a) 90% confident that the true population mean μ is between 91.01 and 102.99.
b) 90% confident that the true population mean μ is between 92.26 and 101.74.
c) 80% confident that the true population mean μ is between 93.79 and 100.21.
d) Using Central Limit Theorem to approximate the population mean as normally distributed, the confidence intervals can still be computed using the same formula as before.
(a) To construct a 90% confidence interval about μ when n = 27, we use the formula:
x ± z × (s/√n)
where x is the sample mean, s is the sample standard deviation, n is the sample size, and z is the z-score that corresponds to the desired confidence level (in this case, z = 1.645 for a 90% confidence level). Plugging in the values given, we get:
97 ± 1.645 × (13/√27)
which simplifies to:
(91.01, 102.99)
Therefore, we are 90% confident that the true population mean μ is between 91.01 and 102.99.
(b) To construct a 90% confidence interval about μ when n = 18, we use the same formula as before, but with a different z-score (since the sample size is smaller). In this case, we use z = 1.734, and we get:
97 ± 1.734 × (13/√18)
which simplifies to:
(92.26, 101.74)
Therefore, we are 90% confident that the true population mean μ is between 92.26 and 101.74.
(c) To construct an 80% confidence interval about μ when n = 27, we use the same formula as before, but with a different z-score (since the confidence level is different). In this case, we use z = 1.282, and we get:
97 ± 1.282 × (13/√27)
which simplifies to:
(93.79, 100.21)
Therefore, we are 80% confident that the true population mean μ is between 93.79 and 100.21.
(d) The confidence intervals in parts (a)-(c) assume that the population is normally distributed. If the population is not normally distributed, we cannot rely on these intervals to accurately estimate the true population mean. However, if the sample size is sufficiently large (usually n > 30), we can use the Central Limit Theorem to approximate the population mean as normally distributed, and the confidence intervals can still be computed using the same formula as before. If the sample size is small and the population is not normally distributed, other methods may be needed to construct a confidence interval, such as bootstrapping or using a non-parametric approach.
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Do any ingredients make up an equal amount of the smoothie? Construct a math argument to explain why or why not
Without knowing the specific ingredients and proportions used in a smoothie, it is impossible to definitively say whether any ingredients make up an equal amount. However, it is likely that the proportions of ingredients in a smoothie vary based on the desired taste, texture, and nutritional content.
For example, a smoothie may have more fruit than liquid to create a thicker consistency, or more liquid than fruit to create a thinner consistency. Additionally, the amount of sweeteners, such as sugar or honey, may vary based on personal preference and the natural sweetness of the fruit used.
In conclusion, without more specific information about the ingredients and proportions used in a particular smoothie, it is not possible to determine whether any ingredients make up an equal amount. The proportions of ingredients in a smoothie are likely to vary based on various factors and personal preference.
Use the drop-down menus to complete each equation so the statement about its solution is true.
The illustration of the types of solutions of systems of equations is explained below.
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, 5 - 4 + 7x + 1 = ax + b.
7x + 2 = ax = b.
Now, If we put b ≠ 2 and x = 7 we have,
7x + 2 = 7x + b.
2 = b, but b ≠ 2 so no solution.
5 - 4 + 7x + 1 = ax + b
7x + 2 = ax + b.
To have one solution we can assume any x value except 7 and any b value except 2,
7x + 2 = 6x + 3.
x = - 1.
5 - 4 + 7x + 1 = ax + b
To have infinitely many solutions both sides must be some constant multiple of each other.
7x + 2 = k(ax + b).
7x + 2 = 2(7x + 2).
7x + 2 = 14x + 4.
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A triangle has angles that measure 60°, 70°, and z. What is z?
Answer:
z = 50°
Step-by-step explanation:
We know that,
→ Sum of all sides of triangle is 180°.
