The cubic equation cannot have more than one solution whenever α > 0.
Rolle's theorem states that if a function is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in the open interval (a, b) such that the derivative f'(c) = 0.
Now, let's consider the cubic equation x^3 + αx^2 + β = 0. To apply Rolle's theorem, we need to show that this equation satisfies the conditions mentioned above.
Since the cubic equation is a polynomial, it is continuous and differentiable for all real numbers. Now, let's differentiate the equation with respect to x:
f'(x) = 3x^2 + 2αx
For Rolle's theorem to hold, we need f'(x) = 0. Solving this equation for x:
3x^2 + 2αx = 0
x(3x + 2α) = 0
This equation has two solutions: x = 0 and x = -2α/3. Since α > 0, x = -2α/3 is a distinct real number different from 0. Thus, we have two distinct points where the derivative is zero.
However, Rolle's theorem states that there can only be one such point if there's only one solution to the cubic equation. Since we found two points where the derivative is zero, it implies that the cubic equation cannot have more than one solution whenever α > 0.
learn more about the application of Rolle's theorem at: https://brainly.com/question/13972986
#SPJ11
please help! Which equation could you use to find the original amount of ounces of rice in a box, when the current box holds 85 ounces, 25% less rice.
The equation which is use to find the original amount of ounces of rice in a box, when the current box holds 85 ounces, 25% less rice is
y = 0.25*85.
What is percentage?
A number or ratio expressed as a fraction of 100 is known as a percentage in mathematics. Though the abbreviations "pct.", "pct.", and occasionally "pc" are also used, the percent sign, "%," is frequently used to indicate it. An amount that is a percentage is dimensionless and has no assigned measurement.
Given:
The current box holds 85 ounces, 25% less rice.
We have to find the equation which is use to find the original amount of ounces of rice in a box, when the current box holds 85 ounces, 25% less rice.
Let, y be the original amount of ounces of rice in a box.
The current box holds 85 ounces.
The 25% less than 85 ounces are 85*0.25.
So, the equation becomes y = 0.25*85.
Hence, the equation which is use to find the original amount of ounces of rice in a box, when the current box holds 85 ounces, 25% less rice is
y = 0.25*85.
To know more about percentage, click on the link
https://brainly.com/question/24877689
#SPJ1
What is dilation in math example?
The meaning of Dilation is changing the size of an object without changing its shape. The size of the object may be decreased or increased based on the scale factor.
One example of dilation math is a square of side 5 units can be dilated to a square of side 25 units, but the shape of the square remains the same.
Dilation is the method of resizing or transforming an object. It is a transformation that makes the objects larger or smaller with the help of the given scale factor.
The new figure get after dilation is called the image and the real image is called the pre-image. Dilation can be of two types:
1. Expansion - It happens when the size of an object is increased.
2. Contraction - it happens when the size of an object is decreased.
To know more about Dilation:
https://brainly.com/question/29811168
#SPJ4
How do I solve (3.4) (6.7) using standard algorithm?
To solve the equation (3.4)(6.7) using the standard algorithm, first multiply 3.4 and 6.7. Then, the result is 22.78.
How to solve for the standard algorithm?The standard algorithm for multiplication is a step-by-step process that involves breaking down the numbers being multiplied into simpler parts, and then multiplying those parts together.
First, you would need to multiply the ones place of the first number (4) by the ones place of the second number (7), which would give you 28.
Next, you would need to multiply the ones place of the first number (4) by the tens place of the second number (6), which would give you 24.
Then, you would add the two products from steps 1 and 2 to get the result: 28 + 24 = 52.
Alternatively, you can use the distributive property of multiplication to simplify the problem.
(3.4)(6.7) = 3.46 + 3.40.7 = 22.8
Read more about standard algorithm here:
https://brainly.com/question/25232442
#SPJ1
TEST RN What is the equation of the line that passes through the point (6.0) and had a slope of -5/6? HELPP PLEASE TEST RN
An equation of the line that passes through the point (6, 0) and had a slope of -5/6 is y = -5x/6 + 5.
What is the point-slope form?Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁)
Where:
m represents the slope.x and y are the points.Given the following parameters:
Point (x, y) = (6, 0)Slope = -5/6Substituting the given parameters into the point-slope formula, the equation of the line is given by;
y - y₁ = m(x - x₁)
y - 0 = -5/6(x - 6)
y = -5x/6 + 5
Read more on slope here: brainly.com/question/28427398
#SPJ1
The table shows the ingredients, in cup, that Renee used in her smoothie
Mixed berries
2 1/2
Yogurt
1 1/3
Milk
1 1/4
Cómo resolver esta factorizacion
Therefore , the solution of the given problem of factorization comes out to be x=0 and x=1/3.
