Answer:
The 94% confidence interval is : ( 0.0867 , 0.1933 ) = ( 8.67 , 19.33 )%
Step-by-step explanation:
Solution:-
- The sample size, n = 150 visitors
- The proportion of visitors who hiked, p = 0.14
- We are to create a 94% confidence interval for the population proportion.
- We will determine the Z-critical value for the CI : 0.94 or significance level α = 0.06
- The critical value is defined and plucked from Z-score (standardized) tables as:
Z-critical = Z_α/2 = Z_0.03 = 1.88
- The confidence interval for the population proportion (p) is constructed as:
[tex]( p - Z-critical\sqrt{\frac{p*(1-p)}{n} } , p + Z-critical\sqrt{\frac{p*(1-p)}{n} } )\\\\( 0.14 - 1.88\sqrt{\frac{0.14*(1-0.14)}{150} } , p + 1.88\sqrt{\frac{0.14*(1-0.14)}{150} } )\\\\( 0.08673 , 0.19326 )[/tex]
- The 94% confidence interval is : ( 0.0867 , 0.1933 ) = ( 8.67 , 19.33 )%
The frequency table represents data gathered about how much time some farmers spend tending to their land each week. Complete the conditional relative frequency table by row by identifying the values for each letter. a = b = c = d =
Answer:
a= 0.6
b= 0.4
c= 0.2
d= 0.8
Step-by-step explanation:
Ons
The value of each letter in the conditional relative frequency table is
a = 0.6b = 0.4c = 0.2d = 0.8What is the value of each letter?Division is an arithmetic operation that is used to determine the quotient of two or more numbers. Division entails grouping a number into equal groups using another number.
a = 180 / 300 = 0.6
b = 120 / 300 = 0.4
c = 40 / 200 = 0.2
d = 160 / 200 = 0.80
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Help with this math question and i will give you 40 points and mark you the brainliest
Answer:
a) CD = 6a + 4.5b
b) k = 4
Step-by-step explanation:
CD = CA + AB
CA = -AC
CD = -3b + 6a + 7.5b
CD = 6a + 4.5b
BC is parallel to CD
BC = BA + AC
BC = ka + 3b
BC is a scalar multiple of CD
Scale factor:
3/4.5 = 2/3
k = ⅔(6)
k = 4
does finding the volume of a solid means the inside or outside of a solid?
Answer: Inside (depending on your definition).
Step-by-step explanation: Finding the volume of a solid means measuring what space that solid takes up. Volume is a measure of how much matter an object is made up of. Technically, finding the volume of a solid does not mean finding the "inside" or "outside" of a solid. If you are referring to the surface area of a solid as the outside, then the answer to your question would be the inside of the solid.
Write a Linear Equation in Slope Intercept Form given the listed slopw and y-intercept.
m = –5, b = –3
Answer:
The equation is y = -5x - 3.
Step-by-step explanation:
The slope form equation is y = mx+b. So you just have to substitute the m and b value into the equation :
y = mx + b
m = -5
b = -3
y = -5x + (-3)
= -5x - 3
Each month, Jeremy adds the same number of cards to his baseball card collection. In Jeremy, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April
Jeremy will have 72 baseball cards in April.
What is a consecutive pattern?A consecutive pattern refers to a sequence or order of numbers, letters, or other objects in which each follows in direct and uninterrupted succession from the one that precedes it.
We have,
To solve this problem, we need to determine the pattern of how many cards Jeremy adds to his collection each month.
To do this, we can find the difference between consecutive months:
February minus January: 48 - 36 = 12
March minus February: 60 - 48 = 12
We can see that Jeremy is adding 12 cards to his collection each month.
So to find out how many cards he will have in April, we can add 12 to the number he had in March:
April
= 60 + 12
= 72
Therefore,
Jeremy will have 72 baseball cards in April.
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An underground gasoline storage tank is leaking. The tank currently contains 600 gallons of gasoline and is losing 3.1 gallons per day. If the value of the gasoline is $2.55 per gallon, how quickly is the value of the stored gasoline changing?
Answer:
-$7.905 per day
Step-by-step explanation:
In this question, given that a tank is losing its fuel content at a certain rate, we are to calculate the rate at which the value in dollars of the content of the fuel tank
is changing
Please check attachment for complete solution and step by step explanation
An underground gasoline storage tank is leaking. The tank currently contains 600 gallons of gasoline and is losing 3.1 gallons per day. The rate at which the value of the stored gasoline is changing is -$7.905 per day.