Formula we use,
→ <A + <B + <C = 180°
Forming the equation,
→ 60° + 70° + z = 180°
Now the value of z will be,
→ 60° + 70° + z = 180°
→ 130° + z = 180°
→ z = 180° - 130°
→ [ z = 50° ]
Hence, the value of z is 50°.
What was the first step in solving this equation?
Original equation:
9(2p+ 1) = 6p + 15
Fill in the missing work:
Result of the first step:
>
3(2p + 1) = 2p + 5
Write a few sentences describing the missing work.
The missing step is 9(2p+ 1)/3 = 6p/3 + 15/3.
What is division?
The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
Given:
An original equation,
9(2p+ 1) = 6p + 15.
Result of the first step:
3(2p + 1) = 2p + 5
To find the missing step:
Divide the original equation by 3 into both sides,
we get,
9(2p+ 1)/3 = 6p/3 + 15/3.
Simplifying further,
3(2p + 1) = 2p + 5
Therefore, the required expression is 9(2p+ 1)/3 = 6p/3 + 15/3.
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determine if the numerical value describes a population parameter or a sample statistic. 79% 79 % of all students at the local university voted in the last election.
"79 % of all students at the local university voted in the last election" the numerical value describes a population parameter.
What is Population parameterPopulation Parameter is a quantity/measurement that can be used as material for follow-up on population management activities.
About populationPopulation is a generalized area consisting of: objects/subjects that have certain qualities and characteristics determined by the researcher to be studied and then drawn conclusions.
The population or universe is the total number of units or individuals whose characteristics are to be studied. And these units are called units of analysis, and can be people, institutions, things, etc.
finite population : population whose number is known with certainty. for example the number of X students. infinite population : population whose number is not known with certainty. for example the number of people in the world. a group of objects that continues to grow (carries out a process due to life or an event process) is an infinite population.Learn more about population parameters at
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A parachutist whose mass is 100 kg drops from a helicopter hovering 3000 m above the ground and falls under the influence of gravity. Assume that the force due to air resistance is proportional to the velocity of the parachutist, with proportionality constant b3 = 20 N-sec/m when the chute is closed and b4 = 100 N-sec/m when the chute is open. If the chute does not open until 30 sec after the parachutist leaves the helicopter, after how many seconds will he hit the ground? If the chute does not open until 1 min after he leaves the helicopter, after how many seconds will he hit the ground?
If the chute does not open until 1 min after he leaves the helicopter, then the equation to calculate the seconds it will he hit the ground is (1/2)mv² + ∫ F(v) dv = 0
The force due to air resistance is assumed to be proportional to the velocity of the parachutist, and we will use this information to determine the time it takes for the parachutist to reach the ground with and without a chute.
The force of air resistance is proportional to the velocity of the parachutist. This means that the larger the velocity, the larger the force of air resistance.
As the velocity increases, the force of air resistance also increases, which reduces the kinetic energy. When the chute opens, the force of air resistance increases even more, which further reduces the kinetic energy and slows down the parachutist.
To determine the time it takes for the parachutist to hit the ground, we can use the following equation:
PE + KE + work done by air resistance = PE + KE
where PE is the gravitational potential energy, KE is the kinetic energy, and work done by air resistance is the work done by the force of air resistance.
When the chute is closed, we can write:
mgh = (1/2)mv² + ∫ F(v) dv
where m is the mass of the parachutist, g is the acceleration due to gravity, h is the height of the helicopter above the ground, and the integral is taken over the velocity range from 0 to v.
Similarly, when the chute is open, we can write:
mgh = (1/2)mv² + ∫ F(v) dv
where b = b4 and the integral is taken over the velocity range from 0 to v.
To solve for the time it takes for the parachutist to hit the ground, we need to determine the velocity at which he hits the ground. This is the velocity where KE = 0, and we can find it by equating the expressions for KE and work done by air resistance when the parachutist hits the ground:
(1/2)mv² + ∫ F(v) dv = 0
We can solve these equations numerically to find the time it takes for the parachutist to hit the ground with and without the chute.
