The number of elements of event E n(E) is 70.
The selecting of r items out of n items is done by using Combination. The formula for combination is as follows,
[tex]C(n,r)=\frac{n!}{r!(n-r)!}[/tex]
Where n is the total number of elements and out of which r is selected and ! is factorial sign and n is greater than or equal to r.
Now, Finding the number of elements in the event E that at least two 5-dollar bills are drawn from the bag.
Here, three bills are drawn out of which at least two 5-dollar bills must be drawn, so
[tex]n(E)=C(5,2)\times C(6,1)+C(5,3)\\\\=\frac{5!}{2!(5-2)!} \times \frac{6!}{1!(6-1)!} +\frac{5!}{3!(5-3)!}\\\\=10\times6+10\\=70[/tex]
Therefore, the number of elements n(E) in event E is 70.
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Given that 4−4i is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.
f(x)=x^4+4x^3−32x^2+128x+1024
Factor of the following polynomial function is (x-1)(x+7)(x-4+4i)(x-4-4i).
If 4-4i is a zero, then is its conjugate, 4+4i, as well.
Next 4-4i and 4+4i are zeros, this means that
x^4-2x^3-23x^2+248x-224=(x-4+4i)(x-4-4i)g(x), (A)
where g(x) is a degree 2 polynomial.
Since (x-4+4i)(x-4-4i)=x^2-8x+32,(A) is equivalent to
x^4-2x^3-23x^2+248x-224=(x^2-8x+32)g(x) (B)
Now dividing the LHS of (B) by x^2-8x+32 you find that g(x)=x^2+6x-7.
So,
x^4-2x^3-23x^2+248x-224=(x^2-8x+32)(x^2+6x-7) (C)
It is easy to see that x^2+6x-7=(x-1)(x+7).
Finally, you get
x^4-2x^3-23x^2+248x-224=(x-1)(x+7)(x-4+4i)(x-4-4i)
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Giving an arithmetic sequence with a33=66 and a common difference of 2.
Find a1;
Hint: Use an = a1 + (n-1) d
a1= ___________
Find the sum of the first 30 terms:
Hint: Use an = a1 + (n-1) d and Sn = n/2 (a1 + an)
S30 = __________
The first term [tex]a_{1}[/tex] = 2 and the sum of first 30 terms [tex]s_{30}[/tex] = 930
What is an Arithmetic progression?Arithmetic Progression (AP) is a sequence of numbers in order, in which the difference between any two consecutive numbers is a constant value.
Arithmetic sequence is gotten by adding a known term by a constant value.
if [tex]a_{33}[/tex] = 66 and d = 2 and [tex]a_{n}[/tex] = [tex]a_{1}[/tex] + (n -1)d and n = 33
Then
[tex]a_{33}[/tex] = [tex]a_{1}[/tex] + 32d,
making [tex]a_{1}[/tex] subject of the equation we have
[tex]a_{1}[/tex] =66 - 32 x 2
[tex]a_{1}[/tex] = 66 - 64
[tex]a_{1}[/tex] = 2
if n = 30, [tex]a_{1}[/tex] =-2, d = 2
Then
[tex]a_{30}[/tex] = 2 + ( 30 - 1) x 2
[tex]a_{30}[/tex] = 2 + 29 x 2
[tex]a_{30}[/tex] = 60
if [tex]s_{n}[/tex] = n/2([tex]a_{1}[/tex] + [tex]a_{n}[/tex])
Then,
[tex]s_{30}[/tex] = n/2 ( [tex]a_{1}[/tex] + [tex]a_{30}[/tex]) where n = 30
[tex]s_{30}[/tex] = 30/2( 2 + 60)
[tex]s_{30}[/tex] = 930
In conclusion, the sum of he first 30 terms is 930 and the first term is 2
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A straw is placed inside a rectangular box that is 4 inches by 2 inches by 10 inches, as
shown. If the straw fits exactly into the box diagonally from the bottom left corner to
the top right back corner, how long is the straw? Leave your answer in simplest
radical form.