Suppose we make an assumption that the tank initial contains h(x) gallons of gasoline which leak in x days. Then, the rate of change of gasoline per gallon is;
[tex]\mathbf{\dfrac{dh}{dx} =- 3.1 \ \ \ \ \ \ \text{since the gas is leaking (-)}}[/tex]
Also, the rate of change of the gasoline in time(t) per unit change in the quantity can be computed as:
[tex]\mathbf{\dfrac{dt}{dh}=2.55 }[/tex]
Therefore, the value of the stored gasoline changing per unit change of days can be deduced by using the chain rule:
[tex]\mathbf{\dfrac{dt}{dx} = \dfrac{dt}{dh} \times \dfrac{dh}{dx}}[/tex]
[tex]\mathbf{\dfrac{dt}{dx} = -3.1 \times2.55}[/tex]
[tex]\mathbf{\dfrac{dt}{dx} =\$-7.905 \ per \ day}[/tex]
Therefore, we can conclude that the rate at which the value of the stored gasoline is changing is -$7.905 per day.
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Which of the following are equivalent to the function y=3 cos x+2 ? Check all that apply
A. y= 3sin(x-pi/2) +2
B. y= 3sin(x+ pi/2) +2
C. y= 3cos(-x) +2
D. y= -3cos x-2
Options B and C are the correct answer.
The given trigonometric function is y=3cosx+2.
We need to check which of the given options are equivalent to the function y=3 cos x+2.
How to solve a given trigonometric function?The given trigonometric function can be solved using complementary angles and even/odd identities.
Complementary angles are sin x=cos(π/2 -x) and cos x=sin(π/2 -x).
Even/odd identities are sin(-x) =-sin x and cos(-x)= cos x.
Now,
From option A:
y=3 sin(x-π/2) +2
using trigonometry identity, we get
y=3 sin(-(x-π/2)) +2
⇒y=-3 sin(π/2 - x)+2
⇒y=-3 cos x+2
From option B:
y= 3sin(x+ π/2) +2
Using Sin(A+B)= sin A cos B+ cos A sin B, we get
y= 3cosx +2
From option C:
y= 3cos(-x) +2 can be written as y=3cosx+2 (cos(-x)=cosx)
From option D:
y= -3cos x-2
Therefore, options B and C are the correct answer.
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Final answer:
The equivalent functions to y=3 cos x+2 are B: y= 3sin(x+ π/2) +2 and C: y= 3cos(-x) +2, as they use the co-function identity and the even property of the cosine function respectively.
Explanation:
The given function is y=3 cos x+2. We need to find functions that are equivalent to this function. The trigonometric identities necessary for this problem are:
cos(-x) = cos(x), which is the even function property of cosine. sin(x + π/2) = cos(x), which is a co-function identity.sin(-x) = -sin(x), showing that sine is an odd function.Now, considering the given options:
A. y= 3sin(x-π/2) +2: Using the co-function identity sin(x - π/2) = -cos(x), this option is not equivalent because it would yield y = -3cos(x) + 2, which is not the same as the given function.B. y= 3sin(x+ π/2) +2: By the co-function identity sin(x + π/2) = cos(x), this option is equivalent to the given function.C. y= 3cos(-x) +2: Using the even function property of cosine, this option is equivalent to the given function because cos(-x) = cos(x).D. y= -3cos x-2: This is not equivalent because it represents the reflection of the given function over the x-axis and a vertical translation down by 2 units.
Therefore, the equivalent functions are B and C.
What is the value of n when (9)^2n-1 = (27)^n+2
[tex]9^{2n-1} = 27^{n+2} \\ ( {3}^{2} ) ^{2n - 1} = ( {3}^{3} )^{ n + 2} \\ {3}^{4n - 2} = {3}^{3n + 6} \\ 4n - 2 = 3n + 6 \\ 4n - 3n = 6 + 2 \\ n = 8[/tex]
Answer: n=8
PLEASE HELP ASAP!Complete the steps to find the area of the trapezoid.