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You start driving west for 3 miles, turn right, and drive north for
another 13 miles. At the end of driving, what is your straight line
distance from your starting point? Round to the nearest tenth of a
mile.
Answer:
13.3 miles
Step-by-step explanation:
Picture a right triangle with legs of 3 and 13. The hypotenuse is the straight line distance back to the starting point. Use the Pythagorean theorem to find the hypotenuse (c):
c² = 13² + 3² = 178
c = √178 = 13.3 miles
How do you find the volume of the parallelepiped with adjacent edges pq, pr, and ps where p(3,0,1), q(-1,2,5), r(5,1,-1) and s(0,4,2)?
The volume of the parallelepiped is 190.
Parallelepiped VolumeThe volume V of a parallelepiped with adjacent edges pq, pr, and ps can be found using the scalar triple product as:
V = |pq · (pr x ps)|
where · denotes the dot product and x denotes the cross product.
First, we need to find the vectors pq, pr, and ps:pq = q - p = (-1-3, 2-0, 5-1) = (-4, 2, 4)
pr = r - p = (5-3, 1-0, -1-1) = (2, 1, -2)
ps = s - p = (0-3, 4-0, 2-1) = (-3, 4, 1)
Next, we need to find the cross product of pr and ps:pr x ps = det([i, j, k; 2, 1, -2; -3, 4, 1]) = i(-4) - j(7) + k(10) = (-4, -7, 10)
Finally, we can find the volume of the parallelepiped:V = |(-4, 2, 4) · (-4, -7, 10)| = 190
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A) In the sequence for which the general term is t_(n)=(n^(2)-1)/(3n^(2)),
find the value of n, if any, for which t_(n)=0.3325.
B) Find S_(n) for the arithmetic series 16+13+10+cdots and determine the value of n for which S_(n)=-6.
S_(n)=. The value of n for which S_(n)=-6 is .
we can conclude that the value of n for which [tex]t_n[/tex] = 0.3325 is between 3 and 4 and the value of n for which [tex]S_n[/tex]= -6 is 6.
How to find value of n and sum of sequence?A) To find the value of n for which [tex]t_n[/tex] = 0.3325, we can use trial and error method.
Starting with n=1, we have:
[tex]t_1 = (1^2 - 1)/(3 * 1^2) = 0[/tex]
For n=2, we have:
[tex]t_2 = (2^2 - 1)/(3 * 2^2) = (3)/(12) = 0.25[/tex]
For n=3, we have:
[tex]t_3 = (3^2 - 1)/(3 * 3^2) = (8)/(27) = 0.296296...[/tex]
Continuing in this way, we find that t_3 < 0.3325 < t_4, where
[tex]t_4 = (4^2 - 1)/(3 * 4^2) = (15)/(48) = 0.3125[/tex]
Thus, we can conclude that the value of n for which [tex]t_n[/tex] = 0.3325 is between 3 and 4, and it can be found more accurately by using numerical methods such as bisection or Newton's method.
B) To find the sum of the arithmetic series 16 + 13 + 10 + ..., we can use the formula for the sum of an arithmetic series:
[tex]S_n = n/2 * (a_1 + a_n),[/tex]
where n is the number of terms in the series, [tex]a_1[/tex] is the first term (16 in this case), and a_n is the nth term.
Let's assume that the nth term is represented by [tex]a_n = 16 - 3 * (n - 1),[/tex] so that a_1 = 16 and a_n = 16 - 3 * (n - 1).
Then, we have:
S_n = n/2 * (16 + 16 - 3 * (n - 1)) = n/2 * (16 + 16 - 3n + 3) = n/2 * (16 + 16 - 3n + 3) = n/2 * (32 - 3n + 3) = 16n - (3/2)n^2 + (3/2)n
We want to find the value of n for which S_n = -6, so we can substitute the value of S_n into the equation and solve for n:
[tex]S_n = n/2 * (16 + 16 - 3 * (n - 1)) = n/2 * (16 + 16 - 3n + 3) = n/2 * (16 + 16 - 3n + 3) = n/2 * (32 - 3n + 3) = 16n - (3/2)n^2 + (3/2)n[/tex]
[tex]-6 = 16n - (3/2)n^2 + (3/2)n[/tex]
Expanding the equation and solving for n, we find that n = 6.
