The linear system l has one solution. This can be determined by examining the augmented matrix to find the row echelon form. The row echelon form is a matrix with zeroes below the leading entries and all the leading entries are in different columns.
By converting the augmented matrix into the row echelon form, we can identify the number of variables and the number of equations. Since there are three equations and three variables in the row echelon form, the linear system l has one solution. The row echelon form of the given augmented matrix is
[tex]$\begin{bmatrix}1&-3&2&3\\0&-5&-9&-10\\0&0&4&5\end{bmatrix}$[/tex]
By counting the number of equations and variables in the row echelon form, it can be seen that there is only one solution for the linear system l. This is because the number of equations is equal to the number of variables, indicating that there is a single solution.
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the linear system l has an augmented matrix which is a row equivalent to the following. how many solutions does l have?
[tex]$\begin{bmatrix}1&-3&2&3\\-3&8&-11&-14\\2&-7&5&7\end{bmatrix}$[/tex]
Scientists use drones with digital cameras to help them identify plants, predict, flooding, and construct 3-D maps of different landscapes. A team of scientists is using two drones to map a region. The heights of the drones are represented by the equations given, where x is the number of minutes since the drones were released by the scientists and y is the height in meters.
Equation for Drone A / y = 8x + 5
Equation for Drone B / y = 6x + 25
A : Solve the system of equations.
B : What does the solution tell you about the drones?
8th Grade Math Book - Don't Know What Brand.
The solution of the system of equations is given by the ordered pair (10, 85).
The solution tells us that the height of both Drone A and Drone B would be the same after 10 minutes.
How to solve the system of equations?In this scenario, heights of drones are represented by the following system of equations:
y = 8x + 5 .......equation 1.
y = 6x + 25 .......equation 2.
where:
x represents the number of minutes since the drones were released by the scientists.y represents the height the drones of in meters.By equating equation 1 and equation 2, number of minutes since the drones were released can be calculated as follows;
8x + 5 = 6x + 25
8x - 6x = 25 - 5
2x = 20
x = 20/2
x = 10 minutes.
Next, we would determine the height when x = 10 minutes:
y = 8(10) + 5
y = 80 + 5
y = 85 meters.
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A certain mailing service's first class mail rates are $0.37 for the first ounce and $0.11 for each additional ounce or fraction thereof up to 3.5 ounces. A model for the cost C (in dollars) of a first class mailing that weighs x ounces is given below. C(x) = 0.37 0 ≤ x ≤ 1 0.48 1 < x ≤ 2 0.59 2 < x ≤ 3 0.7 3 < x ≤ 3.5 A.) At what values is the function not continuous? B.) Find the cost of mailing a 1.6-ounce letter.
For the given function C(x) = 0.37 + (x - 1)0.11 the cost of mailing a 1.6-ounce letter is $0.436
The term function in math is defined as a relation between a set of inputs having one output each.
Here we have given that certain mailing service's first class mail rates are $0.37 for the first ounce and $0.11 for each additional ounce or fraction thereof up to 3.5 ounces.
Let us consider that the term C refers that cost in dollars of a first class mailing and x refers the weight in ounces.
The the function for the given situation is written as,
=> C(x) = 0.37 + (x - 1)0.11
Here we have given that we need to find the cost of mailing a 1.6-ounce letter which defines that the value of x is 1.6, then the cost is calculated as,
=> C(x) = 0.37 + (1.6 - 1) 0.11
=> C(x) = 0.37 + 0.066
=> C(x) = 0.436.
Complete Question:
A certain mailing service's first class mail rates are $0.37 for the first ounce and $0.11 for each additional ounce or fraction thereof up to 3.5 ounces. Find the function of the situation and the cost of mailing a 1.6-ounce letter.
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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.
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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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.
Find the unknown measurement of the following cuboid
length=10cm
Height=8cm
Volume=440cm,Fibd the breadth
Answer:
5.5cm
Step-by-step explanation:
The volume of a cuboid is given by
V = lbh
where l is the length
b is the breadth
h is the height.
We know that
V = lbh = 440 cm^3
Given l = 10 cm, h = 8 cm
So, we can find b by substituting these values in the above equation
440 = 10 * b * 8
Solving for b, we get:
b = 440 / (10*8)
b = 5.5cm
So, the breadth of the cuboid is 5.5 cm.
how does sleep deprivation affect your ability to drive? a recent study measured the effects on 19 professional drivers. each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. the treatments were assigned in random order. in each session, performance was measured on a variety of tasks including a driving simulation. use key terms from this module to identify the following values for this study this experiment.
The independent variable in this study is sleep deprivation, as the study aims to investigate the effects of sleep deprivation on the ability to drive.
This study is an experimental study that measures the effects of sleep deprivation on the ability to drive. The sample size is 19 professional drivers, and the study uses a within-subjects design as each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order, meaning that each driver has an equal chance of being assigned to either the sleep deprivation or normal sleep condition. The outcome variable in this study is performance on a variety of tasks, including a driving simulation. The independent variable in this study is sleep deprivation, as the study aims to investigate the effects of sleep deprivation on the ability to drive.
Thus, The independent variable in this study is sleep deprivation, as the study aims to investigate the effects of sleep deprivation on the ability to drive.
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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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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
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:
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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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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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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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]
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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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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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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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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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
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 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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Find the value of z.
X
7
y
N
3
Z =
√[?
Give your answer in simplest form.
Enter
Answer:
z = √30
Step-by-step explanation:
Given similar right triangles with hypotenuse segments marked 3 and 7, you want to know the length of short leg z adjacent to the segment marked 3.
Similar trianglesThe fact that all of these triangles are similar gives rise to some "geometric mean" relationships:
hypotenuse/short side = 10/z = z/3
30 = z² . . . . . . . multiply by 3z
z = √30 . . . . . . take the square root
__
Additional comment
The other two geometric mean relations are ...
x = √(7·10) . . . . . hypotenuse/long side: 10/x=x/7
y = √(7·3) . . . . . . short side/long side: y/7=3/y
The triangles are similar by AA: each has a right angle, and an angle in common with the largest triangle. Triangles similar to the same triangle are similar to each other.
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:
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).
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:
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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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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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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three wires are connected at point d, which is located 18 in. below the t-shaped pipe support abc. determine the tension in each wire when a 180-lb cylinder is suspended from point d as shown.
The tension in each wire when a 180-lb cylinder is suspended from point d is 52.8
The term tension is defined as the tensile load in the wire as it is continuously fed between the wire guides that are used to keep the wire straight between said guides.
While we looking into the following diagram we have identified that the following values are known.
Here the width is 180 lb.
And the distance between the two wires on all the sides is 22 inches, 24 inches and 18 inches.
Then the tension is calculated as,
=> (22 x 24 x 18)/180
=> 9504/180
=> 52.8
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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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