Area of rectangle=
Area of triangle 1=
Area of triangle 2=
Area of triangle 3=
Area of trapezoid=
just put them how u normally wuld 1 is
3 is 6
4 is 5
2 is 8
5 is 13
Answer: area of the rectangle = 256
area of triangle 1 = 24
area of triangle 2 = 16
area of triangle 3 = 96
area of the trapezoid = 120
Solving Square Roots Worksheet (x - k)^2 : Part 1
1. 2(x + 7)^2 = 16
2. (x - 3)^2 = -12
3. -5(n - 3)^2 = 10
Answer:
1. x = +/- 2[tex]\sqrt{2}[/tex] - 7
2. x = [tex]3[/tex] +/- [tex]2i\sqrt{3}[/tex]
3. n = [tex]2[/tex] +/- [tex]i\sqrt{2}[/tex]
Step-by-step explanation:
1. Divide both sides by 2: (x + 7)^2 = 8
Square root both sides: x + 7 = +/- 2[tex]\sqrt{2}[/tex]
Subtract 7 from both sides: x = +/- 2[tex]\sqrt{2}[/tex] - 7
2. Square root both sides: x - 3 = [tex]\sqrt{-12}[/tex]
Since there is a negative inside the radical, we need to have an imaginary number: [tex]i=\sqrt{-1}[/tex] . So, [tex]\sqrt{-12} =i\sqrt{12} =2i\sqrt{3}[/tex]
Add 3 to both sides: x = [tex]3[/tex] +/- [tex]2i\sqrt{3}[/tex]
3. Divide by -5 from both sides: (n - 2)^2 = -2
Square root both sides: n - 2 = [tex]\sqrt{-2}[/tex]
Again, we have to use i: [tex]n-2=\sqrt{-2} =i\sqrt{2}[/tex]
Add 2 to both sides: n = [tex]2[/tex] +/- [tex]i\sqrt{2}[/tex]
Hope this helps!
Answer:
1. x = 2sqrt(2) - 7, -2sqrt(2) - 7
2. No real solutions
x = 3 + 2sqrt(3) i, 3 - 2sqrt(3) i
3. No real solutions
n = 3 + sqrt(2) i, 3 - sqrt(2) i
Step-by-step explanation:
1. 2(x + 7)² = 16
(x + 7)² = 8
x + 7 = +/- sqrt(8) = +/- 2sqrt(2(
x = 2sqrt(2) - 7, -2sqrt(2) - 7
2. (x - 3)² = -12
A perfect square can never be negative for real values of x
(x - 3) = +/- i × sqrt(12)
x - 3 = +/- i × 2sqrt(3)
x = 3 +/- i × 2sqrt(3)
3. -5(n - 3)² = 10
(n - 3)² = -2
A perfect square can never be negative for real values of x
n - 3 = +/- i × sqrt(2)
n = 3 +/- i × sqrt(2)
If f(x)=7(x−1)+8 , what is the value of f(1) ?
Answer:
8
Step-by-step explanation:
f(1)=7(1-1)+8
f(1)=0+8
f(1)=8
The question is asking us to find the value of the function f(x) = 7(x−1) + 8 when x equals 1. To find this, we substitute 1 in place of x in the equation. So, f(1) = 7(1-1) + 8. Calculating this, 7 multiplied by 0 equals 0, so the equation simplifies to 0 + 8. Therefore, f(1) = 8.
To solve the problem, we must first substitute the value of x in the function `f(x)` which is `7(x−1)+8`.
Here, substitute x = 1:
So we have f(1) = 7*(1-1) + 8.
Simplify the expression inside the parenthesis first (as according to BIDMAS or PEMDAS - operations in parenthetical expressions should be done before multiplication and addition):
7 * (1 - 1) + 8 simplifies to 7 * 0 + 8.
The multiplication operation takes precedence over addition (again per the rules of BIDMAS/PEMDAS), so multiply 7 and 0:
We get 0 + 8.
Finally, perform the addition operation:
0 + 8 equals 8.
So, f(1) = 8.
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What is the radius of a circle with an area of 28.26 square feet
Answer:
The radius is 3 square feet.
(Working is shown in the picture)
The radius of a circle with an area of 28.26 square feet is 3 feet
What is Radius?A radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length.
What is Area?Area is the quantity that expresses the extent of a region on the plane or on a curved surface.
What is Circle?A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the center.
