According to the given statement, 37.9 cups of tomatoes in each jar.
What is linear equations?A system of linear equations is a set of equations with two or more variables. A linear equation is one that can be expressed as ax + b = c, where x is a variable and a, b, and c are constants.
A group of two or more linear equations that share the same variables is known as a system of linear equations. As an illustration, consider the set of linear equations below:
2x + 3y = 10
4x + 6y = 20
x and y are the same variables, and the equations are linear equations. The solution to this system of linear equations is x = 2, y = 2.
This question can be solved using the system of linear equations.
Solving the system of equations gives:
T = 7.4 cups of tomatoes in each jar
Step 1: Calculate the total amount of tomatoes used.
Total tomatoes = 8 jars x cups of tomatoes per jar
Total tomatoes = 8 x (41.6 - 2.1)
Total tomatoes = 303.2 cups
Step 2: Divide the total amount of tomatoes by the total number of jars.
Cups of tomatoes per jar = Total tomatoes ÷ 8
Cups of tomatoes per jar = 303.2 ÷ 8
Cups of tomatoes per jar = 37.9 cups
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A chest contains 5 balls that each have the number 3 written on them. The chest also contains 5
balls that each have the number 7 written on them, and 5 balls that each have the number 11
written on them. Octi the Octopus takes out a few balls, one at a time, and calculates the sum of
the numbers written on them. How many different sums less than 15 are possible?
There are 15 possible sums less than 15 that can be made by taking out one to three balls from the chest.
How many different sums less than 15 are possible?13 possible sums less than 15 are possible: 3, 7, 11, 6, 10, 14, 9, 13, 12, 5, 8, 4, and 2.There are a total of 15 possible sums less than 15 that Octi the Octopus can calculate.Octi can calculate the sum of 3, 7, or 11. If he takes out one ball, he could calculate 3, 7, or 11.If he takes out two balls, he could calculate 6, 14, 9, or 18.If he takes out three balls, he could calculate 9, 12, 15, 18, 21, or 24.If he takes out four balls, he could calculate 12, 15, 18, 21, 24, 27, or 30.Finally, if he takes out five balls, he could calculate 15, 18, 21, 24, 27, 30, or 33.All of these sums are less than 15 and are the only sums possible for Octi to calculate.There are fifteen possible sums less than 15 that Octi the Octopus can achieve by taking out balls one at a time from the chest. The possible sums are 3, 7, 11, 6, 9, 10, 13, 14, 12, 8, 5, 4, 2, 7 + 3 + 3, 11 + 3 + 3, and 7 + 7.All of the sums are possible because the chest contains five balls with the number three, five balls with the number seven, and five balls with the number eleven. This means that Octi can take out one, two, or three balls at a time and calculate the sum of the numbers written on them.For example, if Octi takes out one ball from the chest, the possible sums he can get are 3, 7, or 11. If he takes out two balls, the possible sums he can get are 6, 9, 10, or 13. And if he takes out three balls, the possible sums he can get are 14, 12, 8, 5, 4, 2, 7 + 3 + 3, and 11 + 3 + 3. Lastly, if he takes out four balls, the only possible sum is 7 + 7. In conclusion, Octi the Octopus can achieve fifteen different sums less than fifteen by taking out balls from the chest one at a time.To learn more about different sums less than 15 are possible refer to:
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V-7>24 inequality simplify your answer as much as possible
To simplify the inequality, we need to perform the same mathematical operation on both sides of the inequality.
V-7 > 24
Add 7 to both sides of the inequality:
V > 31
So, the simplified inequality is V > 31.
Suppose you are working with a data set that is normally distributed, with a mean of 300 and a standard deviation of 41. Determine the value of X from the following information
The value of X that is less than 17% of the values is given as follows:
X = 339.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The mean and the standard deviation for this problem are given as follows:
[tex]\mu = 300, \sigma = 41[/tex]
The value of X that is less than 17% of the measures is X when Z has a p-value of 0.83, so X when Z = 0.955, hence:
0.955 = (X - 300)/41
X - 300 = 0.955 x 41
X = 339.
Missing InformationThe problem asks for the value of X that is less than 17% of the values.
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Use the distributive property to write an equivalent expression. Then evaluate
the expression.
(15+5)
Answer:5(3+1); 20
Step-by-step explanation:
5x3 is 15 and 5x1 is 5 which is equivalent to 15+5, then add to get 20.
What proportion of the variation iny can be explained by the variation in the values of x? Report answer as percentage accurate to one decimal place: (If the answer is 0.84471 _ then it would be 84. 5%6 .. syou would enter 84.5 without the percent symbol:) r^2 =
A. To find the correlation coefficient, we can use the formula:
r = (n * Σxy - Σx * Σy) / √ ((n * Σx² - (Σx)²) * (n * Σy² - (Σy)²))
Where n is the number of data points.
