A rather lengthy solution using a neat method I just learned relying on complex analysis.
First observe that
[tex]e^{-x^2} \cos(2x^2) = \mathrm{Re}\left[e^{-x^2} e^{i\,2x^2}\right] = \mathrm{Re}\left[e^{a x^2}\right][/tex]
where [tex]a=-1+2i[/tex].
Normally we would consider the integrand as a function of complex numbers and swapping out [tex]x[/tex] for [tex]z\in\Bbb C[/tex], but since it's entire and has no poles, we cannot use the residue theorem right away. Instead, we introduce a new function [tex]g(z)[/tex] such that
[tex]f(z) = \dfrac{e^{a z^2}}{g(z)}[/tex]
has at least one pole we can work with, along with the property (1) that [tex]g(z)[/tex] has period [tex]w[/tex] so [tex]g(z)=g(z+w)[/tex].
Now in the complex plane, we integrate [tex]f(z)[/tex] along a rectangular contour [tex]\Gamma[/tex] with vertices at [tex]-R[/tex], [tex]R[/tex], [tex]R+ib[/tex], and [tex]-R+ib[/tex] with positive orientation, and where [tex]b=\mathrm{Im}(w)[/tex]. It's easy to show the integrals along the vertical sides will vanish as [tex]R\to\infty[/tex], which leaves us with
[tex]\displaystyle \int_\Gamma f(z) \, dz = \int_{-R}^R f(z) \, dz + \int_{R+ib}^{-R+ib} f(z) \, dz = \int_{-R}^R f(z) - f(z+w) \, dz[/tex]
Suppose further that our cooked up function has the property (2) that, in the limit, this integral converges to the one we want to evaluate, so
[tex]f(z) - f(z+w) = e^{a z^2}[/tex]
Use (2) to solve for [tex]g(z)[/tex].
[tex]\displaystyle f(z) - f(z+w) = \frac{e^{a z^2} - e^{a(z+w)^2}}{g(z)} = e^{a z^2} \\\\ ~~~~ \implies g(z) = 1 - e^{2azw} e^{aw^2}[/tex]
Use (1) to solve for the period [tex]w[/tex].
[tex]\displaystyle g(z) = g(z+w) \iff 1 - e^{2azw} e^{aw^2} = 1 - e^{2a(z+w)w} e^{aw^2} \\\\ ~~~~ \implies e^{2aw^2} = 1 \\\\ ~~~~ \implies 2aw^2 = i\,2\pi k \\\\ ~~~~ \implies w^2 = \frac{i\pi}a k[/tex]
Note that [tex]aw^2 = i\pi[/tex], so in fact
[tex]g(z) = 1 + e^{2azw}[/tex]
Take the simplest non-zero pole and let [tex]k=1[/tex], so [tex]w=\sqrt{\frac{i\pi}a}[/tex]. Of the two possible square roots, let's take the one with the positive imaginary part, which we can write as
[tex]w = \displaystyle -\sqrt{\frac\pi{\sqrt5}} e^{-i\,\frac12 \tan^{-1}\left(\frac12\right)}[/tex]
and note that the rectangle has height
[tex]b = \mathrm{Im}(w) = \sqrt{\dfrac\pi{\sqrt5}} \sin\left(\dfrac12 \tan^{-1}\left(\dfrac12\right)\right) = \sqrt{\dfrac{\sqrt5-2}{10}\,\pi}[/tex]
Find the poles of [tex]g(z)[/tex] that lie inside [tex]\Gamma[/tex].
