The additional root is 6-i.
Here's why:
Conjugate Pairs for Rational Coefficients: When a polynomial with rational coefficients has a complex root of the form a + bi (where a and b are real numbers and i is the imaginary unit), its conjugate, a - bi, must also be a root. This ensures that the polynomial remains with rational coefficients.
Applying the Conjugate Pair Rule: In this case, we're given that 6 + i is a root. Therefore, its conjugate, 6 - i, must also be a root of the polynomial P(x) = 0.
Other Roots: The problem states that √7 is also a root. However, since it's a real number, it doesn't introduce any complex conjugate pairs.
Quartic Polynomial: A quartic polynomial has four roots in total. We've identified three of them: √7, 6 + i, and 6 - i. The fourth root could be either a real number or another complex conjugate pair, but the information provided is insufficient to determine its exact value.
Write an equation for a function with the given characteristics. A cosine curve with a period of pi, and amplitude of 1, a left phase shift of pi, and vertical transiation up 5/2 of a unit.
The equation for the function with the given characteristics is:
f(x) = (1 * cos((x - pi)/pi)) + (5/2)
We have,
The equation for the function with the given characteristics can be written as follows:
f(x) = (1 * cos((x - pi)/pi)) + (5/2)
Amplitude: The amplitude of the cosine curve is 1, so we multiply the cosine function by 1.
Period: The period of the cosine curve is pi, which means it completes one full cycle over the interval [0, pi]. The period of the cosine function is determined by the argument inside the cosine function, (x - pi)/pi. This shifts the period to start at pi and end at 2pi.
Left phase shift: The left phase shift of pi is achieved by subtracting pi from x inside the argument of the cosine function. This shifts the curve to the left by pi units.
Vertical translation: The vertical translation up 5/2 units is achieved by adding 5/2 to the entire function.
Therefore,
The equation for the function with the given characteristics is:
f(x) = (1 * cos((x - pi)/pi)) + (5/2)
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The equation for the function is y(t) = cos(2t + pi) + 5/2.
Explanation:To write an equation for the given characteristics, we can start with the general equation for a cosine curve: y(t) = A cos(wt + p).
Given that the period is pi, we know that the angular frequency w is equal to 2pi divided by the period, so w = 2pi/pi = 2. The amplitude is 1, so A = 1.
The left phase shift is pi, so p = pi. Finally, there is a vertical translation up 5/2 units, so we add 5/2 to the equation.
Putting all these values into the equation, we get: y(t) = cos(2t + pi) + 5/2.
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An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected, what is the probability that each of the balls will be (a) of the same color?
Answer:
As a fraction, the answer is exactly 86/969
In decimal form, the answer is approximately 0.08875
=====================================================
Work Shown:
The assumption is that no replacements are made for each selection.
5 red, 6 blue, 8 green
5+6+8 = 19 total
A = P(3 red) = (5/19)*(4/18)*(3/17) = 10/969
B = P(3 blue) = (6/19)*(5/18)*(4/17) = 20/969
C = P(3 green) = (8/19)*(7/18)*(6/17) = 56/969
D = P(3 all same color)
D = A+B+C
D = 10/969 + 20/969 + 56/969
D = (10+20+56)/969
D = 86/969
D = 0.08875
John is building a custom wall for a kitchen remodel. He knows the space for the wall is square with an area of 121 feet. How long does his diagonal brace need to be?
Answer:
His diagonal brace need to be 15.55 feet long in the kitchen.
Step-by-step explanation:
Given : The wall of the kitchen is in the shape of a square.
Area of the wall = 121 sq ft
Let us assume the each side of wall = a ft
Now, AREA OF THE SQUARE = Side x Side
⇒ a x a = 121 sq units
[tex]\implies a ^2 = 121\\\implies a = 11[/tex]
So, each side of the kitchen wall is 11 ft.
Now,as we know in a square : EACH ANGLE IN THE SQUIRE IS A RIGHT ANGLE.
So, by Pythagoras Theorem, Diagonal D is:
[tex]D^2 = a^2 + a^2\\\implies D^2 = 11^2 + 11^2 = 121 + 121\\\implies D^2 = 242\\\implies D = 15.55[/tex]
Hence, his diagonal brace need to be 15.55 ft long in the kitchen.