What is factorization, exactly?The breaking or decomposition of an entity (such as a number, a matrix, or a polynomial) into a product of another entity, or factors, whose multiplication results in the original number, matrix, etc., is known as factorization or factoring in mathematics.
Here,
Given:
thus . We do factorization of given expression and
we come on a result
=> 3x - 9x²
=> 3x (1 -3x)
x=0 and x=1/3
Therefore , the solution of the given problem of factorization comes out to be x=0 and x=1/3.
To know more about factorization visit:-
brainly.com/question/29128446
#SPJ1
An office building has two elevators. One elevator starts out on the 4th floor, 35 feet above the ground, and is descending at a rate of 2.2 feet per second. The other elevator starts out at ground level and rising at a rate of 1.7 feet per second
The system of equations is y = 35 - 2.2x and y = 1.7x.
The solution of the system of equations is (8.97,15.25).
At about 8.97 seconds, both elevators will be about 15.25 feet above the ground.
What is an elevator?
An elevator, often known as a lift, is a car that moves vertically in a shaft to transport passengers or freight between the levels of a multistory building. Most modern elevators are propelled by electric motors via a system of wires and sheave with the assistance of a counterweight (pulleys).
Given that there are two elevators in an office. The 4th floor of the office at 35 feet above the ground and is descending at a rate of 2.2 feet per second.
The equation represents the height of the elevator above the ground after x seconds is y = 35 + (-2.2)x. Since the elevator is descending at a rate of 2.2 feet per second.
The other elevator starts out at ground level and rises at a rate of 1.7 feet per second.
The equation represents the height of the elevator above the ground after x seconds is y = 1.7x. Since the elevator is rising at a rate of 1.7 feet per second.
The system of equations is
y = 35 + (-2.2)x .....(i)
y = 1.7x ........(ii)
Solve the equations substitution method.
Putting y = 1.7x in equation (i)
1.7x = 35 + (-2.2)x
1.7x = 35 - 2.2x
Add 2.2x on both sides:
1.7x + 2.2x = 35
3.9x = 35
Divide both sides by 3.9:
x = 35/3.9
x = 8.97
Putting x = 8.97 in equation (ii)
y = 1.7 × 8.97
y = 15.25
After 8.97 seconds, the elevators will be 15.25 feet above the ground.
To learn more system of equations, click on the below link:
https://brainly.com/question/18597707
#SPJ1
lines AB and BC are perpendicular. the dashed rays bisect angles ABD and CBD. explain why the measure of angle EBF is 45 degrees.
Angle EBF is 45 degrees because angle ABC is 90 degrees and line D divides each side into two equal parts.
How to determine angles?Given that lines AB and BC are parallel. Angles ABD and CBD are bisected by the dashed rays. Angle EBF is measured at 45 degrees because angle ABC is 90 degrees, line D divides each side into two equal parts, and half of 90 degrees is 45 degrees.
As a result, the angle EBF is 45 degrees because angle ABC is 90 degrees and line D divides each side into two equal parts, and half of 90 degrees is 45 degrees.
Learn more on angles here: https://brainly.com/question/25716982
#SPJ1
Answer the following questions. Make sure to show all your work. 1. When is it best to solve a system of equations using graphing? 2. When is it best to solve a system of equations using substitution? 3. When it is best to solve a system of equations using elimination? When it is 4. Solve the following system of equations graphically. y=3x+4 y=-2x-5
Andrew is going to invest in an account paying an interest rate of 2.2% compounded quarterly. How much would Andrew need to invest, to the nearest hundred dollars, for the value of the account to reach $93,000 in 19 years?
Andrew would need to invest $61,300 for the account to reach $93,000.
What is Compound Interest?Compound interest is defined as the amount of interest which has been calculated on the principal amount as well as the amount accumulated over the previous period is also included.
The equation to find final amount in compound interest is,
A = P (1 + r/n)^(n t)
Here A is the final amount, P is the initial amount, r is the rate of interest, n is the number of times compounding is done in a year, t is the time in years.
Given, A = $93,000
t = 19 years
r = 2.2% = 0.022
n = 4 (since compounding is quarterly)
93,000 = P [1 + (0.022/4)]^(4×19)
93,000 = P [1.0055]^(76)
93,000 = P (1.517)
P = 61,297.8516
To the nearest hundred dollars, P = $61,300
Hence the principal amount Andrew need to invest is $61,300.