Thus, the value of n for which [tex]S_n[/tex]= -6 is 6.
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Find the radius of a sphere with a volume of 26, 244π cubic inches.
27 inches is the radius of a sphere with a volume of 26, 244π cubic inches.
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
Given that volume of a sphere is 26, 244π cubic inches.
We need to find the radius of sphere.
V=4/3πr³
26, 244π=4/3πr³
26, 244=1.333r³
Divide both sides by 1.333
26,244/1.333=r³
19688=r³
Cube root on both sides
∛19688 = r
27=r
Hence, 27 inches is the radius of a sphere with a volume of 26, 244π cubic inches.
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Cost of headphones: $20.00 Tax: 6% What is the new cost?
Answer:
Step-by-step explanation:
e = 20 * (1+6/100) = 20 * 53 50 = 20 * 1.06 =$ 21.2
A family wants to rent a car to go on vacation. Company A charges $55.50 and 16 cents per mile. Company B charges $50.50 and 9 cents per mile. How much more does Company A charge for x miles than Company B?
Company A will charge $7 more than company B.
What is an expression?Expression in maths is defined as the relation of numbers variables and functions by using mathematical signs like addition, subtraction, multiplication and division.
Given that a family wants to rent a car to go on vacation. Company A charges $55.50 and 16 cents per mile. Company B charges $50.50 and 9 cents per mile.
The expression for the charges by company A is,
C = 16x + 55.50
The expression for the charges by company B,
C = 9x + 55.50
The amount of company A more than company B is calculated as:-
Amount = ( 16 + 55.50) - ( 9 + 55.50 )
Amount = 71.50 - 64.50
Amount = $7
Hence, company A will charge $7 more.
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write an equation of the line passing through the given points. give the final answer in standard form. (1/2,-4) and (3/4,-40
The equation of line in standard form is y = -146x + 69.
A line equation is an algebraic way of representing a set of points that together form a line in a coordinate system. The various points that together form a line of axes can be represented as a set of (x,y) variables and form an algebraic equation, also called the line equation. You can use the straight line equation to determine if a given point lies on a straight line.
Each row of equations is a linear equation of degree 1. Read the entire article to learn about the different forms of linear equations and how to determine them. A line segment can be defined as a connection between two points. Any two points in a 2D geometry can be connected by a line segment or a simple straight line. The equation of a straight line can be obtained in three ways:
slope intercept method
point slope method
Standard method
Given two points lying on a given line, we usually follow the point-gradient method.
The equation for a straight line is y−y1=m(x−x1)
where y1 is the Y-axis coordinate, m is the tilt, x1 Coordinates on the x-axis. m = y2-y1/x2-x1
eqn = y +4 = (-36/1/4)(x-1/2)
y +4 = -146x + 73
y = -146x + 69
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The rate of change of the temperature during the hour and a half after the beginning of a thunderstorm is given by T(h) = 9.43 h^3 − 15.41h^2 + 17.34h − 9.83T(h) = 9.43h^3- 15.41h^2 + 17.34h − 9.83where output is measured in per hour and h is the number of hours since the storm began.(a) Evaluate ∫1.5 T(h)dh. ∫0 Tb(h)dh0 1.5(Round your answer to three decimal places.)(b) Interpret the answer to part (a)
∫1.5 T(h) dh = 9.573, rounded to three decimal places. (a) To evaluate ∫1.5 T(h) dh, we need to integrate T(h) with respect to h from 0 to 1.5: ∫1.5 T(h) dh = ∫0.1.5 (9.43h^3 - 15.41h^2 + 17.34h - 9.83) dh
Using the power rule of integration, we can integrate each term of the polynomial: ∫0.1.5 (9.43h^3 - 15.41h^2 + 17.34h - 9.83) dh
= (9.43/4)h^4 - (15.41/3)h^3 + (17.34/2)h^2 - 9.83h |0.1.5
= [(9.43/4)(1.5)^4 - (15.41/3)(1.5)^3 + (17.34/2)(1.5)^2 - 9.83(1.5)] - [(9.43/4)(0)^4 - (15.41/3)(0)^3 + (17.34/2)(0)^2 - 9.83(0)] = 9.573
Therefore, ∫1.5 T(h) dh = 9.573, rounded to three decimal places.