Simplify the complex expression and show work
The answer is
[tex](-9+3\sqrt{2})+(3\sqrt{2}-2)i[/tex]
=======================================================
Work Shown:
[tex](-3+\sqrt{-2})(3-\sqrt{2})[/tex]
[tex](-3+\sqrt{-1*2})(3-\sqrt{2})[/tex]
[tex](-3+\sqrt{-1}*\sqrt{2})(3-\sqrt{2})[/tex]
[tex](-3+i\sqrt{2})(3-\sqrt{2})[/tex]
[tex]x(3-\sqrt{2})[/tex] let [tex]x=-3+i\sqrt{2}[/tex]
[tex]3x-x*\sqrt{2}[/tex] distribute
[tex]3(x)-\sqrt{2}(x)[/tex]
[tex]3(-3+i\sqrt{2})-\sqrt{2}(-3+i\sqrt{2})[/tex]
[tex]-9+3i\sqrt{2}+3\sqrt{2}-i*\sqrt{2}*\sqrt{2}[/tex]
[tex]-9+3i\sqrt{2}+3\sqrt{2}-2i[/tex]
[tex]-9+3\sqrt{2}+3i\sqrt{2}-2i[/tex]
[tex](-9+3\sqrt{2})+(3\sqrt{2}-2)i[/tex]
The use of parenthesis in the last step is to help separate out the terms.
The last expression shown above is in the form a+bi where
[tex]a=-9+3\sqrt{2}[/tex]
[tex]b=3\sqrt{2}-2[/tex]
Name at least 2 major similarities between the sheep and human brains that you did not name in your hypothesis, but that you observed or learned about during the dissection.
Answer:
Step-by-step explanation: The brains are very similar insofar that they function in almost the same way when it comes to controlling physical aspects of our existence. Our brains regulate our organs and our body parts just like a sheep's brain.
A researcher wants to determine whether the number of minutes adults spend online per day is related to gender. A random sample of 315 adults was selected and the results are shown below. Calculate the chi-square test statistic x2 to determine if there is enough evidence to conclude that the number of minutes spent online per day is related to gender. Minutes spent online per day Gender 0-30 30-60 60-90 90+Male 75 45 25 35 Female 30 45 45 15 A) 19.874 B) 18.146 C) 20.912 D) 21.231
Answer:
There is enough evidence to conclude that the number of minutes spent online per day is related to gender
Step-by-step explanation:
Given that a researcher wants to determine whether the number of minutes adults spend online per day is related to gender. A random sample of 315 adults was selected and the results are shown below.
Gender 0-30 30-60 60-90 90+ total
Males 75 45 25 35 180
Females 30 45 45 15 135
H0: there is no association between gender and spending time online
Ha: There is association
(Two tailed test)
Expected for each cell is calculated as row total*grand total/column total
<30 30-60 60-90 >90 Row Totals
Males 75 (60.00) [3.75] 45 (51.43) [0.80] 25 (40.00) [5.62] 35 (28.57) [1.45] 180
Females 30 (45.00) [5.00] 45 (38.57) [1.07] 45 (30.00) [7.50] 15 (21.43) [1.93] 135
chi square -= 27.125
p value <0.0001
Since p is less than 1% (significance level) we reject the null hypothesis
There is enough evidence to conclude that the number of minutes spent online per day is related to gender
Answer:
The answer would be 18.146
Step-by-step explanation:
Given that a researcher wants to determine whether the number of minutes adults spend online per day is related to gender. A random sample of 315 adults was selected and the results are shown below.Gender 0-30 30-60 60-90 90+ totalMales 75 45 25 35 180Females 30 45 45 15 135H0: there is no association between gender and spending time onlineHa: There is association(Two tailed test)Expected for each cell is calculated as row total*grand total/column total<30 30-60 60-90 >90 Row TotalsMales 75 (60.00) [3.75] 45 (51.43) [0.80] 25 (40.00) [5.62] 35 (28.57) [1.45] 180Females 30 (45.00) [5.00] 45 (38.57) [1.07] 45 (30.00) [7.50] 15 (21.43) [1.93] 135chi square -= 27.125p value <0.0001Since p is less than 1% (significance level) we reject the null hypothesisThere is enough evidence to conclude that the number of minutes spent online per day is related to gender
Using your calculator and rounding your answers to the nearest hundredth, find the remaining trigonometric ratios of θ based on the given information. sin θ = 0.26 and θ is in QI
Answer: 15.07°
Step-by-step explanation:
Sin^-1 0.26= 15.07°
Therefore:
Cos15.07= 0.9656
Tan 15.07 = 0.2692
Csc15.07= 1/(Sin15.07)= 3.846
Sec15.07=1/cos15.07=1.0356
Cot15.07=1/tan15.07= 3.7147
We can use the trigonometric identity sin² θ + cos² θ = 1 to find cos θ in a first quadrant triangle where sin θ = 0.26, and get cos θ = 0.966. Then, we calculate tan θ with the formula tan θ = sin θ / cos θ, and get tan θ = 0.269 after rounding.