Given,
Area of the circle = 28.26 square feet
Area of the circle = [tex]\pi r^{2}[/tex]
Then
[tex]28.26 =\pi r^{2}\\ r^{2}=9\\ r=\sqrt{9}\\[/tex]
r = 3 feet
Hence, the radius of a circle with an area of 28.26 square feet is 3 feet
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These numbers are either whole numbers ,or negative of whole numbers ,they are called __
Answer:
integers
Step-by-step explanation:
Answer:
Intergers
Step-by-step explanation:
Intergers are a whole number meaning a number that's not a fraction.
The diameter of a circle is 2 inches. What is the circle's area?
Use 3.14 for .
Answer:
3.14
Step-by-step explanation:
Answer:3.14
Step-by-step explanation:
help pleases with this question !!!!
Answer: 2.13
Step-by-step explanation:
1.42/2=0.71
1.42+0.71=2.13
Answer:
The answer to your question is £ 2.13
Step-by-step explanation:
Data
Cost of 1 kg or oranges = £1.42
Cost of the oranges in the scales = ?
Process
1.- Determine the weight of the oranges in the scales.
As we can see the weight of the three oranges is 1.5 kg
2.- Use proportions and cross multiplication to find the cost.
1 kg of oranges --------------------- £ 1.42
1.5 kg of oranges ------------------ x
x = (1.5 x 1.42) / 1
x = 2.13 / 1
x = £ 2.13
You are feeding a room that is 25‘ x 25‘ there is a window in the room there's 4‘ x 3‘ x 6‘ the paint cost $12 per gallon 1 gallon of paint covers 100 ft.² what is the area of the room
Answer:
With an assumption of room height = 8 ft we have;
The area of the room is 800 ft²
Area to be painted = 794.667 ft²
Paint required ≈ 7.95 gallons of paint
Cost of paint = $95.36
Step-by-step explanation:
Here we have if the height of the room is taken as 8 ft,
Therefore the total area in the room is 25 × 8 × 4 = 800 ft²
since the room has a window with three sides, therefore a triangular window of dimensions
4 ft by 3 ft by 6 ft
The area of the window is then
[tex]A = \sqrt{6.5(6.5-4)(6.5-3)(6.5-6)}[/tex]
Where the semi perimeter = (4 + 3 + 6)/2 = 6.5
Area of window, A = 5.333 ft²
Therefore, the total area to be painted = 800 ft² - 5.333 ft² = 794.667 ft².
Therefore the number of gallons of paint required ≈ 7.95 gallons
Cost of paint= $12 × 7.95 = $95.36.
Arie has $46 in his account and is saving $8.50 each week. Olga has $14 in her savings account and is saving $12.50 each week. After how many weeks will Arie and Olga have the same in their accounts?
Answer:
5 weeks until they have the same
Arie and Olga have the same amount in their accounts in 8 weeks.
What is an expression?An expression is a number, or a variable, or a combination of numbers and variables and operation symbols.
Now it is given that,
Amount in Arie's account = $46
Arie's each week savings = $8.50
Amount in Olga 's account = $14
Olga 's each week savings = $12.50
Let x be the number of weeks when Arie and Olga will have the same amount in their accounts
Total savings of Arie = 8.5x
Total savings of Olga = 12.5x
Amount in Arie's account = 8.5x + 46
Amount in Olga's account = 12.5x + 14
For, Arie and Olga have the same amount in their accounts
8.5x + 46 = 12.5x + 14
Taking alike terms together,
12.5x - 8.5x = 46 - 14
⇒ 4x = 32
Dividing both side by 4 we get,
x =8 weeks
which is the required number of weeks.
Thus, Arie and Olga will have the same amount in their accounts in 8 weeks.
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PLEASE HELP
A radius of 4 millimeters. What is the area?
Answer:
50.24 mm²
Step-by-step explanation:
Area = pi × r²
3.14 × 4²
50.24 mm²
Answer:
The area equals 50.2654 rounded up 50.27
Step-by-step explanation:
A= pi times radius squared equals pi times 4 squared equals 50.2654 (50.27)
I need help asap my friends
Answer:
The first one 24/4
Step-by-step explanation:
Into usually means divide when it's used as a math term,
therefore 24 into fourths means 24 divide by 4.
Answer:
the answer is 24 divided by 4
Step-by-step explanation:
charles needs to fill a large fish tank gallons per min, and what is the rate? a total of 110 gallons of water can flow through hose b in 10 minutes. which hose has a faster water flow rate, in gallon per minute, and what is that rate?