Learn more about Compound Interest here :
https://brainly.com/question/29335425
#SPJ1
Periodical Express is a newspaper printing company. They printed 450 copies of The Informer in just 15 seconds! They next need to print 300 copies of the Sport Report.
If they print at the same rate, how many seconds will it take to print the Sport Report?
Answer: To find out how many seconds it will take to print 300 copies of the Sport Report, we can use the formula:
time = (number of copies) / (rate of printing)
We know that the rate of printing for The Informer is 450 copies in 15 seconds. To find the rate of printing per second, we can divide the number of copies by the time in seconds:
rate of printing = 450 / 15 = 30 copies per second
So to find out how long it will take to print 300 copies of the Sport Report, we can use this rate of printing and plug it into the time formula:
time = (number of copies) / (rate of printing)
time = 300 / 30
time = 10 seconds
Therefore, it will take 10 seconds to print 300 copies of the Sport Report at the same rate as The Informer.
Step-by-step explanation:
Please help! Please! Thank you! :) 30+ points
Answer:
x=6 z=73
Step-by-step explanation:
9x+53=6x+71
3x+53=71
3x=18
x=6
9(6)+53
54+53
107
107+z=180
z=73
Amir operates a large lobster boat. The operating cost for the boat is $2.250 each day. At the end of each day, he sells all his freshly caught lobster to either the local restaurant or the local grocery store with the following conditions:
• The price per pound that the restaurant is willing to pay follows a triangular distribution with minimum value $1.50, maximum value $5.50, and likeliest value $3.50.
• The price per pound that the grocery store is willing to pay is decreasing with more lobsters: $3.85 - 50.0005 * y, where y is the total lobster amount sold in pounds.
• The amount of lobster that Amir catches in a single day follows a normal distribution with mean 1,500 pounds and standard deviation sqrt(12,500) pounds.
Amir decides to sell a fixed percentage of lobster to the local restaurant and the rest to local grocery stores. Using either math or simulation, can you help Amir determine what percentage he should choose in order to maximize his expected profit in the long run?
Previous question
The expected profit in the long run can be determined by calculating the expected value of the total revenue from selling the lobsters. The expected total revenue is the sum of the expected revenue from selling the lobsters to the restaurant and the from selling the lobsters to the grocery store.
The expected value of the revenue from selling the lobsters to the restaurant can be calculated by multiplying the expected price per pound from the restaurant by the expected amount of lobsters caught in a day. The expected price per pound from the restaurant follows a triangular distribution with minimum value $1.50, maximum value $5.50, and likeliest value $3.50. The expected amount of lobsters caught in a day follows a normal distribution with mean 1,500 pounds and standard deviation sqrt(12,500) pounds.
The expected value of the revenue from selling the lobsters to the grocery store can be calculated by multiplying the expected price per pound from the grocery store by the expected amount of lobsters caught in a day. The expected price per pound from the grocery store is decreasing with more lobsters: $3.85 - 50.0005 * y, where y is the total lobster amount sold in pounds. The expected amount of lobsters caught in a day follows a normal distribution with mean 1,500 pounds and standard deviation sqrt(12,500) pounds.
Given the expected values of the total revenue from selling the lobsters to the restaurant and the grocery store, Amir can use simulation to determine the optimal percentage of lobsters to sell to the restaurant in order to maximize his expected profit in the long run. He can run several simulations with different percentages of lobsters to be sold to the restaurant and compare the expected profits for each simulation. The percentage that produces the highest expected profit is the optimal percentage for Amir to maximize his expected profit in the long run.
Amir can use simulation to determine the optimal percentage of lobsters to sell to the restaurant in order to maximize his expected profit in the long run. He can calculate the expected value of the total revenue from selling the lobsters to the restaurant and the grocery store, and then run several simulations with different percentages of lobsters to be sold to the restaurant to compare the expected profits for each simulation. The percentage that produces the highest expected profit is the optimal percentage for Amir to maximize his expected profit in the long run.
Learn more about profit here
https://brainly.com/question/33585
#SPJ4
How do you solve y =- 2x 2?
The slope is 2 using the slope-intercept form. As the slope of the line in slope-intercept form is the coefficient of x, the slope of the line is -2.
what is slope ?The slope of a line governs its steepness. Using mathematics, the gradient is expressed as "gradient overflow" (the change in y divided by the change in x). The slope is the ratio of the vertical (rise) to the horizontal (run) change in distance between any two sites. Every time the equation for a straight line is written as y = mx + b, the slope-intercept representation of the equation is utilized. At the location of the y-intercept, the line has a slope of m, b, and c. (0, b). As an example, the y-intercept of the equation y = 3x - 7 has a value of 3. (0, 7).
given
The slope is 2 using the slope-intercept form.
