What is the remainder when (4x3 + 2x2 − 18x + 38) ÷ (x + 3)?
2
12
96
110
Answer: First Option is correct.
Step-by-step explanation:
Since we have given that
[tex](4x^3+2x^2-18x+38)\div(x+3)[/tex]
We will apply the "Remainder Theorem ":
So, first we take
[tex]g(x)=x+3=0\\\\g(x)=x=-3\\\\and\\\\f(x)=4x^3+2x^2-18x+38[/tex]
So, we will put x=-3 in f(x).
[tex]f(-3)=4\times (-3)^3+2\times (-3)^2-18\times (-3)+38\\\\f(-3)=-108+18+54+38\\\\f(-3)=2[/tex]
So, Remainder of this division is 2.
Hence, First Option is correct.
The remainder of the given expression is 2 and this can be determined by using the factorization method.
Given :
Expression -- [tex]\rm \dfrac{4x^3+2x^2-18x+38}{x+3}[/tex]
The factorization method can be used in order to determine the remainder of the given expression.
The expression given is:
[tex]\rm =\dfrac{4x^3+2x^2-18x+38}{x+3}[/tex]
Try to factorize the numerator in the above expression.
[tex]\rm =\dfrac{4x^3+12x^2-10x^2-30x+12x+36+2}{x+3}[/tex]
[tex]\rm = \dfrac{4x^2(x+3)-10x(x+3)+12(x+3)+2}{(x+3)}[/tex]
Simplify the above expression.
[tex]\rm = (4x^2-10x+12) + \dfrac{2}{(x+3)}[/tex]
So, the remainder of the given expression is 2. Therefore, the correct option is A).
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Which equation does the graph below represent?
y = fraction 1 over 4x
y = 4x y
fraction negative 1 over 4x
y = −4x
Answer:
y=-4x
Step-by-step explanation:
WE need to write the equation for the given graph
In the graph y intercept is (0,0)
The equation of linear graph is y=mx+b
where m is the slope and b is the y intercept
From the given graph y intercept is 0 so b=0
to find slope pick two points from the graph
(0,0) and (1, -4)
slope = [tex]\frac{y_2-y_1}{x_2-x_1} = \frac{-4-0}{1-0} = -4[/tex]
m=-4 and b=0
So the equation becomes
y= -4x+0
y= -4x
If there is a 0.3% chance of something happening one day, what is the possibility of it happening throughout twenty days?
If a rope 36 feet long is cut into two pieces in such a way that one piece is twice as long as the other piece, how long must the long piece be
rope = 36 feet
first piece = x
2nd piece = 2x ( twice as long as the first piece)
3x=36
x=12
first piece = 12 feet
2nd piece = 2 x 12 = 24 feet
Choice #1: Describe each of the following properties of the graph of the cosine function, f(theta) = cos(theta), and relate the property to the unit circle definition of cosine. Amplitude Period Domain Range x-intercepts
The graph of the cosine function, f(theta) = cos(theta), has properties such as amplitude, period, domain, range, and x-intercepts, and these properties can be related to the unit circle definition of cosine.
Explanation:The graph of the cosine function, f(theta) = cos(theta), has several properties:
Amplitude: The amplitude of the cosine function is 1, which means that the graph oscillates between a maximum value of 1 and a minimum value of -1.Period: The period of the cosine function is 2π, which means that the graph repeats itself every 2π units of theta.Domain: The domain of the cosine function is all real numbers, as there are no restrictions on the values of theta for which the cosine function is defined.Range: The range of the cosine function is the interval [-1, 1], as the values of the cosine function range from -1 to 1.X-intercepts: The x-intercepts of the cosine function occur when the value of theta is equal to π/2 + nπ, where n is an integer. In the unit circle, these x-intercepts correspond to the points where the terminal side of theta intersects the x-axis.These properties can be related to the unit circle definition of cosine. The amplitude corresponds to the distance from the origin to the maximum or minimum value of cosine on the unit circle. The period corresponds to the distance traveled along the unit circle to complete one cycle of the cosine function. The x-intercepts of the cosine function correspond to the angles at which the terminal side of theta intersects the x-axis on the unit circle.
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Find the equation of the tangent line of f^-1(x) at the point where it intersects the x-axis
Find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p. n equals 123n=123, p equals 0.85
The mean of the binomial distribution is [tex]\( 104.55 \)[/tex], the variance is [tex]\( 15.6825 \)[/tex], and the standard deviation is approximately .
