The volume of one marble is 0.5 ml and the water level before dropping the marbles is 8 ml.
Finding Volume by Water Displacement Method:The volume of an object is the amount of space occupied by the object within all boundaries. The volume of any geometrical shape can be calculated by the respective formula.
The Water Displacement Method is a method that is used to calculate the volume of any object without knowing the dimensions. If we drop the object into the water and the amount that can be displaced will be equal to the volume of the object.
Here we have
After dropping 4 marbles, the height of the water level is 10 ml
After dropping 6 marbles, the height of the water level is 11 ml
This means for 2 extra marbles the water level increase up to 1 ml
So, the Volume of the marble = 1 ml/2 = 0.5 ml
From the given data,
After dropping 4 marbles the water level is 10 ml
As we know the volume of 1 marble = 0.5ml
=> Volume of 4 marbles = 4 × 0.5ml = 2ml
The water level before dropping 4 marbles = 10 ml - 2 ml = 8 ml
Therefore,
The volume of one marble is 0.5 ml and the water level before dropping the marbles is 8 ml.
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r Analysis A student was asked to find the tient (4x³ + x2 + 2x + 1) = (x + 2). The student's kis shown at right. Identify the student's error provide the correct quotient.
When you divide two numbers the answer is called the quotient.
How do I calculate the quotient?Divide the dividend by the divisor to find the quotient of a division problem.
This may be accomplished by using repeated subtraction or lengthy division.
The quotient of two numbers, two fractions, or two algebraic expressions can be found.
According to the remainder theorem, when a polynomial p(x) is divided by (x – a), the remainder Equals f. (a). This is demonstrated by Euclid’s Division Lemma. Using this, p(x) = q(x) (x – a) + r if q(x) is the quotient and r is the remainder.
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You are given the following information about a simple regression model fit to 10 observations: 10 10 Στη = 20, = , Συ = 100, Sy=8. S = 2, k=1 You are also given that the sample correlation coefficient r = -0.98. Determine the predicted value of y when I = 5. Answer to the nearest integer. A. -10 B. -2 C. 11 D. 30 E. 37
The predicted value of y when I = 5 is equal to D) 30.
To determine the predicted value of y when I = 5, we can use the equation of the simple linear regression model:
y = b0 + b1x
where b0 and b1 are the regression coefficients and x is the predictor variable.
We can find b1 using the formula:
b1 = r(Sy/Sx)
where r is the sample correlation coefficient, Sy is the standard deviation of the y variable, and Sx is the standard deviation of the x variable.
Plugging in the given values:
b1 = -0.98(8/2) = -4
We can find b0 using the formula:
b0 = ybar - b1*xbar
where ybar and xbar are the mean of the y variable and the x variable respectively.
Plugging in the given values:
b0 = 10 - (-4)*10 = 50
Therefore, the equation of the simple linear regression model is:
y = 50 - 4x
So when I = 5, the predicted value of y is:
y = 50 - 4(5) = 50 - 20 = 30
The closest answer choice is D. 30.
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Julie is trying to lose weight. She now weighs 180 pounds. Every week for eight weeks, she was able to lose 2 pounds. a. List Julie's weight for each week. b. Write a recursive definition for this sequence.
a. The table for the weekly weight of Julie is illustrated below.
b. The recursive definition for this sequence is y = 180 - 2x.
What is an equation?An equation is written in the form of variables and constants separated by the operation of multiplication and division,
An equation states that terms in different forms on both sides of the equality sign are equal.
Multiplication and division do not separate the terms of an equation.
Given, Julie is trying to lose weight.
She now weighs 180 pounds and every week for eight weeks, she was able to lose 2 pounds.
The list of her weight every week for eight weeks is,
week 1 = 180 - 2 = 178 pounds.
week 2 = 178 - 2 = 176 pounds.
week 3 = 176 - 2 = 174 pounds.
week 4 = 174 - 2 = 172 pounds.
week 5 = 172 - 2 = 170 pounds.
week 6 = 170 - 2 = 168 pounds.
week 7 = 168 - 2 = 166 pounds.
week 8 = 166 - 2 = 164 pounds.
The equation that represents her weekly weight loss is,
y = 180 - 2x, Where 'x' represents the number of weeks and y represents the final weight.