10 in.
4 in.
2 in.
Answer:
The length of the straw in the simplest radical form is [tex]2\sqrt{30}[/tex] inches
Step-by-step explanation:
The length of the straw can be calculated as follow
The first step is to calculate the diagonal of the base of the rectangular box by using Pythagorean theorem
[tex]c^{2}=a^{2}+b^{2}[/tex]
Where [tex]c[/tex] is the base diagonal or the hypotenuse, [tex]a[/tex] is the short leg, [tex]b[/tex] is the long leg
Substitute [tex]2[/tex] for [tex]a[/tex], [tex]4[/tex] for [tex]b[/tex] in the Pythagorean theorem
[tex]c^{2}=2^{2}+4^{2}[/tex]
Simplify the right side of the equation
[tex]c^{2}=4+16[/tex]
[tex]c^{2}=20[/tex]
Take the square root of the right side to get the value of [tex]c[/tex]
[tex]c=\sqrt{20}[/tex]
Factor [tex]20[/tex] by [tex]4[/tex]
[tex]c=\sqrt{4\times 5}[/tex]
Substitute [tex]2^{2}[/tex] for 4
[tex]c=\sqrt{2^{2}\times 5}[/tex]
Rewrite the value of [tex]c[/tex] in the simplest radical form
[tex]c=2\sqrt{5}[/tex]
Since the value of [tex]c[/tex] is [tex]2\sqrt{5}[/tex], then the length of the diagonal of the base is
[tex]2\sqrt{5}[/tex] inches
The second step is to calculate the length of the straw by using Pythagorean theorem
[tex]c^{2}=a^{2}+b^{2}[/tex]
Where [tex]c[/tex] is the diagonal from the bottom left corner to the top right back corner or the hypotenuse, [tex]a[/tex] is the short leg, [tex]b[/tex] is the long leg
Substitute [tex]2\sqrt{5}[/tex] for [tex]a[/tex], [tex]10[/tex] for [tex]b[/tex] in the Pythagorean theorem
[tex]c^{2}=(2\sqrt{5})^{2}+10^{2}[/tex]
Simplify the right side of the equation
[tex]c^{2}=20+100[/tex]
[tex]c^{2}=120[/tex]
Take the square root of the right side to get the value of [tex]c[/tex]
[tex]c=\sqrt{120}[/tex]
Factor [tex]120[/tex] by [tex]4[/tex]
[tex]c=\sqrt{4\times 30}[/tex]
Substitute [tex]2^{2}[/tex] for 4
[tex]c=\sqrt{2^{2}\times 30}[/tex]
Rewrite the value of [tex]c[/tex] in the simplest radical form
[tex]c=2\sqrt{30}[/tex]
Since the value of [tex]c[/tex] is [tex]2\sqrt{30}[/tex], then the length of the diagonal from the bottom left corner to the top right back corner or the hypotenuse is
[tex]2\sqrt{30}[/tex] inches
Since the length of the diagonal from the bottom left corner to the top right back corner is [tex]2\sqrt{30}[/tex] inches, then the length of the straw is [tex]2\sqrt{30}[/tex] inches
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helppppppppppppppppppp
Answer:
f(-7) = 18
f(-3) = 14
f(4) = 7
Step-by-step explanation:
f(-7)
l-7-8l + 3
l-15l + 3
15 + 3
18
F(-3)
l-3-8l + 3
l-11l + 3
11 + 3
14
F(4)
l4-8l + 3
l-4l + 3
4 + 3
7
The graph shows the cube root parent function.
-5
5.
-5-
5
Which statement best describes the function?
A. The function is always increasing.
OB. The function is increasing when x < 0.
C. The function is increasing when x > 0.
D. The function is never increasing.
The graph of the function is increasing when x>0.
What is graph?
Graph a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.
What is cube root parent?