Explanation:In trigonometry, there are three primary ratios called trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios can be defined using the sides of a right triangle, and they relate the angles of a triangle to the lengths of its sides. If we know one of these ratios in a right triangle, we can find the others.
In this case, we know that sin θ = 0.26 and θ is in Quadrant I. In the first quadrant, all the trigonometric ratios are positive.
So, firstly we find cos θ. Remember that sin² θ + cos² θ = 1, we can solve for cos θ using the calculator with the square root of (1 - sin² θ), which results in cos θ = 0.966 approximately.
Next, to find tan θ, we use the relationship tan θ = sin θ / cos θ. So, tan θ = 0.26 / 0.966 = 0.269 approximately, after rounding to the nearest hundredth.
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In horse racing, a trifecta is a bet that the first three finishers in a race are selected, and they are selected in the correct order. Does a trifecta involve combinations orpermutations? Explain.Choose the correct answer below.A. Because the order of the first three finishers does not make a difference, the trifecta involves permutations.B. Because the order of the first three finishers does make adifference, the trifecta involves permutations.C. Because the order of the first three finishers does not make a difference, the trifecta involves combinations.D. Because the order of the first three finishers does make adifference, the trifecta involves combinations.
Answer:
its a combination
Step-by-step explanation:
because three horses is a combination of horses
Write an expression for the sequence of operations.
Subtract 4 from x, double, and add 8.
The expression for the given sequence of operations is 2(x-4)+8.
The given statement is "subtract 4 from x, double, and add 8".
We need to write an expression for the sequence of operations.
What is an expression?An expression is a combination of terms that are combined by using mathematical operations such as subtraction, addition, multiplication, and division.
Now, subtract 4 from x=x-4
Double the difference that is 2(x-4)
Add 8=2(x-4)+8
Therefore, the expression for the given sequence of operations is 2(x-4)+8.
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The lines below are perpendicular. If
the slope of the solid line is , what
is the slope of the dashed line?
Answer:
Correct answer: s₁ = - 2
Step-by-step explanation:
The relationship between the slope of two lines that are perpendicular is:
s₁ = - 1/s = - 1/(1/2) = - 2
God is with you!!!
Please help asap i really need this. Brainliest. Using Square Roots to Solve Equations.
Answer:
The answer to your question is below
Step-by-step explanation:
a) x² = 196
x = ±√196
x = ± 14 or x = -14 and x = 14
b) 9x² = 49
x² = 49/ 9
x = √(49/9)
x = ±7/3
x = ± 2.3 or x = 2.3 and x = -2.3
c) 4x² - 8 = 56
4x² = 56 + 8
4x² = 64
x² = 64/4
x² = 16
x = ±√16
x = ±4 or x = 4 and x = -4
d) x² = 160
x = ± √160
Find the prime factors of 160
160 2
80 2
40 2
20 2
10 2
5 5
1 160 = 2⁵5
x = √2⁴10
x = ±4√10
e) x² = 72
Find the prime factor of 72
72 = 2²3²6
x = √2²3²6
x = ±6√6 or
x = ±8.5
A gardener is planting two types of trees: Type A is 5 feet tall and grows at a rate of 13 inches per year. Type B is 3 feet tall and grows at a rate of 21 inches per year. Algebraically determine exactly how many years it will take for these trees to be the same height.