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4 ) 316 with work
(Division)
The value of dividing 316 by 4 will be 79.
How to calculate the expression?An expression is simply used to show the relationship between the variables that are provided or the data given regarding an information.
In this case, it is vital to note that they have at least two terms which have to be related by through an operator. Some of the mathematical operations that are illustrated in this case include addition, subtraction, etc.
It should be noted that dividing 316 by 4 will be:
= 316 / 4
= 79
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Complete question
Divide 316 by 4
Lisa earns $215.20 a week. She is married and claims 3 allowances. Starting next week she will receive a $40 per week raise. After the raise, how much will she pay in federal income taxes Annually?
After the raise, Lisa much will pay $9 in federal income taxes Annually.
Explain the term federal income taxes?The federal tax is a tax that is levied on the yearly profits of people, companies, and some other legal entities. Federal income taxes are due on all compensation, including salaries, monetary gifts from employers, business profits, tips, winnings from gaming, bonuses, and unemployment benefits.For the stated question:
Lisa's salary = $215.20 a week.
After increment of $40 per week.
$215.20 + $40 = $255.2
She gets 3 allowances and is married.
After raise,
Federal income taxes = $9. (taken from the table)
Thus, After the raise, Lisa much will pay $9 in federal income taxes Annually.
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A block of mass 2.20 kg is placed against a horizontal spring of constant k = 895 N/m and pushed so the spring compresses by 0.0750 m.
(a)
What is the elastic potential energy of the block-spring system (in J)?
J
(b)
If the block is now released and the surface is frictionless, calculate the block's speed (in m/s) after leaving the spring.
m/s
a) The block-spring system has an elastic potential energy equal to 2.517 joule.
b) The block has speed of approximately 1.513 meters per second.
How to analyze a spring-block system
In this problem we must analyze the energy and kinematics of spring-block system with no irreversibilites (no friction). Part A - Elastic potential energy is determined when spring is entirely deformed:
U = (1 / 2) · k · x²
Where:
k - Spring constant, in newtons per meter.x - Spring deformation, in meters.U - Elastic potential energy, in joule.If we know that k = 895 N / m and x = 0.0750 m, then the elastic potential energy of the block-spring system is;
U = (1 / 2) · (895 N / m) · (0.0750 m)²
U = 2.517 J
The elastic potential energy of block-spring system is 2.517 joule.
Part B - And the speed can be determined by principle of energy conservation and work-energy theorem:
K = U
Where K is translational kinetic energy, in joules.
(1 / 2) · m · v² = U
v² = 2 · U / m
v = √(2 · U / m)
If we know that U = 2.517 J and m = 2.20 kg, then the speed of the block is:
v = √[2 · (2.517 J) / (2.20 kg)]
v ≈ 1.513 m / s
The speed of the block after leaving the spring is approximately 1.513 meters per second.
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Annual Tuition and Fees $6,125.00
Annual Room and Board $2,112.00
Annual Cost of Books and Supplies $1,250.00
Other One-Time Fee $750.00
Annual Scholarship and Grants $5,000.00
Using the information from the table, identify the equation in slope-intercept form that models the total cost of attendance. (4 points)
y = 5,237
y = 10,237x
y = 9,487x + 750
y = 4,487x + 750
The equation in slope-intercept form that models the total cost of attendance is y = 4487x + 750.
What are lines and their slopes?We know lines have various types of equations, the general type is
Ax + By + c = 0, and the equation of a line in slope-intercept form is
y = mx + b.
Where slope = m and b = y-intercept.
the slope is the rate of change of the y-axis with respect to the x-axis and the y-intercept is the (0,b) where the line intersects
the y-axis at x = 0.
To find the equation in the slope-intercept form,
We'll sum all the annual fees and subtract the scholarship amount and 'x' will represent the number of years while the one-time fee will be our constant term.
So, (6125 + 2112 + 1250 - 5000).
= 4487.
Therefore, The equation in slope-intercept form is,
y = 4487x + 750.
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show that, given one nth root of z, the others are obtained by multiplying it by the nth roots of unity. (z is complex number)
It is proven that the other nth roots of z can be found by multiplying the initial one by the nth roots of unity.