(b) The answer to part (a) represents the total change in temperature during the hour and a half after the beginning of the thunderstorm. This means that the temperature increased by an average of 9.573 degrees per hour during this time period.
This integral gives us a measure of the total effect of the thunderstorm on the temperature, which is useful for predicting the weather and understanding the behavior of natural phenomena.
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NO LINKS!! URGENT HELP PLEASE!!
For #4-6, find the area of each figure, round your answer to one decimal point if necessary.
Problem 4
Answer: 2400 square inchesExplanation:
Draw a horizontal divider line to split the figure into two rectangles.
The rectangle up top is 30 by 30, so it has area 30*30 = 900 square inches.
The rectangle on the bottom has area 50*30 = 1500 square inches.
total = 900+1500 = 2400 square inches
========================================================
Problem 5
Answer: 12 square inchesExplanation:
Draw a horizontal divider line to split the figure into two rectangles.
A = area of rectangle on top = 2*2 = 4 square inches
B = area of rectangle on bottom = 4*2 = 8 square inches
C = total = A+B = 4+8 = 12 square inches
========================================================
Problem 6
Answer: 2592 square yardsExplanation:
Draw a horizontal divider line to get two rectangles.
A = area of rectangle on top = 36*36 = 1296 square yards
B = area of rectangle on bottom = 72*18 = 1296 square yards
C = total = A+B = 1296+1296 = 2592 square yards.
You have the correct answer. Nice work.
Answer:
4) 2400 in²
5) 12 in²
6) 2592 yd²
Step-by-step explanation:
To calculate the area of each given composite figure, divide the figure into two rectangles and sum the area of the two rectangles.
[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]
Question 4Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 30\cdot 30+50\cdot 30\\&=900+1500\\&=2400\; \sf in^2\end{aligned}[/tex]
Question 5Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&=2 \cdot 2+ 4\cdot2 \\&=4+8\\&=12\; \sf in^2\end{aligned}[/tex]
Question 6Separate the figure into two rectangles by drawing a horizontal line (see attachment).
[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&=36 \cdot36 +72 \cdot18 \\&=1296+1296\\&=2592 \; \sf yd^2\end{aligned}[/tex]
Please helpppppppppppppppp
24% of attempts hit the bull's eye
of attempts hit the bull's eye
25
0.26 of attempts hit the bull's eye
1 out of every 5 attempts hit the bull's eye
00
Which contestant won the competition?
2 of 10 <
1 2 3
4
Four friends entered an archery contest at summer camp. The winner is the contestant who hits the bull's eye the most frequently. Each contestant takes the same number of attempts. The
results are in the table.
5
6 7 8
9
Answer:
Robin.
Step-by-step explanation:
Clinton scored 24% of his attempts which means 24/100.
Then Katniss scored 6/25 of his attempts which means 24/100.
After that Robin scored 0.26 of his attempts which means 26/100.
Finally William scored 1/5 of his attempts which means 20/100.
Therefore, Robin has the highest score and wins the competition.
The average of four numbers is 10.three of the numbers are 17, 13 and 6.find the fourth numberthe average of four numbers is 10 three of the numbers are 1713 and 6 find the fourth number
The fourth number is equal to -6.