Answer:it will take 0.25 years
Step-by-step explanation:
Let x represent the number of years that it will take for type A and tree B to be the same height.
Let y represent the height of type A after x years.
Let z represent the height of type B after x years.
Type A is 5 feet tall and grows at a rate of 13 inches per year. This means that
y = 5 + 13x
Type B is 3 feet tall and grows at a rate of 21 inches per year. This means that
z = 3 + 21x
To determine the number of years it will take both trees to be of the same height, we would equate equation y to z. It becomes
5 + 13x = 3 + 21x
21x - 13x = 5 - 3
8x = 2
x = 2/8 = 0.25 years
The phenomenon that involves the misperception of the length of two identical lines, one with arrows pointing outward and one with arrows pointing inward is called the:
Müller-Lyer illusion. (T/F)
Answer:
True
Step-by-step explanation:
Müller-Lyer illusion - In this illusion two-line having the same length appear to be different from one another.
the line showing in the file attached, the line with arrow moving outward appears to have large length and line having the arrow pointing inward appear to have a short length. But given lines are equal in length, it is just an illusion that might force to believed you that one is shorter or larger than others.
Adnan made "x" calculators this week, but Maya made twice as many calculators. Write an algebraic expression for the number of calculators that Maya made this week. A) 2x B) x + 2 C) x 2 D) x − 2
Answer:
A) [tex]2x[/tex]
Step-by-step explanation:
Given:
Number of calculator made by Adnan in this week = [tex]'x'[/tex]
We need to write an algebraic expression for the number of calculators that Maya made this week.
Solution:
Now given:
Maya made twice as many calculators.
So we can say that;
Number of calculators made by Maya this week is equal to 2 times Number of calculator made by Adnan this week.
framing in equation form we get;
Number of calculators made by Maya this week = [tex]2\times x = 2x[/tex]
Hence an algebraic expression for the number of calculators that Maya made this week is [tex]2x[/tex].
A delivery truck has 40,000 miles on its odometer and travels an average of 22,000 miles per year.
Which function best models the linear relationship?
Answer:
y=22000x+40000
The concession stand at football games sells pizza and drinks. You buy 4 drink and 6 slices of pizza for $6.70. Your friend buys 3 drinks and 4 slices of pizza for $4.65. How much is each drink and slice of pizza?
Answer:
drinks: $0.55pizza slices: $0.75Step-by-step explanation:
Let d represent the cost of a drink, and p represent the cost of a pizza slice. Then the two purchases can be represented by ...
4d +6p = 6.703d +4p = 4.65To solve these equations by elimination, choose a variable to eliminate and look at the coefficients of that in the two equations. If we choose to eliminate p, we see the coefficients of p are 6 and 4. The least common multiple of these numbers is 12. We can multiply the first equation by -2 and the second equation by +3 and the resulting coefficients of p will be -12 and +12. Adding the results of these multiplications will make the p terms add to zero.
-2(4d +6p) +3(3d +4p) = -2(6.70) +3(4.65)
-8d -12p +9d +12p = -13.40 +13.95 . . . . . . . . . eliminate parentheses
d = 0.55 . . . . . . . . . collect terms
Now, we can substitute this value into either equation to find the value of p. Using the first equation, we get ...
4(0.55) +6p = 6.70
6p = 4.50 . . . . . . . . . subtract 2.20
p = 0.75 . . . . . . . . . . divide by 6
The cost of a drink is $0.55; the cost of a slice of pizza is $0.75.
Answer:each drink costs $0.55
Each slice of pizza costs $0.75
Step-by-step explanation:
Let x represent the cost of each drink.
Let y represent the cost of each slice of pizza.
You buy 4 drink and 6 slices of pizza for $6.70. This means that
4x + 6y = 6.70 - - - - - - - - - - - - 1
Your friend buys 3 drinks and 4 slices of pizza for $4.65. This means that
3x + 4y = 4.65 - - - - - - -- - - 2
Multiplying equation 1 by 3 and equation 2 by 4, it becomes
12x + 18y = 20.1
12x + 16y = 18.6
Subtracting, it becomes
2y = 1.5
y = 1.5/2 = 0.75
Substituting y = 0.75 into equation 1, it becomes
4x + 6 × 0.75 = 6.70
4x + 4.5 = 6.70
4x = 6.7 - 4.5 = 2.2
x = 2.2/4 = 0.55
WILL GIVE BRAINLIEST NEED HELP ASAP THANK YOU <3
Answer:
False
Step-by-step explanation: Are you asking whether or not its false? If so then yes f(x) is not a function because the x's on the left side repeat. Example the six points to two different numbers on the right side. I hope i helped!