A complex number satisfying the equation [tex]z^{n-1} = 0[/tex] is known as the nth root of unity. There are about n nth roots of a unit, according to the fundamental theorem of algebra. We know that,[tex]1=e^{2\pi i}[/tex]. Then, the nth root is written as [tex]\omega=e^{2\pi i/n}[/tex].
The integer of powers of this root between 0 and n-1 provides the other nth roots. Then, the roots are [tex]1, \omega, \omega^2 ,\cdots \cdots, \omega^{(n-1)}[/tex]. Now, factorize the equation [tex]z^{(n-1)} = 0[/tex] we get the required identity as, [tex]z^{(n-1)} = (z- 1)(z^{(n-1)}+ z^{(n-2)} + \cdots \cdots+ z + 1) = 0[/tex].
This indicates that the equation [tex]1+\omega+ \omega^2+\cdots \cdots+ \omega^{(n-1)}[/tex] is true for all nth roots of unity other than 1.
Roots of unity also have the great quality of being periodic, which results from the fact that n = 1. It is equal to a power of that power mod n if the power is bigger than n.
When ω is a third root of unity, [tex]\omega^5 + \omega^4 + 1 = \omega^2 + \omega + 1 = 0[/tex].
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99
2
12
. At the library, Herb spent 1/6 hour looking for a book,1/4 hour of
reading, and 1/2 hour doing research on the computer. How man
hours did Herb spend at thelibrary?
2
3
Herb spent 55 hours at the library.
What is a fraction?Fractions defines a part of a whole and are represented as a numerical value. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.
Parts of a FractionThe denominator indicates the number of parts in which the whole has been divided into. It is placed in the lower part of the fraction below the fractional bar.The numerator indicates how many sections of the fraction are represented or selected. It is placed in the upper part of the fraction above the fractional bar.Total time spent in the Library = 1/6 + 1/4 + 1/2
Total time spent = 11/12 x 60 mins
Total time spent = 55 hours
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You use beads to make design. Of the beads, 1/3 are red and 1/6 are blue. The rest are white. What fraction of the beads are red red or blue?
Answer:
One Half
Step-by-step explanation:
1/3 = 2/6
2/6 + 1/6 = 3/6 = 1/2
A shelf can support 5 5/8 pounds. if a book weights 5/8 of a pound, how many books can the shelf hold?
Answer:
9 books
Step-by-step explanation:
complete the table by using your knowledge of the formula for simple interest. principle amount saved (dollars) 425, interest rate 7% time (years) 5 how many interest Earned (dollars)
Answer:
$ 148.75
Step-by-step explanation:
Simple interest:Principle = $ 425
Rate = 7%
Time = 5 yeats
[tex]\sf \boxed{Simple \ Interest = \dfrac{Principle*rate*time}{100}}[/tex]
[tex]= \dfrac{425*7*5}{100}\\\\=\dfrac{17*7*5}{4}\\\\\\=\dfrac{595}{4}\\\\=148.75[/tex]
A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height h feet is given by the function A where A(h) is measured in square feet. The function A is continuous and decreases as h increases. Selected values for Ach) are given in the table. (a) Use a left Riemann sum with the three subintervals indicated to approximate the integral. Indicate correct units of measure. (b) Interpret your answer from part (a). What is being measured?
(C) Is your estimate of the integral in part (a) an overestimate or underestimate of the actual integral?! Justify your answer
So. the left Riemann sum is 23.74 ft^2.
Given the information in the table:
(a) To use a left Riemann sum with three subintervals to approximate the integral of A(h) with respect to h, we need to divide the interval of integration [0,10] into three equal subintervals. The width of each subinterval is (10-0)/3 = 3.33 ft. The left Riemann sum can be calculated as:
(A(0) * 3.33) + (A(2) * 3.33) + (A(5) * 3.33) = (50.3 * 3.33) + (14.4 * 3.33) + (6.5 * 3.33) = 16.80 + 4.78 + 2.16 = 23.74 ft^2
(b) This approximation represents the total area of the horizontal cross section of the tank up to a specific height h.