The question is incomplete, the correct question is:
Charles needs to fill a large fish tank with water using a hose. he has two hoses from which to choose. Water flows through each hose at a constant rate. The graph below shows the amount of water, in gallons , that flows through holes A based on the number of minutes used. A total of 110 gallo s of water can flow through Hose B in 10 minutes. Which hose has a faster water rate, in gallons per minute, and what is that rate?
Answer:
Hose B has a faster water flow rate in comparison to Hose A
Step-by-step explanation:
For Hose B,
From the information above, it is said that a total of 110 gallons of water can flow through hose b in 10 minutes. Therefore, divide 110 by 10 to get the amount of water that flows through hose b in gallons in one minute.
So,
110/10=11
y=11 x
This implies that 11 gallons of water flows through hose b in 1 minute.
For Hose A,
Select two points to get the rate of change,and then subtraction of y2 from y1 and x2 from x1 is done
Points selected: (2,12) , (4,24)
(x1,y1)=(2,12)
(x2,y2)=(4,24)
So, Subtract 24 from 12 and 2 from 4
M=y2-y1/x2-x1= 24-12 / 2-4= 12/2=6
This implies that in hose a a total of 6 gallons of water flows in one minute.
Therefore, the equation for hose a is y=6 x
In conclusion, hose b has a faster water flow rate since y=6 x < y=11 x.
Answer:
is b the correct
Step-by-step explanation:
A circle with radius of 3cm sits inside a circle with radius of 11cm
What is the area of the shaded region?
Round your final answer to the nearest hundredth.
The larger circle is shaded.
Answer:
Circular shapes - They are those planner shapes that represent the locus of all the points that has a constant distance from a fixed point on the plane. This constant distance is termed as the radius of the circle and the fixed point is known as the center of the circle.
The center of the circle is enclosed by all the points on its periphery.
The circumference of the circle is the total length of its periphery around the center.
Concentric circles are two circles that have the same center
Step-by-step explanation:
Zad.1 12 pracowników spytano o liczbę dni wykorzystanego urlopu. Oto odpowiedzi: 2, 5, 14, 3, 6, 5, 8, 2, 5, 4, 6, 8. Podaj dominantę i medianę. Zad. 2 13 pracowników spytano o liczbę dni wykorzystanego urlopu. Oto odpowiedzi: 2, 4, 14, 3, 6, 7, 8, 2, 5, 4, 6, 8, 9. Podaj dominantę i medianę. Zad.3 Oblicz dominantę i medianę a)Odległość zamieszkania w km. b) Liczba pracowników a) b) 0-5 5 5-10 25 10-15 30 15-20 55 20-35 30 25-30 20 30-35 15
Answer:
Taak 1.12
Mediaan is 5 en het gemiddelde is 5,67
Taak 2 13
De mediaan is 6
Het gemiddelde is 6
Standaarddeviatie is 3,32
Variantie is 11
Aantal werknemers is 13
Step-by-step explanation:
Taak.1 12 medewerkers is gevraagd naar het aantal verlofdagen. Hier zijn de antwoorden: 2, 5, 14, 3, 6, 5, 8, 2, 5, 4, 6, 8. Geef de dominante en de mediaan. Taak 2 13 medewerkers is gevraagd naar het aantal vakantiedagen. Hier zijn de antwoorden: 2, 4, 14, 3, 6, 7, 8, 2, 5, 4, 6, 8, 9. Geef de dominante en mediaan. Zad.3 Bereken de dominante en de mediaan a) Verblijfsafstand in km. b) Aantal werknemers a) b) 0-5 5 5-10 25 10-15 30 15-20 55 20-35 30 25-30 20 30-35 15
Taak. 1 12, wordt de mediaan gegeven door
2, 5, 14, 3, 6, 5, 8, 2, 5, 4, 6, 8 herschikken, we krijgen
2, 2, 3, 4, 5, 5, 5, 6, 6, 8, 8, 14
Daarom is de mediaan 5 en het gemiddelde 5,67
Taak 2 13
2, 4, 14, 3, 6, 7, 8, 2, 5, 4, 6, 8, 9 herschikken, we krijgen
2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 14
De mediaan is het zevende nummer in de rij = 6
Het gemiddelde is ook = 6
Standaarddeviatie = 3,32
Variantie = 11
Aantal werknemers = som van frequentie of aantal gegevens = 13.