[tex]g(z_p) = 1 + e^{2azw} = 0 \implies z_p = \dfrac{(2k+1)\pi}2 e^{i\,\frac14 \tan^{-1}\left(\frac43\right)}[/tex]
We only need the pole with [tex]k=0[/tex], since it's the only one with imaginary part between 0 and [tex]b[/tex]. You'll find the residue here is
[tex]\displaystyle r = \mathrm{Res}\left(\frac{e^{az^2}}{g(z)}, z=z_p\right) = \frac12 \sqrt{-\frac{5a}\pi}[/tex]
Then by the residue theorem,
[tex]\displaystyle \lim_{R\to\infty} \int_{-R}^R f(z) - f(z+w) \, dz = \int_{-\infty}^\infty e^{(-1+2i)z^2} \, dz = 2\pi i r \\\\ ~~~~ \implies \int_{-\infty}^\infty e^{-x^2} \cos(2x^2) \, dx = \mathrm{Re}\left[2\pi i r\right] = \sqrt{\frac\pi{\sqrt5}} \cos\left(\frac12 \tan^{-1}\left(\frac12\right)\right)[/tex]
We can rewrite
[tex]\cos\left(\dfrac12 \tan^{-1}\left(\dfrac12\right)\right) = \sqrt{\dfrac{5+\sqrt5}{10}}[/tex]
so that the result is equivalent to
[tex]\sqrt{\dfrac\pi{\sqrt5}} \cos\left(\dfrac12 \tan^{-1}\left(\dfrac12\right)\right) = \boxed{\sqrt{\frac{\pi\phi}5}}[/tex]
While hiking down a mountain, your rate of decent is - 500 feet per hour. Your decent begins at an elevation of 3.000 leet. What your elevation after hiking 3 hours?
The expression for this scenario is:
[tex]\begin{gathered} E=3000-500t \\ E\text{ is the elevation } \\ t\text{ is the time} \\ \text{From the question, t = 3 hours} \end{gathered}[/tex]Now substitute the value of t into the elevation expression above, to get the elevation reached after 3 hours.
[tex]\begin{gathered} E=3000-500t \\ E=3000-500(3) \\ E=3000-1500 \\ E=1500\text{ feet.} \\ \text{The elevation after 3 hours is 1,500 feet.} \end{gathered}[/tex]If a dragon can eat an entire cow weighing 500 kilograms in 12 seconds, how long will it take to eat a human weighing 90kilograms and how would I write it out mathematically with the answer
The dragon would take 2.16 seconds to eat a human weighing 90 kilograms.
What is a fraction?Fraction number consists of two parts, one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.
Let x seconds it would take to eat a human weighing 90kilograms
500 kg → 12 seconds
90 kg → x
500/90 = 12/x
x = (90×12)/500
x = 2.16 seconds
Thus, the dragon would take 2.16 seconds to eat a human weighing 90 kilograms.
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write a quadratic function in standard form containing the point (5,-6) and x-intercepts -7 and 3
The quadratic equation in standard form that contains the points (5, - 6), (- 7, 0) and (3, 0) is y = - (1 / 4) · x² - x + 21 / 4.
How to determine a quadratic function in standard form from a point and two x-interceptsHerein we find the equation of a parabola that contains a points and its x-intercepts (two real roots). According to fundamental theorem of algebra, we can derive a quadratic function with real coefficients if we know three points of the parabola. The procedure is shown below.
First, use the quadratic function in product form and substitute on all known variables to determine the lead coefficient:
y = a · (x + 7) · (x - 3)
- 6 = a · (5 + 7) · (5 - 3)
- 6 = a · 12 · 2
- 6 = 24 · a
a = - 6 / 24
a = - 1 / 4
Second, expand the quadratic function into its standard form:
y = - (1 / 4) · (x + 7) · (x - 3)
y = - (1 / 4) · (x² + 4 · x - 21)
y = - (1 / 4) · x² - x + 21 / 4
The quadratic equation in standard form is y = - (1 / 4) · x² - x + 21 / 4.
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Calculate the volume of the cuboid shown below. Give your answer in cm³. If your answer is a decimal, then round it to 1 d.p.