(c) The Riemann sum is an underestimate of the actual integral. This is because the Riemann sum uses the left endpoint of each subinterval to approximate the area under the curve, but the actual area under the curve is always greater than or equal to the area of the rectangles.
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Three vertices of a parallelogram are shown in the figure below.
Give the coordinates of the fourth vertex.
(-5,-3)
(2,7)
(0, -1)
Step-by-step explanation:
it is a parallelogram.
in comparison to a rectangle one side is shifted a bit, but the endpoints of the shifted side are still positioned the same way relative to each other.
and particularly, their relative offset from each other must be the same as for the endpoints of the other, parallel side.
let's look at
(0, -1) and (2, 7).
their relative offset is given by
x + 2 (from 0 to 2)
y + 8 (from -1 to 7)
the missing point must have the same offset from (-5, -3), as this line is parallel to the one above :
-5 + 2 = -3
-3 + 8 = 5
so, the missing point is (-3, 5).
240 people are going to a charity event.
3535 of the guests have ordered chicken for their meals.
Of the remaining guests, 12.5% have ordered gluten-free meals.
How many people haven't ordered their meals yet?
Answer:
First, find out how many guests have ordered chicken: 240 guests * 3535/10000 = 84 guests
Then subtract that number from the total number of guests to find the number of guests who haven't ordered chicken: 240 guests - 84 guests = 156 guests
Then find how many of those guests have ordered gluten-free meals: 156 guests * 12.5% = 19.5 guests
Then round down the number of guests who have ordered gluten-free meals to the nearest whole number to find the exact number of guests who ordered gluten-free meals: 19 guests
Then subtract that number from the number of guests who haven't ordered chicken to find the number of guests who haven't ordered their meals yet: 156 guests - 19 guests = 137 guests
So 137 people haven't ordered their meals yet.
Determine whether the lines L₁ and L2 given by the vector equations are parallel, perpendicular, or neither.
L₁: r(t) = (-5+ 3t)i + (3 + t)j
L₂: r(s) = (2+2s)i + (4 - 6s)j
parallel
Operpendicular
neither
If the lines are not parallel, find their point of intersection. (If the lines are parallel, enter PARALLEL.)
The lines L₁ and L₂ given by the vector equations is perpendicular.
How are the lines perpendicular?In order to prove that the lines are perpendicular,
L₁: r(t) = (-5+ 3t)i + (3 + t)j
L₂: r(s) = (2+2s)i + (4 - 6s)j
To determine whether the lines L₁ and L2 are parallel, perpendicular, or neither, we need to compare their direction vectors.
The direction vector for L₁ is given by <3,1> and the direction vector for L₂ is given by <2,-6>.
The dot product of the direction vectors is (32) + (1-6) = -8. Since the dot product is zero, the vectors are perpendicular, and therefore the lines are perpendicular.
Therefore, the lines are PERPENDICULAR.
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Can someone please help me with these equations ??
The value for x is 1 or -6. The solution has been obtained by using logarithm.
Describe the logarithm.A logarithm is the power to which a number must be raised in mathematics in order to produce other numbers. It is the most useful way to express really large numbers.
We are given [tex]log_3 \frac{3(2-x)}{x(x+2)} = 0[/tex]
Solving the equation, we get
[tex]log_3 \frac{3(2-x)}{x(x+2)} = 3^0[/tex] (Using the property of log)
On multiplying both sides by x(x+2), we get
[tex]log_3 \frac{3(2-x)}{x(x+2)} x(x+2) = 3^0 x(x+2)[/tex]
⇒3 (2−x) = x(x+2)
⇒ 6 - 3x=x² +2x
= x²+5x-6=0
⇒ x²+6x-x-6=0
⇒ (x+6)(x-1)=0
⇒ x = 1,-6
Hence, the value for x is 1 or -6.
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Explain weather there is a possible transformation of f in the form g(x) =f(x)+k, where k is a constant for which the graph y=g(x) will have no x-intercepts. Enter your answer and your justification in the space provided.
There is a possible translation of the form g(x) = f(x) + k for which the graph y=g(x) will have no x-intercepts.
What is a translation?A translation is a movement to a graph or figure in which only the position of the figure changes, either left, right, up or down, keeping the inclination, orientation and congruence.