Plugging in the values for the given data set:
r = (12 * 712.94 - 1451.9 * 511.6) / √ ((12 * 1297.14 - 1451.9²) * (12 * 636.35 - 511.6²))
r = -0.636
So, the correlation coefficient is -0.636.
B. To find the proportion of the variation in y that can be explained by the variation in x, we can use the formula:
r² = r²
r² = (-0.636) ²
r² = 0.40496
So, 40.5% of the variation in y can be explained by the variation in the values of x.
C. To find the regression line,we can use the formula:
y = a + bx
Where a is the y-intercept and b is the slope of the line.
To find b, we can use the formula:
b = (n * Σxy - Σx * Σy) / (n * Σx² - (Σx)²)
Plugging in the values for the given data set:
b = (12 * 712.94 - 1451.9 * 511.6) / (12 * 1297.14 - 1451.9²)
b = -0.856
To find a, we can use the formula:
a = (Σy - b * Σx) / n
Plugging in the values for the given data set:
a = (511.6 - -0.856 * 1451.9) / 12
a = 55.54
So, the regression line is:
y = 55.54 - 0.856x
The negative slope of -0.856 means that as x increases, y decreases, which is consistent with the negative correlation coefficient of -0.636. The y-intercept of 55.54 means that when x is 0, y is predicted to be 55.54.
Complete question:
Run a regression analysis on the following bivariate set of data with y as the response variable.
x y
43.8 54.1
41.3 51.2
35.3 60.1
48.1 44.9
42.8 51.8
44.7 50.8
39.4 56.6
38.2 56.6
40.7 53.5
45.3 51.1
45 46.7
41.1 52.2
A. Find the correlation coefficient and report it accurate to three decimal places.
r = ________________
B. What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place. (If the answer is 0.84471, then it would be 84.5%...you would enter 84.5 without the percent symbol.)
r² =_______________ %
C. Based on the data, calculate the regression line (each value to three decimal places)
y =___________ x + _______________
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please answer thank you !
The volume of the squared pyramid is 9 cubic yards, so the correct option is D.
What is the volume of the ice pyramid?For a square pyramid of sidelength S (S is the sidelength of the square base), the volume is given by the formula:
V = (1/3)*S^3
Here we want to find the volume if S = 3 yards, so we just need to replace that in the formula above to get the volume of the sculpture, we will get:
V = (1/3)*(3 yd)^3
V = (1/3)*(27 yd^3)
V = 9 yd^3
The volume is 9 cubic yards, then the correct option is D.
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sketch an example of two different triangles, triangle `abc` and triangle `def`, that fit the given criteria (they share two sets of congruent sides and one pair of congruent angles) but are not congruent.
Congruency is said to be defined on two triangles when they have exact same length of sides and equal angles between them. The shape and size are supposed to be the same even after rotation or flipping of triangles for them to remain congruent.
There are 4 criteria to test the congruency of the triangle and they are as follows:
SAS - This proves congruency when two sides and the angle between them are the same w.r.t both trianglesSSS - Two triangles are congruent if all three sides of both of them are equal w.r.t to each otherAAS- When Two pairs of corresponding angles and one pair of corresponding sides (not between the angles) are equal then this is trueASA- When Two pairs of corresponding angles and one pair of corresponding sides between the angles are equal then this is trueIn this set of triangles, (ABC, def) AB=DE, AC=DF, and ∠ABC= ∠DEF i.e., two sides and one angle are equal but it is not congruent because according to SAS criteria, the angle should be between AB and AC (or DE and DF)
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How to make a square with this equation y2+18y+
Answer: The missing value is 81.
Step-by-step explanation: In a square, the formula is [tex]y^2 + 2y + y^2[/tex],
where the first y^2 is staying there. 18/2 is 9, and 9 squared is 81.
Square ABCD has sides of length 3. Segments CM and CN divide the square's area into three equal parts. How long is segment CM?
A √10
B √12
C √13
D√ 14
E √ 15
The length of segment CM is equal to the square root of 10. This is determined by dividing the area of the square ABCD, which has sides of length 3, into three equal parts, resulting in a segment of length √10.
The area of square ABCD can be calculated by squaring the length of one side, which is 3, to get 9. To divide this area into three equal parts, each part must have an area of 3. To find the length of segment CM, the square root of this area, 3, must be taken, resulting in a length of √10.
To find the length of segment CM in square ABCD, the area of the square must first be determined. This can be done by squaring the length of one side, which is 3, resulting in an area of 9. To divide this area into three equal parts, each part must have an area of 3. This means the length of segment CM must be the square root of 3, which is equal to √10. Therefore, the length of segment CM is equal to √10. This is the result of dividing the area of the square ABCD into three equal parts. This process can be used to determine the length of any segment dividing an area into equal parts
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Which describes 4x and −7x
in this expression?
