?
A. Economics and mathematics
B. Technology and citizenship
C. Art and Latin
D. Vocational skills and English
The Partnership for 21st Century Learning identifies Economics and Mathematics as two core subjects necessary for employee knowledge, essential for developing critical thinking and problem-solving skills in the global economy. Hence the correct answer is option A
Explanation:The Partnership for 21st Century Learning lists Economics and Mathematics as two core subject areas that are essential for all employees to have knowledge about. These subjects are foundational to understanding the global economy and are associated with the skills needed by "knowledge workers" such as engineers, scientists, doctors, teachers, financial analysts, and computer programmers. A strong grounding in Economics and Mathematics equips students with valuable skills such as critical thinking, problem-solving, and the ability to analyze complex data, which are highly sought after in the modern workforce.
As per the Partnership for 21st Century Learning, to support the quality of American education, it's crucial to prepare students with a well-rounded understanding of core disciplines, including Economics and Mathematics. This preparation is significant in light of increasing global competition and the need for American students to improve in reading, math, and critical thinking to match or exceed the capabilities of their peers in other industrialized nations.
Hence the correct answer is option A
Find the area of a regular hexagon whose side length is 16 in. and the apothem is 8 square root of 3 in
Write an expression for the number of hours in an unknown number of minutes.