The basic parent cube root function is of the form f (x) = ∛x. It increases on the interval (-∞, ∞). It increases on the interval [0, ∞). It is positive on (0, ∞) and negative on (-∞, 0). It is a non-negative function always (on [0, ∞)). Its absolute min is 0 but no absolute max. Its domain and range is the set of all real numbers.
Given,
The function f(x) = ∛x
Let x = 1, we get f(x) = 1,
x =2, f(x) = 8
if x =3, f(x) = 27.
As we know, with increase in the value of x, the value of f(x) is also increasing.
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A line that includes the points (n, -9) and (5, 7) has a slope of 8. What is the value of n
n =
Submit
K!
Value of n is 3 when a line that includes the points (n, -9) and (5, 7) has a slope of 8.
Define slope.A line's steepness is indicated by its slope. Any line's slope is constant along its length. The direction of the line on the coordinate plane can also be determined by the slope. A line's graph can be used to determine slope, or the coordinates of any two points on a line can be used.
Given,
Slope = 8
Points = (n,-9) and (5,7)
Formula:
Slope = y₂ - y₁/ x₂ - x₁
8 = 7 -(-9)/ 5-n
Cross multiplying,
8(5 - n) = 7 +9
(5-n) = 16/8
5 -n = 2
n = 3
Value of n is 3 when a line that includes the points (n, -9) and (5, 7) has a slope of 8.
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write an equation of the passing through the point (4,-1) and perpendicular to line y=2x-5
Answer: Line perpendicular to y = 2x - 5 is y = (-1/2)x.
Step-by-step explanation:
f ( x) = 24x 3 + Ax 2 − 3x + B where A and B are constants. When f ( x) is divided by (2x − 1) the remainder is 30 (a) Show that A + 4B = 114
The function [tex]24x^{3}[/tex] + [tex]Ax^{2}[/tex] -3x + B results to the equation A + 4B = 114 hen f(x) is divided by (2x - 1) and with a remainder 30.
What is a remainder?The Remainder is the value left after the division. If a number (dividend) is not completely divisible by another number (divisor) then we are left with a value once the division is done. This value is called the remainder.
For example, 10 is not exactly divided by 3. Since the closest value, we can get 3 x 3 = 9.
Hence, 10 ÷ 3 → 3 R 1, where 3 is the quotient and 1 is the remainder.
if ( 2x - 1) divides [tex]24x^{3}[/tex] + [tex]Ax^{2}[/tex] -3x + B with a remainder 30, it means if x = 1/2 when substituted should give 30.
The substitute x = 1/2 into the given function and equate it to 30
24[tex](\frac{1}{2}) ^{3}[/tex] + A[tex](\frac{1}{2}) ^{2}[/tex] - 3(1/2) + B = 30
24[tex](\frac{1}{8})[/tex] + A[tex](\frac{1}{4})[/tex] - 3/2 + B = 30
3 + A/4 -3/2 + B = 30
multiply through by 4
12 + A -6 + 4B = 120
keeping A + 4B at the left we have
A + 4B = 120 -12 + 6
A + 4B = 114
Proved
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The solution to the system of equations x + y = 40 and x = 4y is:
(32, 8).
(29, 11).
(8, 32).
Answer:
(32, 8)
Step-by-step explanation:
Substitute 4y for x in x + y = 40 and solve for y. Then find x.
[tex]4y + y = 40[/tex]
[tex]5y = 40[/tex]
[tex]y = 8[/tex]
[tex]x = 4(8) = 32[/tex]
Four lengths have a mode of 12 km and a range of 5 km. One of the lengths is 14 km. What are the other three lengths?
The other three length are 9km,12km,14km as mode is 12km
What is mode?
The value that appears most frequently in a set of values is referred to as the mode. It is the value that shows up the most frequently. The value that consistently appears in a given set is known as the mode in statistics. The mode or modal value is the value or number that appears most frequently in a data set and has a high frequency. Along with mean and median, there are three other ways to measure central tendency.
The mean, median, and mode all have the same values for this distribution. This indicates that this value—the one that appears the most frequently in the data—is the average, the centre, and the mode.