The translations are represented as follows:
Left a units: g(x) = f(x + a).Right a units: g(x) = f(x - a).Up a units: g(x) = f(x) + a.Down a units: g(x) = f(x) - a.Hence, for the function to have no x-intercepts, the function must be shifted up as though the minimum value of the function is greater than zero.
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Identify whether or not each pair of terms can be combined to simplify the expression (−4t − 7) + (3t+8).
-4r and -7
3r and 8
-41 and 3r
Answer: (-4t - 7) + (3t + 8) cannot be simplified further as the terms have different variables.
(-4r and -7) and (3r and 8) cannot be combined as these terms have different variables.
(-41 and 3r) cannot be combined as these terms have different variables.
Step-by-step explanation:
Let g(t) be the population in thousands of people) of Calculus City, as a function the time t (in years), where t = 0 corresponds to the year 2019. In 2019, the population was 10 thousand, and each year the population increases by by 30%. (a) Model the population by finding a formula for g(t), using the appropriate choice of a linear, polynomial, power, trigonometric, or exponential function, or a piecewise contbination thereof. (b) Evaluate the expression 9(6), and write a sentence explaining what this means. (c) Solve the equation g(t) = 16.9, and write a sentence explaining what this means.
The population of Calculus City can be modeled by an exponential function, g(t) = 10(1.3) ^t, where t is the number of years since 2019. Evaluating g (6) means calculating the population of Calculus City after 6 years, which is 10(1.3) = 39.5 thousand people.
(a) The population of Calculus City can be modeled by an exponential function, g(t) = 10(1.3) ^t, where t is the number of years since 2019.
(b) g (6) = 10(1.3) = 46.6 thousand. This means that the population of Calculus City will be 46.6 thousand in the year 2025.
(c) Solving the equation g(t) = 16.9 yields t = log1.3(16.9/10) = 4.7 years. This means that the population of Calculus City will be 16.9 thousand in the year 2023.7.
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Set B is the set of positive two-digit even numbers less than 36 that do not contain the digit 2 .
The required set of positive two-digit even numbers less than 36 that do not contain the digit 2 is 11,13,14,15,16,17,18,19,30,31,33,34,35.
What is an integer?An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Zero is not a fraction or decimal of any number. It is neither positive nor negative.
Given:
Set D is the set of positive two-digit even numbers less than 36 that do not contain the digit 2.
According to the given question, we have
2-digit numbers are numbers that have two digits and they start with the number 10 and end with the number 99.
Starting at 11 and going up the two-digit numbers even numbers less than 36 that do not contain the digit 2.
11,13,14,15,16,17,18,19,30,31,33,34,35.
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11) The dimensions, in feet, of a rectangle are represented by expressions, as shown in the
diagram below. Explain your work.
(2.5m) ft.
(4n+ 5) ft.
The area of the rectangle is and using the length and breadth
What is the area of the rectangle?In a two-dimensional plane, a rectangle area is a space it encloses. Known for having 4 sides and 4 vertices, a rectangleis a closed shape. The sum of the spaces occupied by a rectangle three sides is known as its area. The product of a rectangle breadth and length is the usual formula for calculating its area.The area that is included within the perimeter of a flat object or figure is generally considered to be the definition of the word "area." The square units used for measuring are square metres as the standard unit.area of rectangle = length × breadth
area of rectangle = 2.5m × (4n +5)
area of rectangle = 10mn + 12.5m
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Question:
The dimensions, in feet, of a rectangle, are represented by expressions, as shown in the diagram below. The area of the rectangle is 10mn + 12.5m. Explain your work.
Candice and Joel each wrote a proof for the statement: If m ∠ DOX = m ∠ BOX , then m ∠ AOX = m ∠ COX . Diagram shows two lines, AB and CD, intersecting at point O. Ray OX extends between rays OA and OC. Candice's Proof Statements Reasons 1. m ∠ AOX ≠ m ∠ COX 1. by supposition 2. ∠ AOD ≅ ∠ COB 2. vertical angles theorem 3. m ∠ AOD = m ∠ COB 3. definition of congruence 4. m ∠ AOD + m ∠ AOX ≠ m ∠ COB + m ∠ COX 4. additional property of equality 5. m ∠ AOX ≠ m ∠ COX 5. angle addition postulate Joel's Proof Statements Reasons 1. m ∠ DOX = m ∠ BOX 1. given 2. m ∠ AOD + m ∠ AOX = m ∠ COB + m ∠ COX 2. angle addition postulate 3. ∠ AOD ≅ ∠ COB 3. vertical angles theorem 4. m ∠ AOD = m ∠ COB 4. definition of congruence 5. m ∠ AOX = m ∠ COX 5. subtraction property of equality What type of proofs did they use? Candice's proof is because . Joe's proof is because .