Answer:
your answer is correct
Step-by-step explanation:
When a relation has an input that maps to more than one output, it is not a function. Both 6 and 12 map to two different outputs, so the relation is not a function. Your answer choice is correct.
Find the midpoint of (0,4) and (-4,-12)
Needing help here don’t have a clue
Answer:
v(9)=2000
V(15)=8000
Step-by-step explanation:
Answer:
Step-by-step explanation:
You are given a function V(t) and asked to find V(9) and V(15). In each case, the number in parentheses replaces "t" in the function expression. After that, it is just arithmetic. (You may want to use a calculator, but it isn't really necessary.)
Given:
V(t) = 250·2^(t/3)
Find:
V(9) = 250·2^(9/3)
= 250·2^3
= 250·8
V(9) = 2000
__
V(15) = 250·2^(15/3)
= 250·2^5
= 250·32
V(15) = 8000
_____
As you know, an exponent signifies repeated multiplication. The exponent tells you the number of times the base is a factor in the product:
2^3 = 2·2·2 = 8 . . . . . . the exponent of 3 means 2 is a factor 3 times
Which choice correctly expresses the solution set in the inequality in words?
Answer:
The answer to your question is letter A.
Step-by-step explanation:
Inequality x + 2 ≥ 5
Solve the inequality, remember that inequalities and equations are solved in similar ways.
- Subtract 2 in both sides
x +2 - 2 ≥ 5 - 2
- Simplify
x ≥ 3
- Remember that this sign "≥" means that x is greater or equal to a number, and this sign "≤" means that x is lower or equal to a number.
The option that satisfies our inequality is the first option.
It cost $525 for 7adult tickets and 12 youth tickets to the hockey game. If the difference in price between adult ticket Nd youth ticket is $18, find the cost of a youth ticket
Final answer:
To find the cost of a youth ticket, you can set up a system of equations using the given information. Solve the system of equations to find the cost of a youth ticket.
Explanation:
To find the cost of a youth ticket, we can set up a system of equations using the given information. Let x be the cost of an adult ticket and y be the cost of a youth ticket. From the problem, we know that 7x + 12y = 525 and x - y = 18. We can solve this system of equations using substitution or elimination.
First, let's solve for x in terms of y using the second equation. Adding y to both sides gives x = y + 18.
Next, substitute this expression for x in the first equation: 7(y + 18) + 12y = 525. Simplifying this equation gives 7y + 126 + 12y = 525. Combining like terms, we get 19y + 126 = 525. Subtract 126 from both sides: 19y = 399. Finally, divide both sides by 19 to solve for y: y = 21.
Therefore, the cost of a youth ticket is $21.
PLEASE HELP!!!! (100 PTS)
Answer:
(x) free throw = 8
(y) 2-point = 6
(z) 3-point = 2
x+(y*2)+(z*3)
8+(6*2)+(2*3)
8+12+6
=26
Find an equation for those points P such that the distance from P to A(0, 1, 2) is equal to the distance from P to B(6, 4, 2). What is this set?
Answer: The required equation for points P is [tex]4x+2y=17.[/tex]
Step-by-step explanation: We are give two points A(0, 1, 2) and B(6, 4, 2).
To find the equation for points P such that the distance of P from both A and B are equal.
We know that the distance between two points R(a, b, c) and S(d, e, f) is given by
[tex]RS=\sqrt{(d-a)^2+(e-b)^2+(f-c)^2}.[/tex]
Let the point P be represented by (x, y, z).