Historically, the mean yield of corn in the United States has been 120 bushels per acre with a standard deviation of 12. A survey of 40 farmers this year gives a sample mean yield of 125 bushels per acre . Let p be the mean yield of corn nationally for this year. Supposing that the past standard deviation is still correct, what is the p-value for testing a null hypothesis of mu=120 against an alternative of mu not equal to 20 ? A 0.0041 B 0.0082 C 2.64 D 125
Answer:
[tex]z=\frac{125-120}{\frac{12}{\sqrt{40}}}=2.64[/tex]
P-value
Since is a two sided test the p value would be:
[tex]p_v =2*P(z>2.64)=0.0082[/tex]
And the best answer would be
B 0.0082
Step-by-step explanation:
Data given and notation
[tex]\bar X=125[/tex] represent the mean height for the sample
[tex]\sigma=12[/tex] represent the population standard deviation
[tex]n=40[/tex] sample size
[tex]\mu_o =120[/tex] represent the value that we want to test
[tex]\alpha[/tex] represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the true mean is 120 or not, the system of hypothesis would be:
Null hypothesis:[tex]\mu = 120[/tex]
Alternative hypothesis:[tex]\mu \neq 120[/tex]
If we analyze the size for the sample is > 30 and we know the population deviation so is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:
[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]z=\frac{125-120}{\frac{12}{\sqrt{40}}}=2.64[/tex]
P-value
Since is a two sided test the p value would be:
[tex]p_v =2*P(z>2.64)=0.0082[/tex]
And the best answer would be
B 0.0082
You have 80 dollars and play the following game. An urn contains two white balls and two black balls. You draw the balls out one at a time without replacement until all the balls are gone. On each draw, you bet half of your present fortune that you will draw a white ball. What is your expected final fortune
Answer:
$45
Step-by-step explanation:
In order to find this we need to understand that every time we bet we are selecting white ball. Because of this we have the chance of getting two times right and two times wrong irrespective of the order. For example we can any of the following orders:
White - White - Black - Black
White - Black - White - Black
Black - White - Black - White
Black - Black - White - White
Irrespective of any of the orders above every time our bet is right (we get white urn) the amount of dollars that we have get increased by half of the current amount. For example on the first turn we are betting $40 (half of initial amount = $80) and if we are right we now have $120 with us. This can be written as
1.5*$80 = $120
Similarly if on first turn we get wrong (black ball) then our amount is half of initial which can be written as
0.5*$80 = $40
So now we know that exactly two times we will be right and exactly two times we will be wrong irrespective of the order because the balls are not replaced hence final solution is:
E(X) = 80*1.5*1.5*0.5*0.5
= $45
f(x) = x4 + x3 + 2x2 + ax + b,
where a and b are constants.
When f(x) is divided by (x - 1), the remainder is 7
(a) Show that a + b = 3
When f(x) is divided by (x + 2), the remainder is -8
(b) Find the value of a and the value of b
Answer:
see explanation
Step-by-step explanation:
The remainder theorem states that if f(x) is divided by (x - h)
The the remainder is f(h)
(a)
Given f(x) is divided by (x - 1) then remainder is f(1), thus
f(1) = [tex]1^{4}[/tex] + 1³ + 2(1)² + a + b = 7, that is
1 + 1 + 2 + a + b = 7
4 + a + b = 7 ( subtract 4 from both sides )
a + b = 3 ← as required → (1)
(b
Given f(x) is divided by (x + 2) then the remainder is f(- 2), thus
f(- 2) = [tex](-2)^{4}[/tex] + (- 2)³ + 2(- 2)² + 2a + b = - 8, that is
16 - 8 + 8 + 2a + b = - 8
16 + 2a + b = - 8 ( subtract 16 from both sides )
2a + b = - 24 → (2)
Multiply (1) by 2
2a + 2b = 6 → (3)
Add (2) and (3) term by term to eliminate the term in a
3b = - 18 ( divide both sides by 3 )
b = - 6
Substitute b = - 6 into (1)
a - 6 = 3 ( add 6 to both sides )
a = 9
Thus a = 9 and b = - 6
Final answer:
To solve for the constants a and b in the polynomial function f(x), we apply the remainder theorem to the given division scenarios. Setting up and solving a system of linear equations using the remainders allows us to find that a is 9 and b is -6.
Explanation:
The given function is f(x) = x^4 + x^3 + 2x^2 + ax + b, and we need to find the constants a and b given that when f(x) is divided by (x - 1), the remainder is 7, and when divided by (x + 2), the remainder is -8.