Both Candice and Joel used a proof by contradiction.
What is the proof about?Candice began her proof by assuming the statement was false (by supposition) and then used the vertical angles theorem, definition of congruence, and additional property of equality to arrive at a contradiction.
By demonstating that assuming a proposition to be false results in a contradiction, proof by contradiction establishes the truth or validity of a proposition in logic.
Joel began his proof by assuming the statement was true and used the angle addition postulate, vertical angles theorem, definition of congruence, and subtraction property of equality to arrive at a conclusion that the statement was true.
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Example 1. 15+6x3 This problem is modeled after example 1. Find the requested pattern. (Enter the first three numbers of the pattern as a comma-separated list.) three pattern
Therefore, the 3 patterns are
3,6,9,3,6,9
Explanation:
Given that,
we will carry out the multiplications of succession counting number by 3 and if there is more than a single digit answer .we add the digits.
we are looking for is a pattern for there single digit numerals.
carryout the plan
1x3=3→3
2x3=6→6
3x3=9→9
4x3=12→1+2=3
5x3=15→1+5=6
and so on
continue with some additional at terms.
6x3=18→1+8=9
7x3=21→2+1=3
∵ Therefore the 3 patterns are
3,6,9,3,6,9
A pattern is defined in mathematics as a sequence of repeating objects, shapes, or numbers. A pattern can be associated with any type of event or object. A pattern has a rule that tells us which objects belong to the pattern and which objects do not belong to the pattern.
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1.) A husband 3 years older than his wife. The Sum of their ages is 73 years. Calculate the difference between the square of their ages.
Answer:
The difference between the square of their ages is 219----------------------------
Let the husband's age be h and wife's age be w.
The difference of their age is 3 and sum is 73:
h - w = 3h + w = 73The difference between the squares is:
h² - w² = Difference of squares(h + w)(h - w) = Identity for difference of squares73*3 = Substitute219 AnswerAnswer:
219
Step-by-step explanation:
H-W=3
H+W =73
2H=76
H=38
W=35
38² -35² = 219
The sum of four consecutive integers; n = first integer of the four
The sum of four consecutive integers is,
⇒ 4n + 6
What is Addition?The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.
We have to given that;
⇒ n = first integer of the four.
Now, Let four consecutive integers with n is first integer of the four are,
⇒ n, n + 1, n + 2, n + 3
Hence, The sum of four consecutive integers is,
⇒ S = n + (n + 1) + (n + 2) + (n + 3)
= 4n + 6
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find the order of the group generated by a and b under matrix multiplication (use the convention a^n b^m)
The order of the group generated by a and b under matrix multiplication is a^2b^3.
Start with the identity matrix:
I = [1 0 ; 0 1]
Multiply it by a:
a*I = [1 1; 0 1]
Multiply it by b:
b*a*I = [1 2; 0 1]
Multiply it by a:
a*b*a*I = [2 3; 0 1]
Multiply it by b:
b*a*b*a*I = [2 5; 0 1]
Multiply it by a:
a*b*a*b*a*I = [4 3; 0 1]
Therefore, the order of the group generated by a and b is a^2b^3.
The order of the group generated by a and b under matrix multiplication is a^2b^3.
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Tyrone went to a basketball game and bought a jersey that was on sale at a 30% discount. He also bought a hat that cost $37.89. Tyrone spent a total of $83.39. Write an equation to find the original cost of the jersey.
Answer: Let x be the original cost of the jersey.