According to the given information, we have
[tex]PA=PB\\\\\Rightarrow \sqrt{(x-0)^2+(y-1)^2+(z-2)^2}=\sqrt{(x-6)^2+(y-4)^2+(z-2)^2}\\\\\Rightarrow x^2+y^2-2y+1+z^2-4z+4=x^2-12x+36+y^2-8y+16+z^2-4z+4~~~~~~~[\textup{Squaring both sides}]\\\\\Rightarrow -2y+1=-12x-8y+52\\\\\Rightarrow 12x+6y=51\\\\\Rightarrow 4x+2y=17.[/tex]
Thus, the required equation for points P is [tex]4x+2y=17.[/tex]
What is the horizontal asymptote of f (x) =StartFraction negative 2 x Over x + 1 EndFraction? y = –2 y = –1 y = 0 y = 1
Answer:
y = -2
Step-by-step explanation:
Any asymptotes of a rational function will be described by the quotient of the numerator and denominator (excluding any remainder).
[tex]f(x)=\dfrac{-2x}{x+1}=-2+\dfrac{2}{x+1}[/tex]
The horizontal asymptote is ...
y = -2
Answer:
y = –2Step-by-step explanation:
The given function is
[tex]f(x)=\frac{-2x}{x+1}[/tex]
An asymptote refers to a restriction in the domain or range set of the function. This happens to rational functions, because there's a scenario where the function is undetermined: when its denominator is zero.
So, in this case, the value that makes the denominator zero is
[tex]x+1=0\\x=-1[/tex]
That means, we need to restrict the domain for [tex]x=-1[/tex]. But this is a vertical asymptote.
To find horizontal asymptotes, we need to find the restrictions for the range of the function, we do that isolating [tex]y[/tex].
[tex]y=\frac{-2x}{x+1}\\yx+y=-2x\\yx+2x=-y\\x(y+2)=-y\\x=-\frac{y}{y+2}[/tex]
So, if we analyse the denominator
[tex]y+2=0\\y=-2[/tex]
Therefore, the restriction is [tex]y=-2[/tex]. In other words, the vertical asymptote is the first choice.
Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon and medals are awarded for first, second, and third place, what is the probability that at least two of the triplets will win a medal?
Answer: Probability that at least two of the triplets will win a medal is19/24
Step-by-step explanation:
Total number of cmpetitors=9
Total number of selecting 3 winners out of 9 competitors=9C3=84
Number of ways of selecting 2 out of 3 triplets=3C2
Number of selecting 1 out of 6 others =6C1
Number of selecting 2 out of 3 triplet and o1 out of 6 others= 3C2 ×6C1=18
Number of selecting 3 out of 3 triplets=3C3=1
Total ways of selecting less than 2 or 2 triplets= 18+1=19
Probability that at least 2 of the triplets will win is 19/84
A particle that moves along a straight line has velocity v(t) = t 2 e −t meters per second after t seconds. How far will it travel during the first t seconds?
The distance traveled by the particle is determined as [tex]\int\limits^0_t {v(t)} \, dt=-e^{-t}t^{2}+2(-e^{-t}t-e^{-t})+2[/tex].
Given data:
To determine the distance traveled by the particle during the first t seconds, integrate the velocity function with respect to time over the interval [0, t].
Given the velocity function:
[tex]v(t) = t^2 * e^{(-t)}[/tex]
To find the distance traveled, integrate the absolute value of the velocity function over the interval [0, t]:
[tex]\text{distance} = \int\limits^0_t {v(t)} \, dt[/tex]
Since the velocity can be negative (indicating a change in direction), taking the absolute value ensures that we consider the magnitude of the velocity.
Now, calculate the integral:
[tex]D = \int\limits^0_t {t^2 * e^{(-t)}} \, dt[/tex]
To simplify this integral, split it into two parts, considering the cases when the velocity is positive and negative:
[tex]D=[-e^{-t}t^2-\int\limits^0_t {-2e^{-t}} \, dt]_0^t[/tex]
The first integral represents the distance traveled when the velocity is positive, and the second integral represents the distance traveled when the velocity is negative.
Integrating each term separately:
[tex]\int\limits^0_t {v(t)} \, dt=-e^{-t}t^{2}+2(-e^{-t}t-e^{-t})+2[/tex]
Hence, the distance is [tex]D=-e^{-t}t^{2}+2(-e^{-t}t-e^{-t})+2[/tex].