Four lengths have a mode of 12 km and a range of 5 km.
The range is the largest number minus the smallest number.
As mode is 12 so one more number is 12 as well.
Now another number given 14km, so the last number is (14-5 = 9)
Hence the other three lengths are 9km,12km,14km.
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Find the value of x.
I really need help. Please help
Step-by-step explanation:
Using Triangle Intersection Theorem,
2(LM)=2+x
[tex]2(2x - 11) = 2 + x[/tex]
[tex]4x - 2 2 = 2 + x[/tex]
[tex]3x = 24[/tex]
[tex]x = 8[/tex]
Help me please!!!!!!
Answer:
the answer is A
An oil tanker ruptures at 12:00 AM and leaks oil at
the rate R(t) =[tex]\sqrt{t}[/tex] barrels of oil per minute where t is the
number of minutes past 12:00 AM. Specialized divers repair
the rupture in 20 minutes, completely stopping the leakage
at 12:20 AM. How many barrels of oil were leaked in total?
Answer:
R = 4.47
Step-by-step explanation:
Simply just R = sqrt(20)
In a class A of 25 students, 20 passed with 60% or more marks; in another class B of 30 students, 24 passed with 60% or more marks. In which class was a greater fraction of students getting with 60% or more marks?
Answer:
class A .................
Sat question solve
2^x=x^2
The logarithm function can be used to convert large problem of multiplication and division into simple addition or subtraction. The value of x for the given equation is 2.
What is Logarithm?Logarithm function is defined as the inverse of the exponential function.
If a^(b) = c, then b = log(c) at the base of a.
The given equation is as below,
2^x=x^2
Take logarithm both sides,
x log(2) = 2 log(x)
=> x / log(x) = 2 / log(2)
Compare the denominator and numerator at both sides of the equation to get,
x = 2.
Hence the solution of the given equation is x = 2.
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Use the recipe to find how much hair conditioner Helena
needs to make a half batch and a double batch?
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Explain how you could estimate 22% of 78 to check if your answer is reasonable.
17.16 is the estimated 22% of 78.
What is estimation?
To make calculations simpler and more realistic, estimation of a number refers to a plausible assumption of the actual value.
Estimation is the process of approximating a quantity with the necessary accuracy.
The result is quickly and roughly determined by rounding off the numbers used in the calculation.
Assumedly, 22% of 78.
78 comes out to 80.
Therefore, cut 22% to 1/5.
78 through 80 is increased.
80/5 = 16( close to 17.16) ( approximate to 17.16)
Or
22% = 22/100
22% of 78 = 22 × 78 /100 = 1716/100
= 17.16.
As a result, 17.16 is the estimated 22% of 78.
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Help find the slope of the line passing through the points??
Answer: [tex]\frac{11}{5}[/tex]
Step-by-step explanation
(y2-y1)/(x2-x1)
which means (5-(-6))/-4-(-9)=[tex]\frac{11}{5}[/tex]
Money is accepted as final payment for
O costs, benefits
O minting, exchange rates
goods, services
and
Money is anything generally accepted as final payment for goods, services, and debt.
First, let us understand the money supply:
The money supply is the total amount of currency and liquid assets in a country's economy on the day measured. Cash and deposits that can be utilized virtually as quickly as cash are roughly included in the money supply.
Money supply refers to the total amount of money held by the public for a particular point.
Governments print paper money and coins through a combination of central banks and treasuries. Bank regulators impact the money supply available to the public by requiring banks to retain reserves, determining how to grant credit, and other monetary matters.
We can say that we are using the money for goods purchasing, service requirements, and for the debt taken.
Thus, money is anything generally accepted as final payment for goods, services, and debt.
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there are 3 towns, ainsly broking and Cinderford
Check the picture below.