Answer:
Can't see sh## ur photo is crazy low quality
After sitting out of a refrigerator for a while, a turkey at room temperature (70°F) is
placed into an oven. The oven temperature is 325°F. Newton's Law of Heating
explains that the temperature of the turkey will increase proportionally to the
difference between the temperature of the turkey and the temperature of the oven, as
given by the formula below:
T=Ta +(To-Ta)e-kt
Ta = the temperature surrounding the object
To the initial temperature of the object
=
t the time in hours
T= the temperature of the object after t hours
k = decay constant
The turkey reaches the temperature of 116°F after 2.5 hours. Using this
information, find the value of k, to the nearest thousandth. Use the resulting
equation to determine the Fahrenheit temperature of the turkey, to the nearest
degree, after 5.5 hours.
The value of k = -0.079
The Fahrenheit temperature of the turkey, to the nearest degree, after 5.5 hours = T(5.5) = 159.865
What is Temperature?
Temperature is a numerical expression of how hot a substance or radiation is. There are three different types of temperature scales: those that are defined in terms of the average, like the SI scale;
Newton's law of cooling states that T=Ta + (To-Ta)e-kt
T = Ta + (To-Ta)e-kt
Ta is the object's ambient temperature.
To equal the starting temperature.
t = the number of hours.
after t hours, temperature equals T.
decay constant is k.
According to the formula,
To = 70 and
Ta = 325,
therefore
(To -Ta) = 70 - 325.
(To -Ta) = -255
T = 325 + (-255 e^kt) or
325 -255e^kt.
After 2.5 hours, the turkey reaches a temperature of 116°F.
In other words,
T(2.5) = 116 F.
Consequently,
116 = 325 - 255e^k2.5
Following that,
116 - 325 = - 255e^2.5k
Now,
-209 = -255e^2.5k
e^2.5k = -209/-255
Using the function 'ln,'
now
ln(e2.5k) = ln(209 / 255)
Then, 2.5k = -0.198929
The value of k is now given by k = -0.198929 / 2.5, which is -0.079571.
The result is k = -0.079.
After 5.5 hours, the temperature is
T(5.5) = 325 - 255e^(-0.079 x 5.5)
Then, = 325 - 255e^(-0.4345)
Now,
= 325 - (165.1350) (165.1350)
T(5.5) = 159.865 is the result.
Consequently, T(5.5) = 159.865 degrees Fahrenheit is the turkey's exact temperature after 5.5 hours.
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can someone please help me solve and help me graph for this equation?
Graph both equations. The coorinates of the point where the graphs intersect is the solution to the system of equations.
To graph them, notice that each equation corresponds to a line. A straight line can be drawn if two points on that line are given. Replace two different values of x into each equation to find its corresponding value of y, then, plot the coordinate pairs (x,y) to draw the lines.
First equation:
[tex]y=2x-3[/tex]For x=2 and x=5 we have that:
[tex]\begin{gathered} x=2 \\ \Rightarrow y=2(2)-3 \\ =4-3 \\ =1 \end{gathered}[/tex][tex]\begin{gathered} x=5 \\ \Rightarrow y=2(5)-3 \\ =10-3 \\ =7 \end{gathered}[/tex]Then, the points (2,1) and (5,7) belong to the line:
Second equation:
[tex]x+3y=12[/tex]For x=0 and x=6 we have:
[tex]\begin{gathered} x=0 \\ \Rightarrow0+3y=12 \\ \Rightarrow3y=12 \\ \Rightarrow y=\frac{12}{3} \\ \Rightarrow y=4 \end{gathered}[/tex][tex]\begin{gathered} x=6 \\ \Rightarrow6+3y=12 \\ \Rightarrow3y=12-6 \\ \Rightarrow3y=6 \\ \Rightarrow y=\frac{6}{3} \\ \Rightarrow y=2 \end{gathered}[/tex]Then, the points (0,4) and (6,2) belong to the line:
Solution:
The lines intersect at the point (3,3).
Then, the solution for this system of equations, is:
[tex]\begin{gathered} x=3 \\ y=3 \end{gathered}[/tex]find the coordinates of the midpoint of ab for a(2 5) and b(6 9)
The coordinates of the midpoint of a,b is given as;
[tex]\lbrack\frac{1}{2}(a_1+a_2),\text{ }\frac{1}{2}(_{}b_1+b_2)\rbrack[/tex]Use point-slope form to write the equation of a line that passes through the point (8,17) with slope 3/2 .