As the slope of the line in slope-intercept form is the coefficient of x, the slope of the line is -2.
Divide the difference in elevation between two points by the distance between them, multiply the result by 100, and that is the percent slope. The term "rise" refers to the elevational difference between two places. The run is the distance separating the points. The formula for percent slope is (rise / run) x 100.
To know more about slope visit:
https://brainly.com/question/3605446
#SPJ4
Peter has 18 dimes. He uses 12 dimes to buy a drink. The number of dimes he has left is one-half the number of Pennie’s he has. How many Pennie’s does Peter have? *Put this in a 5th grade understanding*
18-12=6, peter now has 6 dimes left after buying a drink. Since the number he has left is ONE HALF of the number of pennies he has. 6x2=12.
peter has 12 pennies. hoped that helped
A group of researchers investigated the effect of media usage (whether or not subjects watch television or use the Internet) in the bedroom on "Tiredness" during the day (measured on a 50 point scale
a. Identify each of the variables, state whether the variables are quantitative or categorical, and identify each variable as either explanatory or response variables. nterviewed
b. To collect these data, the researchers randomly selected homes to visit and i the adult member of the household whose birthday was nearest. Is this an experiment or an observational study? Briefly explain.
YES, This is an observational study.
The variables in this study are:
Media usage: categorical (explanatory variable)
Tiredness: quantitative (response variable)
This is an observational study. The researchers are observing and measuring the relationship between media usage and tiredness, but they are not manipulating or controlling any variables. They are collecting data by randomly selecting homes to visit and interviewing the adult member of the household whose birthday was nearest.
An observational study draws conclusions from a sample of a population in disciplines like epidemiology, social sciences, psychology, and statistics when the independent variable is not under the researcher's control due to ethical considerations or practical limitations. One typical observational study examines how a therapy may affect its participants, but the researcher has no control over which group the subjects are assigned to—the treated group or the control group. Unlike experiments like randomized controlled trials, when each subject is arbitrarily assigned to either a treated group or a control group, this does not happen. Observational studies naturally create challenges for inferential analysis because they lack an assignment mechanism.
To know more about observational study visit :
https://brainly.com/question/15938198
#SPJ4
In 1967 a pair of slacks sold for $8.00. Ten years later, the price increased by 65%. How much more did a pair of slacks sell for ten years later?
The LCM of two numbers is 18. One of the numbers is 9. The other number is less than 9. What could the other number be? Show your work.
Answer:6
Step-by-step explanation: 6,12,18
9,18
there are two drama groups in school one group were 36 boys and 48 girls in the other 3/7 of students are boys others girls
ann says "the ratio of the number of boys to the number of girls is the same for both groups"
is ann correct
Answer:
UMMMM Wait IM sorry What grade are you in? I'm in fifth.
Step-by-step explanation:
Chad bought a box of donut hole that contain 60% glazed and the ret powdered ugar. The box ha 12 powdered ugar donut hole. How many glazed donut hole are in the box Chad bought?
The box has 18 glazed donut hole that Chad bought.
The percentage of glazed donut holes = 60%
The percentage of powdered sugar donut hole = (100 - 60) % = 40%
Let us consider the total number of donut hole in the box bought by Chad to be 'x'
Number of powdered sugar donut hole = 12
Hence, we can say that the number of powdered sugar donut hole in the box Chad bought is:
(40% of x) = 12
Therefore, x = (12 x 100) / 40 = 30
Since the total number of donut hole in the box = 30
The number of glazed donut hole = 30 - 12 = 18
Thus, the box that Chad bought had 18 glazed donut hole.
To know more about donuts:
https://brainly.com/question/29476313
#SPJ4
What is an example of inequality math?
Answer: 5x − 4 > 2x + 3
Step-by-step explanation:
The expression 5x − 4 > 2x + 3 looks like an equation but with the equals sign replaced by an arrowhead. It is an example of an inequality. This denotes that the part on the left, 5x − 4, is greater than the part on the right, 2x + 3.