How to calculate the mean?Mathematically, the arithmetic average for a given data set can be calculated by using this formula:
Average = [F(x)]/n
Next, we would calculate the arithmetic average for the four numbers (data set) as follows:
Let the fourth number be represented by the variable n.
Total numbers = 17 + 13 + 16 + n
Total numbers = 46 + n
10 = 46 + n/4
40 = 46 + n
n = 40 - 46
n = -6
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What is 4 to the 4th power
Answer:
4 to the 4th power is 256
Explanation:
4 to the 4th power is a mathematical expression indicating that the number 4 is multiplied by itself four times. Mathematically, it is represented as 4^4.
To calculate 4^4, you would multiply 4 by itself four times: 4 x 4 x 4 x 4.
The result of this multiplication is 256, which means that 4 to the 4th power is equal to 256.
The number 4 to the 4th power is equal to 256.
When you raise a number to a power, it means multiplying that number by itself the specified number of times.
In the case of 4⁴, you are multiplying 4 by itself four times:
4⁴ = 4 × 4 × 4 × 4
When you perform the multiplication, you get:
4 × 4 = 16
16 × 4 = 64
64 × 4 = 256
Therefore, 4 to the 4th power is equal to 256.
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Complete the table:
0
Cos(8)
a. 1
b. -1
?
90
10
C. 2
d.
0
a. 1
b. -1
?
90
10
c. 0
d. -2
Let g be the function given by g(x)=limh→0sin(x+h)−sinx/h. What is the instantaneous rate of change of g with respect to x at x=π3?
Answer:
The function g(x) is the derivative of the function sin(x). So, g(x) = cos(x).
Therefore, the instantaneous rate of change of g with respect to x at x = π/3 is cos(π/3), which is approximately equal to 0.5.
Step-by-step explanation:
np and pls mark brainlist:)
find the value of x 2×+× +x+2×+3×+30+(2×-10)
Answer:
= 11x + 20.
Step-by-step explanation:
2x + x + x+2x+3x+30+(2x-10)
= 2x+x+x+2x+3x+30+2x-10
=2x + x + x + 2x + 3x + 2x + 30 - 10
=11x + 20.
You need to rent a moving truck for a day. You have identified two companies that rent trucks. Company A charges $45 per day plus $0.10 per mile. Company B charges $25 per day plus $0.30 per mile. For how many miles will the cost of renting be the same?
Answer:
the cost of renting a truck from both companies will be the same for 100 miles.
Step-by-step explanation:
Let's assume that the cost of renting a truck from Company A for x miles is C_A(x) and the cost of renting a truck from Company B for x miles is C_B(x).
From the given information, we know that:
C_A(x) = 45 + 0.10x
C_B(x) = 25 + 0.30x
We need to find the value of x for which C_A(x) = C_B(x). So we can set the two equations equal to each other and solve for x:
45 + 0.10x = 25 + 0.30x
20 = 0.20x
x = 100
Therefore, the cost of renting a truck from both companies will be the same for 100 miles.
find two numbers whose product is 15 and sum is 16
Answer:
Step-by-step explanation:
l
Answer:
15+1
Step-by-step explanation:
Determine if //,
I
y+3=
-
F
or neither.
4
(x − 1)
y=-x-1
Select the correct response:
o Perpendicular
Parallel
O Neither
If y+3=-4/3(x-1) and y=-4/3x-1 are two lines then these are parallel.
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
The two equations are
y+3=-4/3(x-1)
y=-4/3x-1
When we compare both the equations to slope intercept form.
we get slope -4/3 for both the equations.
If the lines have same slope then the lines are parallel.
Hence, the two lines are parallel lines.
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Determine where f is continuous. If possible, extend f as in example 4.2 to a new function that is continuous on a larger domain.F(x) = {sin x/x if x ≠ 0; 1 if x= 0Continuous everywhere since limx→0 sinx/x=1,and f(0) = 1.