To find a + b, we apply the remainder theorem. When a polynomial f(x) is divided by (x - p), the remainder is f(p). Therefore, for (x - 1):
f(1) = 1 + 1 + 2 + a + b = 7
4 + a + b = 7
a + b = 3
Similarly, for (x + 2):
f(-2) = (-2)^4 + (-2)^3 + 2(-2)^2 + a(-2) + b = -8
16 - 8 + 8 - 2a + b = -8
16 - 2a + b = -8
We already know that a + b = 3, so we can set up a system of equations:
a + b = 3
-2a + b = -24
Solving for a and b:
Subtracting the first equation from the second: -3a = -27
Divide by -3: a = 9
Substitute a into the first equation: 9 + b = 3
Solve for b: b = -6
Dennis has three identical cylinders filled with water. How many cones should he be able to fill with the water if the cones have the same radius and the same height as the cylinders?
Answer:
3 cylinder will fill 9 cones with water
Step-by-step explanation:
This problem bothers on mensuration of slides, cone and cylinder
We know that the
Volume of a cylinder = πr²h
Volume of a cone = 1/3(πr²h)
Given that both cylinder and cone has same height and radius
From the given expression we can deduce that the cone is 3 times smaller than the cylinder in volume
So if 1 cylinder will fill 3 cones
Then 3 cylinders will fill x cones
By cross multiplication we have
x= 3*3cones
x= 9 cones
Hence 3 cylinder will fill 9 cones with water
Answer:
9 cones
Step-by-step explanation:
The formula to find a volume of a cylinder is:
V1 = pi*r^2*h
Where r is the base radius and h is the height
The formula to find a volume of a cone is:
V2 = (1/3)*pi*r^2*h
So, if they have the same base radius and same height, we have that:
V1/V2 = 1/(1/3) = 3
The volume of the cylinder is 3 times bigger than the volume of the cone, so each cylinder of water can fill 3 cones.
Is Dennis has 3 cylinders, he is able to fill 3*3 = 9 cones with water.
The regular price of a computer is $1200 and the regular price of a printer is $300. The regular price of a computer is 1200 dollars and the regular price of a printer is 300 dollas. An electronics store has a promotion that offers a 40% 40 percent discount on the printer when the computer is purchased at the regular price. What is the total cost of the computer and the printer at the promotional price?
Answer:
$1380
Step-by-step explanation:
The regular price of a computer is $1200The regular price of a printer is $300.The electronics store offers a 40% discount on the printer only when the computer is purchased at the regular price.
Discount on the Printer = 40% of $300 =0.4 X 300 =120
Promotional Price of the Printer = $300-120 =$180
Therefore, the total cost of the computer and the printer at the promotional price
=Regular price of a computer + Promotional Price of the Printer
=1200+180
=$1380
Simplify the algebraic expression
6(-4x-7) -4(x-6)
A -20x-18
B -20x-66
C -28x+66
D -28x-18
6(-4x - 7) - 4(x - 6) Distribute 6 into (-4x - 7) and -4 into (x - 6)
(6)(-4x) + (6)(-7) + (-4)x + (-4)(-6) (two negative signs cancel each other out and become positive)
-24x - 42 - 4x + 24 Combine like terms(terms that have the same variable and power/exponent)
-24x - 4x - 42 + 24 (I rearranged the terms to be next to their like term)
-28x - 18 Your answer is D
12 rotten tomatoes are usually found in every four boxes.how many rotten tomatoes would likely be found in 14 boxes?
Answer:
42 tomatoes
Step-by-step explanation:
tomatoes/boxes = tomatoes/boxes
12/4 = x/14
4x/4 = 168/4
x=42
Answer:
42 rotten tomatoes would likely be found in 14 boxes
Step-by-step explanation:
1) Find how many rotten tomatoes are normally found in one box by dividing the amount of rotten tomatoes by the number of boxes they are found in. [tex]12/4=3[/tex]
2) So, now you know that 3 rotten tomatoes are normally found in one box, but the question is asking how many tomatoes would likely be found in 14 boxes which means you need to multiply the amount of rotten tomatoes found in one box by the amount of boxes you want to know about. [tex]3*14=42[/tex]
PLEASE HELP
factor 12n - 18.
I think the answer is 6(2n-3)