The equation of the line with the given properties is y - 17 = 3/2(x - 18)
What are linear equations?Linear equations are equations that have constant average rates of change.
How to determine the equation of the line in point-slope form?The points are given as
(8, 17)
The slope is given as
Slope = 3/2
Calculate the slope of the points using
m = (y2 - y1)/(x2 - x1)
Where
Slope = m = 3/2
(x1, y1) = (8, 17)
So, we have
3/2 = (y - 17)/(x - 8)
Cross multiply in the above equation
So, we have
y - 17 = 3/2(x - 18)
A linear equation is represented as
y = m(x - x1) + y1
In point-slope form, we have
y - y1 = m(x - x1)
By comparing y - y1 = m(x - x1) and y - 17 = 3/2(x - 18), we can see that
y - 17 = 3/2(x - 18) is in point-slope form
Hence, the equation in point-slope form of the line that passes through the point (8, 17) is y - 17 = 3/2(x - 18)
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On Monday, a baker made cookies. He had enough cookies to completely fill 2
equal-sized trays. He sells the cookies for $3 each.
2 3 4 5
12
At the end of the day on Monday, the trays are pictured above. How much mone
did the baker earn selling cookies on Monday?
10
4 78910
12
Answer:
Step-by-step explanation:
it is my first time doing dis so it is 12.
Vertical angles are two angles which are congruent. Below is an example of vertical
angles. Write an equation and solve for x.
(9y+7)°
go on play store and download symbolab it can help you
Let x = any negative rational number. Select all statements that are true.
Answer:
please provide options
Step-by-step explanation:
Reflection across y=3
In the picture, There is graph with a triangle SNZ. The reflected triangle of y=3 as (1,2),(5,3) and (5,5).
Given that,
In the picture, There is graph with a triangle.
The triangle is SNZ.
We have to find the reflection across y=3.
We have to draw a line on y=3.
On the line y=3,
Z point is there so it will be same that is (5,3).
Now, the point S is on (5,1)
Here, y is 1 that is 3+2=5
So, we take the reflected S point as (5,5)
Now, the point N is on (1,4)
Here, y is 4 that is 3-1=2
So, we take the reflected N point as (1,2).
Therefore, we get the reflected triangle of y=3 as (1,2),(5,3) and (5,5).
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which expression is equal to (-7)^2 x (-7)^5 c (-7)^-9
The simplified expression of (-7)^2 x (-7)^5 x (-7)^-9 is (-7)^-2
What are expressions?Expressions are mathematical statements that are represented by variables, coefficients and operators
How to evaluate the expression?The expression is given as
(-7)^2 x (-7)^5 x (-7)^-9
The base of the above expression are the same
i.e. Base = -7
This means that we can apply the law of indices
When the law of indices is applied, we have the following equation:
(-7)^2 x (-7)^5 x (-7)^-9 = (-7)^(2 + 5 - 9)
Evaluate the sum in the above equation
So, we have
(-7)^2 x (-7)^5 x (-7)^-9 = (-7)^(7 - 9)
Evaluate the difference in the above equation
So, we have
(-7)^2 x (-7)^5 x (-7)^-9 = (-7)^-2
Hence, the simplified expression of the expression given as (-7)^2 x (-7)^5 x (-7)^-9 is (-7)^-2
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can anyone help pleasee
Answer:
a) x³
b)y⁵
Step-by-step explanation:
Kaitlin baked 56 cookies. Her family ate d of them. Using d, write an expression for the number of cookies that remained.
WILL GIVE BRAINLYEST 100 POINTS 1 WUESTION
All the options occurred as a result of Roman expansion following the Punic Wars except; B: It allowed many Romans to buy large farming estates
What happened in history after the the Punic Wars?The three Punic Wars between Carthage and Rome took place over about a century, starting in 264 B.C. and it ended with the event of the destruction of Carthage in the year 146 B.C.