To find the mean, variance, and standard deviation of a binomial distribution with given values of [tex]\( n \)[/tex] and [tex]\( p \)[/tex], we use the following formulas:
Mean [tex](\( \mu \)) = \( n \times p \)[/tex]
Variance [tex](\( \sigma^2 \)) = \( n \times p \times (1 - p) \)[/tex]
Standard Deviation [tex](\( \sigma \)) = \( \sqrt{\text{Variance}} \)[/tex]
Given:
[tex]\( n = 123 \)[/tex]
[tex]\( p = 0.85 \)[/tex]
Let's calculate each of these:
Mean [tex](\( \mu \))[/tex]:
[tex]\( \mu = n \times p \)[/tex]
[tex]\( \mu = 123 \times 0.85 \)[/tex]
[tex]\( \mu = 104.55 \)[/tex]
Variance [tex](\( \sigma^2 \))[/tex]:
[tex]\( \sigma^2 = n \times p \times (1 - p) \)[/tex]
[tex]\( \sigma^2 = 123 \times 0.85 \times (1 - 0.85) \)[/tex]
[tex]\( \sigma^2 = 123 \times 0.85 \times 0.15 \)[/tex]
[tex]\( \sigma^2 = 104.55 \times 0.15 \)[/tex]
[tex]\( \sigma^2 = 15.6825 \)[/tex]
Standard Deviation [tex](\( \sigma \))[/tex]:
[tex]\( \sigma = \sqrt{\sigma^2} \)[/tex]
[tex]\( \sigma = \sqrt{15.6825} \)[/tex]
[tex]\( \sigma \approx 3.9599 \)[/tex]
Therefore, the mean of the binomial distribution is [tex]\( 104.55 \)[/tex], the variance is [tex]\( 15.6825 \)[/tex], and the standard deviation is approximately .
The diameter of a large lawn ornament in the shape of a sphere is 16 inches. What is the approximate volume of the ornament? Use 3.14 for (PIE SYMBOLE) Round to the nearest tenth of a cubic inch.
Answer:
2143.6 in
Step-by-step explanation:
Find all solutions in the interval [0, 2pi): sin5x+sinx=sin3x
Four photographers are taking pictures at a school dance. Photographer A takes 2/5 of the pictures, Photographer B takes 4%, Photographer C takes 0.29, and Photographer D takes 27/100.
Which choice lists the photographers in order from least to greatest by the amount of pictures they take?
A) 2/5 = 0.4
B) 4% = 0.04
C) 0.29
D) 27/100 = 0.27
least = 0.04, then 0.27, then 0.29, then 0.4
so B, D, C A is the order
Use these values to solve this problem. X=2,y=3,z=4. 21xy/x+z
A shipping company offers various sized shipping boxes to its customers. Some of these boxes are cube-shaped, with equal height, width, and depth. As part of an upcoming sales promotion, the company will offer two cube-shaped boxes for the price of one.
a. Write an expression to represent the total volume of two different sized boxes as a sum of cubes if one of the boxes has sides with a length of 1 foot and the other has sides with a length of x feet.
b. Factor the sum of cubes.
c. Calculate the total volume of the two boxes if x = 3 feet.
Which function represents g(x), a reflection of f(x) = 6(1/3)^x across the y-axis?
Answer:
its b
Step-by-step explanation:
i got u homies
Assume that adults have iq scores that are normally distributed with a mean of 100100 and a standard deviation of 15. find the third quartile upper q 3q3, which is the iq score separating the top 25% from the others.
Final answer:
The third quartile (Q3) of IQ scores, which separates the top 25% of scores, is approximately 110.125 for a normal distribution with a mean of 100 and a standard deviation of 15.
Explanation:
To find the third quartile (Q3) of IQ scores, which is the value that separates the top 25% from the others in a normally distributed data set, we use the properties of the normal distribution. The mean IQ score is 100 and the standard deviation is 15. Q3 corresponds to the 75th percentile in a normal distribution.
To find the third quartile (Q3), we often refer to the z-score table or use a statistical software or calculator that can handle normal distribution calculations. The z-score corresponding to the 75th percentile is approximately 0.675. We can then use the formula for z-scores:
Q3 = mean + z*(standard deviation)
Q3 = 100 + 0.675*15
Q3 = 100 + 10.125
Q3 = 110.125
Thus, the third quartile IQ score, separating the top 25% of scores from the rest, is approximately 110.125.
To find Q3 for IQ scores (mean 100, SD 15), calculate 75th percentile: [tex]\( Q3 = 100 + 0.674 \times 15 = 110.11 \).[/tex]
To find the third quartile (upper Q3) of IQ scores, we need to determine the IQ score that separates the top 25% from the rest. Given that IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, we can use the properties of the normal distribution to find this value.
1. Identify the z-score corresponding to the third quartile (Q3):
- The third quartile (Q3) corresponds to the 75th percentile of the normal distribution.
- Using the standard normal distribution table or a calculator, the z-score for the 75th percentile is approximately 0.674.