4x−5−7x
A. constant term
B. like terms
C. multiplication
D. simplify
======================================
Reason:
The 4x and -7x are like terms because they involve the same variable x. This allows us to combine the like terms to end up with -3x
4x-7x = -3x
-------
Another example of like terms is 2x and 5x.
Adding them gets us 2x+5x = 7x
We can think of it like saying "2 boxes + 5 boxes = 7 boxes". Replace each word "boxes" with the single letter "x" and we arrive at 2x+5x = 7x.
independent random samples of voters from two voting districts, g and h, were selected to investigate the proportion of all voters who favor a proposal to widen a road that runs through both districts. the difference between the sample proportions (g minus h) was used to create the 95 percent confidence interval (0.13,0.47) for the population difference between districts.
The most common level of confidence is 95%, however 90% or 99% have also been used on occasion.
A series of observations that are not reliant on any other samples or data constitute an independent random sample. Random samples selected without regard to sequencing. So, choice C is the proper one.
The essential points that are highlighted below are the followings:
Because the difference between the sample proportions G minus H greater population, according to the question. Due to the interval's all-positive values, it seems expected that district G will have more supporters than district H.
Assurance interval A confidence interval (CI) is a range of estimates for an unknown parameter that is determined by a lower bound and an upper bound. At a certain level of confidence, the interval is calculated.
The most typical level of confidence is 95%, however higher levels, such 90% or 99%, are occasionally employed.
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Using separation of variables, solve the differential equation,
(3+x^10)(????y/????x) = x^9/y
Use C to represent the arbitrary constant.
y^2=_____________
The value of y² is 1/5 ln(3+x¹⁰) + C
What is the separation of variables?
Separation of variables, in mathematics, is any of several strategies for rewriting equations so that each of two variables appears on a different side of the problem. These strategies are used to solve ordinary and partial differential equations.
Here, we have
Given: (3+^10)(/) = ^9/
We have to solve the differential equation by using the separation of variables.
Rewrite as y dy = x⁹/3+x¹⁰ dx
As (3+x¹⁰)' = 10x⁹, we solve
y²/2 = 1/10 ln(3+x¹⁰) + C
y² = 1/5 ln(3+x¹⁰) + 2C
Since C is any constant, y² = 1/5 ln(3+x¹⁰) + C
Hence, the value of y² is 1/5 ln(3+x¹⁰) + C
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−4≤−2(y−1)<2
Step 2 of 2 : Graph the solution set.
The solution set for the inequality -4≤-2(y-1)<2 is the set of all y-values that make the inequality true.
To graph the solution set, we can begin by plotting the inequality as an inequality on the y-axis, and then identifying the solutions that make the inequality true:
First, we can simplify the left side of the inequality: -4≤-2(y-1)
Next, we can solve for y by isolating y: -4/2≤y-1
Then we can add 1 to both sides of the inequality: -2≤y
Now, we can graph the inequality y ≥ -2, which is a line that is equal to or greater than -2.
On the right side, we have -2(y-1)<2
2(y-1)>-2
y-1>-1
y>-1
So we can graph the inequality y>-1, which is a line that is greater than -1.
So the solution set is the region above the line y=-2 and below the line y=-1. The region is a strip between two lines.
It is important to note that the solution set doesn't include the values of the lines, as the inequality is strict, not inclusive.
4x - 5 ≤ 6x + 3
solve
pls show work
Answer: X[tex]\geq[/tex]-4
Step-by-step explanation:
4x-5[tex]\leq[/tex] 6x + 3
-2x [tex]\leq[/tex] 3 + 5
-2x[tex]\leq[/tex] 8
X = [tex]\geq[/tex] - 8/2
Find X
I’m not really sure how to do this
Answer:
x = 121°
Step-by-step explanation:
the shape is a heptagon (7 sides). the interior angles in a heptagon will always add up to 900°, so all you need to do is add all the known angles and subtract from 900.
140 + 133 + 145 + 117 + 119 + 125 = 779
900 - 779 = 121
this means that x = 121°
hope this helped, good luck!
Return to the calculator tab, and click the clear button to begin A new calculation. this time, you'll check for valid triangles given two angles and the side between themUnder sides, enter 5 for a. under angles, enter 30 for b and 50 for c. then click calculate. how many triangles can be created from the given conditions?
There can be one valid triangle created from the given conditions of side a being 5 and angles b and c being 30 and 50 degrees respectively.