Now, at the time the First Punic War broke out, Rome had become the dominant power throughout the Italian peninsula, while Carthage–a powerful city-state in northern Africa–had established itself as the leading maritime power in the world. The First Punic War commenced in the year 264 B.C. when Rome expressed interference in a dispute on the island of Sicily controlled by the Carthaginians. At the end of the war, Rome had full control of both Sicily and Corsica and this meant that the it emerged as a naval and a land power.
In the Second Punic War, the great Carthaginian general Hannibal invaded Italy and scored great victories at Lake Trasimene and Cannae before his eventual defeat at the hands of Rome’s Scipio Africanus in the year 202 B.C. had to leave Rome to be controlled by the western Mediterranean as well as large swats of Spain.
In the Third Punic War, we saw that Scipio the Younger led the Romans by capturing and destroying the city of Carthage in the year 146 B.C., thereby turning Africa into yet another province of the mighty Roman Empire.
Thus, we can see that the cause of the Punic wars is that the Roman republic grew, so they needed to expand their territory by conquering other lands, including Carthage.
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Let Q(x, y) be the statement "student x has become a contestant on quiz event y". Express each of the following sentences in terms of Q(x, y), quantifier, and logical connection, where the domain for x consists of all students in your major and for y consists of all quiz shows on television.
a) There is a student in your major who has become a contestant on a television quiz show.
b) No student in your major has ever been a contestant on a television quiz show.
c) There are students in your department who have been contestants for Family 100 and Super Deal Indonesia.
d) Every television quiz show has a student of your major as a contestant.
e) At least two students from your major have become contestants in Family 100
The statement is (a) [tex]E_{xy}[/tex] Q(x, y); (b) -Q(x, y); (c) [tex]E_{x}[/tex] (Q(x, jeopardy) ∩ Q(x, Wheel of fortune)); (d) ∀[tex]_{y}[/tex][tex]E_{x}[/tex]Q(x, y); (e) [tex]E_{x}E_{y}[/tex](x ≠ y) (Q(x, jeopardy) ∩ Q(y, jeopardy)).
what is probability?Simply put, probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes. Statistics is the study of events that follow a probability distribution.
Given that,
Q(x, y) be the statement "student x has become a contestant on quiz event y",
where the domain for x consists of all students in your major and for y consists of all quiz shows on television.
a) [tex]E_{xy}[/tex] Q(x, y)
b) -Q(x, y)
c) [tex]E_{x}[/tex] (Q(x, jeopardy) ∩ Q(x, Wheel of fortune))
d) ∀[tex]_{y}[/tex][tex]E_{x}[/tex]Q(x, y)
e) [tex]E_{x}E_{y}[/tex](x ≠ y)(Q(x, jeopardy) ∩ Q(y, jeopardy)).
Therefore, the statements are above.
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Write an algebraic expression for two less than the quotient of 15 and a number
The algebraic expression for two less than the quotient of 15 and a number is 15x - 2.
What is algebraic expression ?
At least one variable and one operation must be present in an algebraic expression (addition, subtraction, multiplication, division). One such algebraic expression is 2(x + 8y).
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A delivery truck is transporting boxes of two sizes: large and small. The combined weight of a large box and a small box is 80 pounds. The truck is transporting 65 large boxes and 55 small boxes. If the truck is carrying a total of 4850 pounds in boxes, how much does each type of box weigh?
Large box:____Pounds
Small box:____Pounds
The large box weighs 45 pounds and the small box weighs 35 pounds.
How to calculate the value?Let the weight of the small box = x
Let the weight of large box = y
The combined weight of a large box and a small box is 80 pounds. The truck is transporting 65 large boxes and 55 small boxes. If the truck is carrying a total of 4850 pounds in boxes. This will be illustrated as:
x + y = 80 ...... i
55x + 65y = 4850 .... ii
From equation i x = 80 - y
This will be put into equation ii
55x + 65y = 4850
55(80 - y) + 65y = 4850
4400 - 55y + 65y = 4850
10y = 4850 - 4400
10y = 450
y = 450 / 10 = 45
Large box = 45 pounds
Since x + y = 80
x = 80 - 45 = 35
Small box = 35 pounds.