I need help only have 2 hours left
Answer:
[tex]\begin{array}{|c|c|c|c|c|c|}\cline{1-6} \vphantom{\dfrac12} x&-4&1&2&3&6 \\\cline{1-6} \vphantom{\dfrac12} f(x)&-20&5&10&15&30 \\\cline{1-6}\end{array}[/tex]
Step-by-step explanation:
Given function:
[tex]f(x)=5x[/tex]
To complete the function table, substitute each given value of x into the given function:
[tex]x=-4 \implies f(-4)=5(-4)=-20[/tex]
[tex]x=1 \implies f(1)=5(1)=5[/tex]
[tex]x=2 \implies f(2)=5(2)=10[/tex]
[tex]x=3 \implies f(3)=5(3)=15[/tex]
[tex]x=6 \implies f(6)=5(6)=30[/tex]
Completed table:
[tex]\begin{array}{|c|c|c|c|c|c|}\cline{1-6} \vphantom{\dfrac12} x&-4&1&2&3&6 \\\cline{1-6} \vphantom{\dfrac12} f(x)&-20&5&10&15&30 \\\cline{1-6}\end{array}[/tex]
8. What is the converse of this
statement?
If Sarah keeps practicing gymnastics,
she will win a medal.
Answer:
Step-by-step explanation:
Sara will win a medal if she keeps practicing gymnastics.
Is x² a quadratic equation?
Answer: No, x^2 is not a quadratic equation. It is a monomial equation. A quadratic equation has the form ax^2 + bx + c = 0, where a, b, and c are constants.
Step-by-step explanation:
Show two
different ways to find the answer to the problem below.
Dog and cat food are sold in 20-pound bags. There are 14 bags of dog food and 12 bags of cat food on the store shelves. How many pounds of dog and cat food are on the shelves?
Answer: There are 520 pounds of dog and cat food
Step-by-step explanation:
1. 14+12= 26
2. 26x20=520
A company ha two manufacturing plant with daily production level of 5x 14 item and 3x - 5 item, repectively. The firt plant produce how many more item daily than the econd plant?
As per the given equation, the first plant is produce 2x+21 more items than second plant.
The term equation is known as a mathematical statement that shows that two mathematical expressions are equal.
Here we have given that the first plant produces 5x+14 items daily
And the second plant produces 3x-7 items daily
In order to find the number of items produced on first plant, then we have to calculate the differences between number of items by the two plants daily, then it can be written as by using the equations,
=> (5x+14) - (3x-7)
Now, we have to expand the equations, then we get
=> 5x+14 - 3x+7
=> 2x+21
Therefore, first plant produces 2x+21 more items daily than the second plant.
To know more about Equation here.
https://brainly.com/question/10413253
#SPJ4
If JK=27JK=27, KI=20KI=20, JI=24JI=24, NL=42NL=42, and ML=50.4ML=50.4, find the perimeter of \triangle LMN△LMN. Round your answer to the nearest tenth if necessary. Figures are not necessarily drawn to scale.
Answer:
so basically all you have to do is round.
Step-by-step explanation:
1. Find the measure of the missing angle:
A
EN
D
1320
Ch
MESELE
HERO 4
270
STREETR
10
SAN VIERES
K
MAKAL
MESE THREE
PORTE
SEATTLEFFE
PERTINES
B
The measure of the missing angle is 21°.
How to find a interior and exterior angle of triangle? Interior angles of a triangle are the angles inside the triangle that are formed by the intersection of two sides. To calculate the measure of an interior angle, you can use the formula: interior angle = 180-(angle 1 + angle 2). This formula works for all triangles, regardless of the size or shape. Exterior angles of a triangle are the angles outside the triangle that are formed by extending one of the sides. To calculate the measure of an exterior angle, you can use the formula: exterior angle = angle 1 + angle 2. This formula works for all triangles as well. To find the interior and exterior angles of a triangle, you need to know the measures of the angles of the triangle. If you don’t know the measures of the angles, you can use the following steps:
1. Measure the length of each side of the triangle.
2. Using the triangle inequality theorem, determine the angles of the triangle.
3. Use the formula to calculate the measure of each interior and exterior angle.
By following these steps, you can easily calculate the interior and exterior angles of a triangle.Solution:
To measure a missing angle of triangle,
∠ABC + ∠ACB + ∠CAB = 180°
x + 132° + 27° = 180°
x + 159° = 180°
x = 180° - 159°
x = 21°
To learn more about missing angle refer to:
https://brainly.com/question/24509783
#SPJ1
Solve for y.
y−58.8=62
Enter your answer in the box.
y =
Answer:
y = 120.8
Step-by-step explanation:
y − 58.8 = 62
Add 58.8 to both sides
y = 120.8
Let's check
120.8 - 58.8 = 62
62 = 62
So, y = 120.8 is the correct answer.
Answer:y= 120.8 I hope this helps
Step-by-step explanation:
y-58.8=62
Move the constant to the right-hand side and change its sign
Like this
Y=62+58.8
Then Add the numbers 62 and 58.8
Which will give you
Plus bring down your y
Your answer is y=120.8
what is 999,999,999 x 600277798128
Answer:
6.000277798 x 10^20