We know that the jersey was on sale at a 30% discount, so the sale price is x - (30/100)*x = x(1- 0.3) = 0.7x
We also know that Tyrone spent a total of $83.39 and the hat cost $37.89, so the jersey cost is 83.39 - 37.89 = 45.5
So we can write the equation:
0.7x = 45.5
To find the original cost of the jersey, we can divide both sides by 0.7:
x = 45.5 / 0.7
x = 65
So the original cost of the jersey is $65.
Step-by-step explanation:
Following is a statement of a theorem which can be proven using calculus or precalculus mathematics. For this theorem, a, b, and c are real numbers. Theorem If f is a quadratic function of the form f (x) = ax^2 + bx + c and a < 0, then the function f has a maximum value when x = -b/2a. Using only this theorem, what can be concluded about the functions given by the following formulas? ? (a) g (x) = -8x^2 + 5x - 2 (b) h (x) = -(1/3)x^2 + 3x (c) k (x) = 8x^2 - 5x - 7 (d) j (x) = -(71/99)x^2 + 210 (e) f (x) = 4x^2 - 3x + 7 (f) F (x) = -x^4 + x^3 + 9
(a) The maximum value of g(x) = -8[tex]x^2[/tex] + 5x - 2 is 5/16.
(b) The maximum value of h(x) = -(1/3)[tex]x^2[/tex] + 3x is 9/2.
(c) The maximum value of k(x) = 8[tex]x^2[/tex] - 5x - 7 is 5/16.
(d) The maximum value of j(x) = -(71/99)[tex]x^2[/tex] + 210 is 0.
(e) The maximum value of f(x) = 4[tex]x^2[/tex] - 3x + 7 is 3/8.
(f) The maximum value of F(x) = -[tex]x^4[/tex] + [tex]x^3[/tex] + 9 is 0/0. We cannot say anything about that function.
The general form of a quadratic function;
f(x) = a[tex]x^2[/tex] + bx + c and a < 0,
Then the function f has a maximum value when x = -b/2a
(a) The function is g(x) = -8[tex]x^2[/tex] + 5x - 2
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -8 and b = 5
Now putting the value at x = -b/2a
x = -5/2(-8)
x = (-5)/(-16)
x = 5/16
(b) The function is h (x) = -(1/3)[tex]x^2[/tex] + 3x
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -(1/3) and b = 3
Now putting the value at x = -b/2a
x = (-3)/2(-1/3)
x = (3×3)/2
x = 9/2
(c) The function is k(x) = 8[tex]x^2[/tex] - 5x - 7
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 8 and b = -5
Now putting the value at x = -b/2a
x = -(-5)/(2×8)
x = 5/16
(d) The function is j(x) = -(71/99)[tex]x^2[/tex] + 210
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -(71/99) and b = 0
Now putting the value at x = -b/2a
x = 0/2(-(71/99))
x = 0
(e) The function is f(x) = 4[tex]x^2[/tex] - 3x + 7
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 4 and b = -3
Now putting the value at x = -b/2a
x = -(-3)/(2×4)
x = 3/8
(f) The function is F(x) = -[tex]x^4[/tex] + [tex]x^3[/tex] + 9
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 0 and b = 0
Now putting the value at x = -b/2a
x = 0/0
We cannot say anything about that function.
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The complete question is given below:
At a factory, Jin assembles 12 parts each minute. He has assembled 156 parts when Summer starts on the line, assembling at a pace of 15 parts per minute. If their assembly rates continue, will summer ever catch up to jin? When?
I need the substitution and elimination method
If their assembly rates continue, based on an equation, Summer will catch up with Jin in 52 minutes.
What is an equation?An equation refers to a statement claiming the equality of two or more mathematical expressions.
Using the equation sign (=) mathematicians show that expressions are equal or equivalent.
Jin's assembling rate per minute = 12 parts
Summer's assembling rate per minute = 15 parts
The number of parts already assembled by Jin before Summer started = 156 parts
Let the minutes used by Jin or Summer = n
The total parts assembled by Jin in n minutes = 156 + 12n
The total parts assembled by Summer in n minutes = 15n
For Summer to catch up to Jin, 15n = 156 + 12n
Grouping like terms:
15n - 12n = 156
3n = 156
n = 52
Thus, in 52 minutes, the number of assembly parts by Summer will equal Jin's.
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