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Preform each operation. (1/8)(9/11)= 5/6+4/7=
Perfoming each operation, we have:
[tex]\frac{1}{8}\cdot\frac{9}{11}=\frac{9}{88}[/tex][tex]\begin{gathered} \frac{5}{6}+\frac{4}{7} \\ =\frac{7\cdot5+6\cdot4}{42} \\ =\frac{35+24}{42} \\ =\frac{59}{42} \end{gathered}[/tex]A salesperson earns a commission of $624 for selling $3900 in merchandise. find commission rate
Answer:
16%
Step-by-step explanation:
A salesperson earns a commission of $624 for selling $3900 in merchandise. Find the rate of commission:
624 is ?% of 3,900
624/3,900 = 16% of 3,900
check answer:
3,900 * 0.16 = 624
Please help me on my hw
We can find the x and y-intercept by substituting zero for x and y respectively.Part A
iven the eequation below;
[tex]y=x+7[/tex]When x=0
[tex]\begin{gathered} y=0+7 \\ y=7 \end{gathered}[/tex]when y=0
[tex]\begin{gathered} 0=x+7 \\ x=-7 \end{gathered}[/tex]Answer 1
[tex]\begin{gathered} x-\text{intercept}=(-7,0) \\ y-\text{intercept}=(0,7) \end{gathered}[/tex]Part B
[tex]y=x^2-3[/tex]When x=0
[tex]\begin{gathered} y=0^2-3 \\ y=-3 \end{gathered}[/tex]When y=0
[tex]\begin{gathered} 0=x^2-3 \\ x^2=3 \\ x=\pm\sqrt[]{3} \\ x=-\sqrt[]{3}\text{ or x=}\sqrt[]{3} \end{gathered}[/tex]Answer 2:
[tex]\begin{gathered} x-\text{intercept}=(-\sqrt[]{3},0) \\ x-\text{intercept}=(\sqrt[]{3},0) \\ y-\text{intercept}=(0,-3) \end{gathered}[/tex]THE RAFFLE QUEEN
1000 raffle tickets are sold for $3.00 each.
There is one grand prize for $750 and two
consolation prizes of $200 each. What is the
expected value of one ticket?
ST
co
W
Answer:
$1.15
Step-by-step explanation:
750 + 400 = 1,150
1/1000 * 1,150 = 1,150/1000 = $1.15 expected value of one ticket.
Since he paid more than $1.15 for the ticket he has a bad bet.
The perimeter of a rectangle is to be no greater than 70 centimeters
and the width must be 5 centimeters. Find the maximum length of the
rectangle.
Answer:
Maximum length = 30 cm
Step-by-step explanation:
Perimeter of a rectangle = 2 × (length + width)
According to the question,
2 × (length + width) < 71 cm (It can be 70 cm at maximum)
length + width < 71/2 cm
length + width < 36 cm
Since, width = 5 cm,
length + 5 cm < 36 cm
length < 36 - 5 cm
length < 31 cm
Therefore, the maximum length can be 30 cm
Part a and part B help please its all one question that goes together incase the pic is confusing
The winning average of the Varsity football team is a non-terminating decimal.
The winning average of the Junior Varsity football team is a terminating decimal.
Which team had a better season? Varsity team
How is the winning average calculated?
a ) Part A
1. Team Varsity
Number of total matches won = 8
Number of total matches lost = 3
Total number of matches = 11
The winning average [tex]=\frac{\text{total number of matches won}}{\text{total matches}}[/tex]
=[tex]\frac{8}{11} \\\\[/tex]
= 0.72727
0.72727 is a non-terminating decimal
2. Team Junior Varsity
Number of total matches won = 7
Number of total matches lost = 3
Total number of matches = 10
The winning average [tex]=\frac{\text{total number of matches won}}{\text{total matches}}[/tex]
=[tex]\frac{7}{10} \\\\[/tex]
= 0.7
0.7 is a terminating decimal.