[tex]\textit{Law of Cosines}\\\\ c^2 = a^2+b^2-(2ab)\cos(C)\implies c = \sqrt{a^2+b^2-(2ab)\cos(C)} \\\\[-0.35em] ~\dotfill\\\\ b = \sqrt{7^2+11^2~-~2(7)(11)\cos(92^o)} \\\\\\ b = \sqrt{ 170 - 154 \cos(92^o) }\implies b\approx 13.24~km[/tex]
now, to find the bearing from C to A hmmm, let's use the Law of Cosines again, to find the angle at A, once we get that, we'll simply add 37 to it :)
[tex]\textit{Law of Cosines}\\\\ \cfrac{a^2+b^2-c^2}{2ab}=\cos(C)\implies \cos^{-1}\left(\cfrac{a^2+b^2-c^2}{2ab}\right)=\measuredangle C \\\\[-0.35em] ~\dotfill\\\\ \cos^{-1}\left(\cfrac{7^2+13.24^2-11^2}{2(7)(13)}\right)\approx\measuredangle A \implies \cos^{-1}\left(\cfrac{ 103.3 }{ 185.36}\right)\approx\measuredangle A \\\\\\ 56.13^o\approx \measuredangle A\hspace{15em} \underset{\textit{Bearing of C from A}}{\stackrel{56.13~~ + ~~37}{\boxed{\approx 93.13^o}}}[/tex]
In interval notation 6< x < 7 is written as
The interval notation of the inequality 6<x<7 is:
(6,7)
Given the inequality is:
6<x<7
the interval notation is:
(6,7)
The leftmost number of the set, a comma, and the rightmost number of the set are written using interval notation. Depending on whether those two integers are part of the set, we either place parentheses around the pair or square brackets around it (sometimes we use one parenthesis and one bracket!).
Hence we get the required interval notation.
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Meredith is decorating a picture frame to give to her grandmother the frame has an area of 80 in² and a length of 10" Meredith wants to glue a piece of ribbon around the edge of the frame how many inches ribbon does Meredith need
Answer:36
Step-by-step explanation:
A toy rocket is shot vertically into the air from a launching pad 7 feet above the ground with an initial velocity of 40 feet per second. The height h, in feet, of the rocket above the ground at t seconds after launch is given by the function h(t)=-16 t^2+40 t+7. How long will it take the rocket to reach its maximum height? What is the maximum height?
The rocket
Answer:
The maximum height of the rocket with respect to the ground can be obtained by evaluating the function h(t) for a time of 1.25 seconds. The rocket reaches a maximum height of 32 ft.
Find the arc length of the curve y =
1/2(e^x + e^-x) from x = 0 to x = 4.
The arc length of the given curve is L= 1/2([tex]e^{4} - \frac{1}{e^{4} }[/tex]).
What is arc length?
The distance between two places along a segment of a curve is known as the arc length.
Main Body:
So, we have [tex]y=\frac{1}{2} (e^{x} +e^{-x} )[/tex]
This is in the form y=f(x), so we know arc length L on [0,4] is given by,
L=[tex]\int\limits^4_0{\sqrt{1+(\frac{dy}{dx} })^{2} \, dx[/tex]
The derivative y=
dy/dx= [tex]\frac{1}{2} (e^{x} +e^{-x} )[/tex]= sinhx
([tex]\frac{dy}{dx}^{2}[/tex])= sinh²x
L=[tex]\int\limits^4_0{\sqrt{1+sinh^{2}x \, dx[/tex]
L= [tex]\int\limits^4_0{\sqrt{cosh^{2}x \, dx[/tex]
L=[tex]\int\limits^4_0{coshx \, dx[/tex]
Now putting the limits as given=
L= [tex]sinhx\left \{ {{y=4} \atop {x=0}} \right.[/tex]
L= [tex]\frac{1}{2} (e^{x} -e^{-x} )[/tex]|⁴₀
L=1/2([tex]e^{4} - \frac{1}{e^{4} }[/tex])
Hence the answer is 1/2( [tex]e^{4} - \frac{1}{e^{4} }[/tex]).
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use implicit differentiation to find y" in terms of x and y.