This function is continuous everywhere, as the limit of sinx/x as x approaches 0 is 1, and f(0) = 1. It cannot be extended to a larger domain, as it is already continuous everywhere.
For example, when x = 0.1, f(x) = 0.998334166, and when x = 0.001, f(x) = 0.998999667.This function is continuous everywhere, as the limit of sinx/x as x approaches 0 is 1, and f(0) = 1. This means that there are no breaks or discontinuities in the function, and it is continuous everywhere. To demonstrate this, we can look at specific values of x. For example, when x = 0.1, f(x) = 0.998334166, and when x = 0.001, f(x) = 0.998999667. This shows that the function is smoothly transitioning from one value to the next, and is therefore continuous everywhere. The function cannot be extended to a larger domain, as it is already continuous everywhere. This means that the function is defined everywhere on its domain, and there is no need to extend it.
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Find all other zeros of P(x)=x^3-6x^2+18x-40 , given that 1+3i is a zero.
Answer:
1 - 3i, 4
Step-by-step explanation:
The polynomial has real coefficients, so if it has a complex root, it must have at least two complex roots which are complex conjugates.
Since 1 + 3i is a root, then 1 - 3i is also a root.
Two of the factors of the polynomial are:
x - (1 + 3i) and x - (1 - 3i)
Simplify the factors above:
x - 1 - 3i and x - 1 + 3i
Find their product:
(x - 1 - 3i)(x - 1 + 3i) =
Rewrite them showing we have the product of a sum and a difference.
= [(x - 1) - 3i][(x - 1) + 3i]
Multiply the factors above noticing they are a sum and a difference which follows the pattern (a + b)(a - b) = a² - b².
= (x - 1)² - (3i)²
= x² - 2x + 1 - 9(-1)
= x² - 2x + 10
Now we divide the original polynomial by the product we just found using long division.
x - 4
------------------------------
x² - 2x + 10 | x³ - 6x² + 18x - 40
- x³ - 2x² + 10x
------------------------
-4x² + 8x - 40
- -4x² + 8x - 40
--------------------------
0 + 0 + 0
Now we know that x³ - 6x² + 18x - 40 factors into (x² - 2x + 10)(x - 4).
(x² - 2x + 10)(x - 4) = 0
From x² - 2x + 10, we have roots x = 1 + 3i and x = 1 - 3i.
x - 4 = 0
x = 4
From x - 4, we have root x = 4.
Answer: 1 - 3i, 4
Carmen planted some corn in the back yard and measures its height regularly. The last time she checked, the stalks were 52.6 inches tall. Now they are the same height. What is the percent of increase in the height of the stalks?
The percent of increase in the height of the stalks is given as follows:
0%.
How to obtain the percent of increase?The percent of increase is obtained applying the proportions in the context of this problem.
A proportion is applied because the percent increase is given by the division of the increase by the initial amount, and multiplied by 100%.
The parameters for this problem are given as follows:
Change of 0, as the plant is of the same height as the initial height.Initial height of 52.6 inches.Hence the percent increase is given as follows:
0/52.6 x 100% = 0%.
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3(6-4)=-3
How do I show work for this
The expression is not true.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ 3 (6 - 4) = - 3
Now, We can check as;
⇒ 3 (6 - 4) = - 3
LHS;
⇒ 3 (6 - 4)
⇒ 3 × 2
⇒ 6
RHS;
⇒ - 3
Hence, The expression is not true.
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What is the slope of the line passing through the points (1, -3) and (1, 0)?
Enter your answer as a number, like this: 42
Or, if the slope is undefined, enter a lowercase letter "u," like this: u
u is the slope of the line passing through the points (1, -3) and (1, 0).
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
We have to find the slope of the line passing through the points (1, -3) and (1, 0)
m=0-(-3)/1-1
=3/0
Undefined
Hence, the slope of the line passing through the points (1, -3) and (1, 0) is u.
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