The winning average of the Varsity football team is a non-terminating decimal.
The winning average of the Junior Varsity football team is a terminating decimal.
b) Part B
Which team had a better season? Varsity team
Varsity team had a better season because the winning average of team Varsity is higher than Junior Varsity.To learn more about non - terminating decimals, refer:
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Seema used compatible numbers to estimate the product of (–25.31)(9.61). What was her estimate?
When Seema used compatible numbers to estimate the product of (–25.31)(9.61), her estimate is A. -250.
How to illustrate the information?From the information, it should be noted that Seema used compatible numbers to estimate the product of (–25.31)(9.61).
It should be noted that -25.31 when rounded will be -25.
It should be noted that 9.61 when rounded will be 10.
Therefore, the multiplication will be:
= -25 × 10
= -250.
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Seema used compatible numbers to estimate the product of (–25.31)(9.61). What was her estimate?
-250
-240
240
250
Two planes start from the same point and fly in opposite directions. The first plane is flying 30 mph slower than the second plane. In 3 h, the planes are 810 mi apart. Find the rate of each plane.
Answer:
120 mph150 mphStep-by-step explanation:
Given two planes flying in opposite directions are 810 miles apart after 3 hours, and the first is 30 mph slower than the second, you want the speed of each plane.
SetupLet s represent the speed of the slower plane. Then faster plane will have a speed of (s+30). The distance between the planes increases at a rate equal to the sum of their speeds. Distance is the product of speed and time, so we have ...
distance = speed × time
810 = (s + (s+30)) × 3
SolutionDividing the equation by 3, we get ...
270 = 2s +30
240 = 2s . . . . . . subtract 30
120 = s . . . . . . . divide by 2
150 = s+30 . . . the speed of the faster plane
The speed of the first plane is 120 mph; the speed of the second plane is 150 mph.
The rate of the two planes flying in opposite direction was found to be
The faster plane = 150 mphThe slower plane = 120 mphHow to find the rate of each plane
given data
The first plane is flying 30 mph slower than the second plane.
time = 3 hours
distance = 810 miles
let the rate of the faster plane be x
then rate if the slower plane will be x - 3
rate of both planes
= x + x - 30
= 2x - 30
Finding the rate of each plane
rate of both planes = total distance / total time
2x - 30 = 810 / 3
2x - 30 = 270
2x = 270 + 30
2x = 300
x = 150
Then the slower plane = 150 - 30 = 120 mph
Hence the rate of the faster plane is 150 mph and the rate of the slower plane is 120 mph
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Cara deposited $200 dollars into her savings account bringing her balance up to $450.Which equation can be used to find, x, the savings account balance before the $200 deposit?
Let x be her saving accounts balance before the $200 deposit
So;
x + 200 = 450
or
x = 450 -200
If x is a solution to the equation 3x−12=24, select all the equations that also have x as a solution. Multiple select question. A) 15x−60=120 B) 3x=12 C) 3x=36 D) x−4=8 E) 12x−12=24
The equations that have x as a solution are 15x - 60 = 120 and 3x = 24.
How to find equations that has the same solution?The equation is as follows:
3x - 12 = 24
The equations that also has x as the solution can be found as follows:
Let's use the law of multiplication equality to find a solution that has x as the solution.
The multiplication property of equality states that when we multiply both sides of an equation by the same number, the two sides remain equal.
Multiply both sides of the equation by 5
3x - 12 = 24
Hence,
15x - 60 = 120
By adding a number to both sides of the equation, we can get same solution for x.
3x - 12 = 24
add 12 to both sides of the equation
3x - 12 + 12 = 24 + 12
3x = 24
Therefore, the two solution are 15x - 60 = 120 and 3x = 24
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x+5y=10 3x + 5y = 0 Graph each equation determine the solution of the system of equations
Answer:
Explanation:
Given the system of equations:
[tex]\begin{gathered} x+5y=10 \\ 3x+5y=0 \end{gathered}[/tex]To