2. Convert the z-score to an IQ score:
- Use the formula for converting a z-score to a value in a normal distribution:
[tex]\[ X = \mu + z \sigma \][/tex]
where:
- [tex]\( \mu \)[/tex] is the mean (100)
- [tex]\( \sigma \)[/tex] is the standard deviation (15)
- [tex]\( z \)[/tex] is the z-score (0.674)
3. Calculate the IQ score:
[tex]\[ Q3 = 100 + (0.674 \times 15) = 100 + 10.11 = 110.11 \][/tex]
Therefore, the third quartile (Q3) IQ score, which separates the top 25% from the others, is approximately 110.
Name a pair of supplementary angles.
A. angle A E B and angle C E D
B. angle A E B and angle B E D
C. angle A E C and angle B E D
D. angle B E A and angle C E B
Answer:
B.angle AEB and angle BED.
Step-by-step explanation:
We are given that a diagram .
We have to find a pair of supplementary angles.
Supplementary angles:The pair of angles whose sum is 180 degrees is called supplementary angles.
From given diagram
[tex]\angle AEB+\angle BED=180^{\circ}[/tex]
[tex]\angle AEC+\angle CED=180^{\circ}[/tex]
Hence, option B is true.
Answer:B.angle AEB and angle BED.
A bond payable is similar to which of the following?
A Bond payable is are likely similar to note payable. They are similar because they have both written premises to pay the interest and the principal amount on a specific futures dates. They are both liability and also the interest is accrued in current liability.
The answer to your question is "Notes Payable."
Which statement about corresponding sides and angles of the two polygons is correct?
Answer:
Option D. is the correct answer.
Step-by-step explanation:
In this graph, polygon PQRST has been dilated by a scale factor of 3 keeping origin as the center of dilation to form P'Q'R'S'T'.
We know when two polygons are similar, their angles will be same and their corresponding sides will be in the same ratio.
Therefore, option D.which clearly says that the ratio of side SR and S'R' is 1 : 3, will be the answer.
Tiffany put a $1550 item on layaway by making a 20% down payment and agreeing to pay $120 a month. How many months sooner would she pay off the item if she increased her monthly payment to $180?
A. 18 months sooner
B. 11 months sooner
C. 4 months sooner
D 7 months sooner
1550*0.8=1240
1240/120 = approximately 11 months to pay off
1240/180=approximately 7 months to pay off
11-7 =4
so it would be paid off 4 months sooner, so C is the answer
Option: C is the correct answer.
C. 4 months sooner.
Step-by-step explanation:Total amount of the item is: $ 1550
Also, Tiffany paid 20% of the amount by down payment.
Hence, the amount left to pay after the down payment is:80% of total amount.
i.e. 0.80 of total amount.
= 0.80×1550
= $ 1240
Now the number of month it will take if she pay $ 120 a month is:
[tex]=\dfrac{1240}{120}=10.3333[/tex]
which is approximately equal to 11 months.
Similarly, the number of month it will take if she pay $ 180 a month is:
[tex]=\dfrac{1240}{180}=6.8889[/tex]
which is approximately equal to 7 months.
Hence, the number of months sooner she will pay off is:
11-7=4 months.
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Determine which ordered pair is NOT a solution of y=3x-8.
a.
(8, 16)
b.
(3, 4)
c.
(–6, –26)
d.
(–10, –38)
If the length of a rectangle parking lot is 10 meters less than twice it's width, and the perimeter is 400 meters, find the length of the parking lot
The length of the rectangular parking lot is 130 meters.
What are the area and perimeter of a rectangle?The area of a rectangle is the product of its length and width.
The perimeter of a rectangle is the sum of the lengths of all the sides.
Given, The length of a rectangular parking lot is 10 meters less than twice it's width.
Assuming the width of the rectangle is x meters, therefore length would be
(2x - 10) meters and the perimeter is 400 meters.
We know the perimeter of a rectangle is 2(length + width).
∴ 2( x + 2x - 10) = 400.
2(3x - 10) = 400.
6x - 20 = 400.
6x = 420.
x = 70 meters and length is (2.70 - 10) = 130 meters.
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How much money should be deposited today in an account that earns 6% compounded monthly so that it will accumulate to $9000 in three years?
Find three consecutive odd integers with the sum of 51.
Sat scores are distributed with a mean of 1,500 and a standard deviation of 300. you are interested in estimating the average sat score of first year students at your college. if you would like to limit the margin of error of your 95% confidence interval to 25 points, how many students should you sample?
To estimate the average SAT score of first-year students at a college with a 95% confidence interval and a margin of error of 25 points, you would need to sample approximately 554 students.