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The complete question is attached below:
A particle that moves along a straight line has velocity [tex]v(t) = t^2e^{-t}[/tex] meters per second after t seconds. How far will it travel during the first t seconds?
The function v(t) = t2 e−t defines the velocity of a particle. The displacement or distance covered by the particle in the first t seconds is given by the integral from 0 to t of the velocity function.
Explanation:
The given velocity function for the particle is v(t) = t2 e−t. The distance (or displacement) traveled by an object is the integral of the velocity over a given interval. So, to find the distance covered by the particle in the first t seconds, you would set up the following integral:
∫0t t2 e−t dt
We won't compute this integral here, as it requires knowledge of methods beyond basic calculus, specifically the use of integration by parts and possibly a tabular method for successive integrations by parts. However, this setup gives you the right starting point to find the distance covered in the first t seconds.
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The sum of four numbers in arithmetic progression is 16. The square of the last number is the square of the first number plus 48. What are the four numbers?
Answer:
four numbers are 1, 3, 5, 7
Step-by-step explanation:
The sum of four numbers in arithmetic progression is 16
a, a+d, a+2d, a+3d are the four arithmetic series
sum of 4 numbers are
[tex]a+a+d+a+2d+a+3d=4a+6d[/tex]
[tex]4a+6d= 16[/tex]
divide both sides by 2
[tex]2a+3d=8[/tex]
[tex]3d= 8-2a[/tex]
The square of the last number is the square of the first number plus 48.
[tex](a+3d)^2=a^2+48[/tex]
solve for a and d
[tex](a+3d)^2=a^2+48\\(a+8-2a)^2=a^2+48\\(8-a)^2=a^2+48\\a^2-16a+64=a^2+48\\-16a=48-64\\-16a=-16\\a=1[/tex]
Now find out 'd'
[tex]3d=8-2a\\3d=8-2\\3d=6\\a=2[/tex]
a, a+d, a+2d, a+3d
four numbers are 1, 3, 5, 7
find the distance between (-6,6)(-5,-7)
The inverse of x y is:
xy
xy
yx
xy
yx
To answer this question, first, we need to understand the concept of an inverse operation, especially in the context of binary operations. A binary operation is a calculation that combines two elements to produce another element. Common binary operations are addition, subtraction, multiplication, and division in mathematics.
The inverse of a binary operation, denoted here as x y, is an operation, which when performed after the initial operation, returns the original value. This means, if we take x and y to produce z through the operation x y (denoted as x y -> z), the inverse operation will bring us back to the original x or y value. In mathematical terms, if we perform z (inverse) x operation, it should give us y, and z (inverse) y should return x.
From the given options, we can clearly observe that only one option obeys the rules of the inverse operation, that is 'yx' (option 3). Because only 'yx' will satisfy, z(inverse) x = y and z (inverse) y = x.
So, option 3, 'yx' is the inverse operation of x y in the sense of the inverse element.
Answer:yx
In 2006 the population of the country was 30.3 million. This represented an increase in population of 4.7% since 2001 . What was the population of the country in 2001
Answer:
The population in 2001 was = 23.95 millions
Step-by-step explanation:
Given
Population in 2006 = 30.3 million
Rate of growth since 2001 = 4.7%
To find the population in 2001.
Solution:
The population growth formula is give as:
[tex]P=P_0\times e^{rt}[/tex]
[tex]P\rightarrow[/tex] New population
[tex]P_o\rightarrow[/tex] Initial population
[tex]r\rightarrow[/tex] rate of growth
[tex]t\rightarrow[/tex] time
Data given to us:
New population [tex]P[/tex] (2006) = 30.3 million
Rate = 4.7% = 0.047
Time = [tex]2006-2001[/tex]= 5 years
Plugging in the data in the formula.
[tex]30.3=P_0\times e^{0.047\times 5}[/tex]
[tex]30.3=P_0\times e^{0.235}[/tex]
[tex]30.3=P_0\times 1.265[/tex]
Dividing both sides by 1.265
[tex]\frac{30.3}{1.265}=\frac{P_0\times 1.265}{1.265}[/tex]
[tex]23.95=P_o[/tex]
∴ [tex]P_o=23.95[/tex]
Thus, population in 2001 was = 23.95 millions