Using implicit differentiation, the value of y in terms of x and y is; y = 0.5/(dy/dx)
How to solve implicit differentiation?In implicit differentiation, what we do is that we differentiate each side of an equation using the two variables (usually x and y) by treating one of the variables to be as a function of the other.
We are given the equation;
4x = 4y² + 1
Thus, first differentiation gives;
4 = 8y(dy/dx)
y(dy/dx) = 0.5
y = 0.5/(dy/dx)
This is the result when using implicit differentiation on the given function where y is the subject of the formula.
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A number y added to 2 is greater than y
Answer:
y+2>y
Step-by-step explanation:
Well, first of all. Adding any sum to y makes it greater then the same variable.
Greater than sign is >. Keeping that in mind now add them. y+2>y is gonna be you’re final answer!
{REPOST: FORGOT TO INCLUDE PICTURE FOR QUESTION 10}
{Question 10} Please help, need this by tonight! Thanks so much :) {Click on picture}
Answer:
X-intercept: (6, 0)
Y-intercept: (0, -9)
Step-by-step explanation:
Required: Find the X and Y intercept of the standard from equation:
-3x + 2y = -18.
We shall write equation in the form [tex]\frac{x}{a} +\frac{y}{b} =1[/tex] and find the intercepts on X and Y axis:
-3x + 2y = -18
∴ -3x + 2y = -18
∴ -3x/-18 + 2y/-18 = 1
∴ x/6 + y/-9 = 1
Comparing this equation with [tex]\frac{x}{a} +\frac{y}{b} =1[/tex], we get:
intercept of X-axis = a = 6
intercept of Y-axis = b = -9
Thanks.
Answer:
The y-intercept for the line is ( 0, -9 ).
Hope this helps!
Step-by-step explanation:
The y-intercept is where the graph intercepts the y-axis and is the y-value of the point, ( x, y ).
If EF measures 12 meters the approximate area of circle b
The EF is the diameter. Then the area of the circle is 113.04 square meters.
circle :
A circle is a circular shape without corners or line segments. In terms of geometry, it has the shape of a closed curve.
EF = diameter
EF = 12 meter
then, radius = d/2 = 12/2 = 6 meter
Area of the circle is calculated as:
area = πr²
area = 3.14 x 6 x 6
area = 113.04 square meter
The given que is incomplete . The correct question is given as :
The EF is the diameter. If EF measures 12 meters, the approximate area of circle B is square meters.
The EF is the diameter. Then the area of the circle is 113.04 square meters.
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Suppose that the number of bacteria in a certain population increases according to a continuous exponential growth model. A sample of 2400 bacteria selected from this population reached the size of 2892 bacteria in four hours. Find the hourly growth rate parameter.
Answer: [tex]\frac{\ln(241/200)}{4}[/tex]
Step-by-step explanation:
If [tex]P(t)[/tex] represents the number of bacteria after [tex]t[/tex] hours, then [tex]P(t)=2400e^{kt}[/tex], where [tex]k[/tex] is the hourly growth rate parameter.
Using the fact that [tex]P(4)=2892[/tex],
[tex]2892=2400e^{4k}\\\\e^{4k}=\frac{241}{200}\\\\4k=\ln(241/200)\\\\k=\frac{\ln(241/200)}{4}[/tex]
The following scatterplot shows two variables, x and y, along with a least-squares model.
Which of the following is a high leverage point with respect to the regression?
A (5,8)(5,8)
B (20,31)(20,31)
C (27,22)(27,22)
D (30,60)(30,60)
E (80,70)
Answer:
D(30,60)
Step-by-step explanation:
It was way outside the other points that are around the line.
The point (30,60) is a high leverage point on the regression plot.
What is High leverage points?High leverage points are those that are extreme but follow the regression equation's trend.
High leverage points are distinct from outliers, which deviate from the graph's or plot's pattern or trend.
Looking closely at the regression plot, the coordinate (30,60) follows the trend of the plot, however, it is farther from the majority of the points on the graph.
Hence, the point (30,60) is a high leverage point on the regression plot.
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