Explanation:This question involves using the concepts of mean, standard deviation, margin of error, and confidence intervals from probability and statistics. To determine the sample size needed to estimate the average SAT score with a certain margin of error, we'll use the formula for the sample size required for estimating a population mean in statistics:
n = (Z*σ/E)^2
Where:
n is the sample size Z is the Z-score (for a 95% confidence interval, Z is 1.96) σ is the standard deviation (300 in this case) E is the margin of error (25 in this case)
Plugging the values into the formula, we get:
n = (1.96*300/25)^2 ≈ 553.47
As we cannot have a fraction of a student, we always round up to ensure our margin of error requirement is met. Therefore, you would need to sample approximately 554 students.
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To limit the margin of error of your 95% confidence interval to 25 points, given the standard deviation is 300, use the formula for the margin of error for a confidence interval and solve for the sample size 'n'. Approximately 553 students should be sampled.
Explanation:For this question, we can use the formula for the margin of error for a confidence interval which is calculated as: Margin of Error = z * (σ/√n). Here, 'z' represents the z-score, 'σ' is the standard deviation, and 'n' is the sample size.
In this case, we wish to have a 95% confidence level. From z-tables, we know that the z-value for a 95% confidence level is approximately 1.96. We want a margin of error of 25 points. The standard deviation (σ) for the dataset is said to be 300.
So, we can rearrange our formula to solve for 'n', giving us: n = (z * σ / Margin of Error)². Substituting in our numbers, we get n = (1.96 * 300 / 25)². So, to limit the margin of error on your 95% confidence interval to 25 points, you will need to sample approximately 553 students.
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What is the formula for a finite geometric series and how can you use it in a practical situation?
The main cables of a suspension bridge are ideally parabolic. the cables over a bridge that is 400 feet logn are attached to towers that 100 feet tall. the lower point of the cable is 40 feet abouve the bdrige. find the equation that can be used to model the cables.
The equation to model the suspension cables of a bridge, where the lowest point of the cable is 40 feet above the bridge, and the towers are 100 feet tall and 200 feet from the center, is y = -0.0015x^2 + 60.
Explanation:To find the equation that models the suspension cable of a bridge, we use the properties of a parabolic shape. Since the cable is attached to towers that are 100 feet tall and the lowest point of the cable is 40 feet above the bridge, we know the vertex of the parabola is 60 feet (100 - 40) below the top of the towers.
Let's define the coordinate system with the origin at the lowest point of the cable. Then the towers are at (-200, 60) and (200, 60) because the bridge is 400 feet long, so each tower is 200 feet horizontally from the center. The parabolic equation takes the general form y = ax^2 + bx + c. Because the vertex is at (0, 60), c = 60.
Using the points (-200, 60), we can substitute into the parabolic equation and write a system to solve for a and b. Since the parabola is symmetric, b = 0. The system becomes 60 = a(-200)^2 + 60, which simplifies to a = -60/40000.
Thus, the equation to model the cables is y = -60/40000x^2 + 60, or simplified, y = -0.0015x^2 + 60.
Line EF has an equation of a line y = −2x + 7. Which of the following could be an equation for a line that is perpendicular to line EF?
y = 2x − 3
y = 1 over 2x − 3
y = −2x − 3
y = −1 over 2x − 3
Answer:
[tex]y=\frac{1}{2} x-3[/tex]
Step-by-step explanation:
Step 1
Find the slope of the line EF
we have
[tex]y=-2x+7[/tex]
The slope of the line EF is equal to
[tex]m=-2[/tex]
Step 2
Find the slope of the line perpendicular to the line EF
we know that
If two lines are perpendicular, then the product of its slopes is equal to minus one
so
[tex]m1*m2=-1[/tex]
we have
[tex]m1=-2[/tex] -----> slope of the line EF
Find the value of m2
substitute
[tex](-2)*m2=-1[/tex]
[tex]m2=1/2[/tex]
therefore
the equation [tex]y=\frac{1}{2} x-3[/tex] could be an equation for a line that is perpendicular to line EF
What is the length of LINE AC?
A. 128
B. 136
C. 144
D. 108
What are the amplitude, period, and midline of f(x) = −7 sin(4x − π) + 2?
This is an example of a sine wave function. A given sine wave function has a standard form of:
y = A sin [B (x + C)] + D
Where,
A = absolute value of amplitude
2 π / B = the period of the sine wave
D = is the midline of y
C = phase of the sine wave
Rewriting the given equation into this form will yield:
f (x) = -7 sin[4 (x – π / 4)] + 2
Therefore from this form, we can get the answers:
Amplitude = 7
Period = 2π / 4 = π / 2
Midline = 2
write the equation of a line that is perpendicular to the line y= (-1/5)x + 2 and has a y-intercept that is 5 units larger that the y-intercept of y= (-1/5)x + 2.
a. y= -5x + 7
b. y= 5x + 7
c. y= (-1/5)x + 7
d. y= (1/5)x + 7
The quotient of a number and 9. Algebraic expression.