To calculate how many triangles can be created from two angles and a side, the user must click the “calculator” tab. Once this is done, it is important to click the “clear” button to begin a new calculation. After this, the user must enter the side (a) under the “sides” section, and the angles (b and c) under the “angles” section. This example used 5 for the side, 30 for angle b, and 50 for angle c. Once the information is entered, the user must click “calculate”. This will result in a valid triangle being created from the given conditions. It is important to note that if the user entered different values, a different result may be obtained. Therefore it is important to ensure that the values entered are the desired ones. Additionally, if the sum of the angles entered are not equal to 180 degrees, then no valid triangle can be created.
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What is the function for f(5x)=4x squared +2x-2
The value of the function for f(5) is 133
What is a function?A function can be defined as an equation, law, rule or expression that explains the relationship between two variables.
These variables are known as;
The independent variableThe dependent variableThe different types of functions in mathematics are;
Polynomial functionLinear FunctionIdentical FunctionQuadratic FunctionFrom the information given, we have that;
f(x) = 4x² + 2x - 2
To determine the function, f(5), substitute the value of x as 5
f(5) = 4(5)² + 2(5) -2
expand the bracket
f(5) = 125 + 10 -2
Add or subtract the values
f(5) = 133
Hence, the value is 133
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¿A qué altura está la cometa de Ana si su cuerda mide 8 metros y tendría que moverse 6 metros para situarse debajo de ella?
The kite is 10 meters high
Solving for the height:To calculate how high Ana's kite is, you can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the length of the string is one side of the triangle and the distance Ana needs to move to get under the kite is the other side. The height of the kite is the hypotenuse.
Therefore, the height of the kite can be calculated by:
height = [tex]\sqrt({string^{2} +distance^{2}) }[/tex]
height = [tex]\sqrt{(8^{2} + 6^{2} )[/tex]
height = √(64 + 36)
= √100
height = 10
Hence, the kite is 10 meters high.
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I need help asap, please rn
Answer:
g(f(- 5)) = - 31
Step-by-step explanation:
to calculate g(f(- 5)), evaluate f(- 5) and substitute the value obtained into g(x)
f(- 5) = (- 5)² + (- 5) - 12 = 25 - 5 - 12 = 25 - 17 = 8 , then
g(8) = - 2(8) - 15 = - 16 - 15 = - 31
Answer:
-31
Step-by-step explanation:
f(x)=x²+x-12
or,f(-5)=(-5)²+(-5)-12
=25-5-12
=25-17
=8
now, g(x)= -2x-15
= (-2)×8-15
= -16-15
=-31
HELP ME OUT PLEASE!!!!!!
What is the range of h?
All real numbers greater than or equal to -5
All real numbers greater than or equal to 0
All real numbers
All real numbers greater than or equal to -2
The range of the function h(x) is -
"all real numbers greater than or equal to -2"
What are functions?In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.Any subset of the Cartesian product of two sets X and Y defines a binary relation R ⊆ X × Y between these two sets.A binary relation is univalent (also called right-unique) if -[tex]${\displaystyle \forall x\in X,\forall y\in Y,\forall z\in Y,\quad ((x,y)\in R\land (x,z)\in R)\implies y=z.}[/tex]
A binary relation is total if -[tex]${\displaystyle \forall x\in X,\exists y\in Y,\quad (x,y)\in R.}[/tex]
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is a table representing a function.
Range is defined as the set of all possible values of f(x). Therefore, all real numbers greater than or equal to -2, will belong to the range of the function h(x).
Therefore, the range of the function h(x) is -
"all real numbers greater than or equal to -2"
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•25 POINTS•
Guessers will be reported.
Answer:
Step-by-step explanation:
If its going by 4 each time then it should be 21 but for n its 6
Morgan and her children went into a grocery store and she bought $10 worth of peaches and mangos. Each peach costs $1 and each mango costs $2. She bought a total of 8 peaches and mangos altogether. Graphically solve a system of equations in order to determine the number of peaches, 2, and the number of mangos, y, that Morgan bought.Morgan and her children went into a grocery store and she bought $10 worth of peaches and mangos. Each peach costs $1 and each mango costs $2. She bought a total of 8 peaches and mangos altogether. Graphically solve a system of equations in order to determine the number of peaches, ~, and the number of mangos, y, that Morgan bought.
pls show the asnwer with a graph clear pleaseee
On solving the linear equations x + y = 8 and x + 2y = 10, the value for peaches and mangoes is 6 and 2 respectively.
What is a linear equation?
A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.
Let the number of peaches be = x
Let the number of mangoes be = y
Total number of peaches and mangoes Morgan bought = 8
The linear equation for Morgan if she has bought 8 fruits -
x + y = 8 ....(1)
The cost of one peach is = $1
The cost of one mango is = $2
Total amount for Morgan to spend is = $ 10
The linear equation for Morgan if she has $10 to spend -
x + 2y = 10 ....(2)
Subtracting equation (1) from (2) -
x + 2y - (x + y) = 10 - 8
x + 2y - x - y = 2
y = 2
Substituting the value of y in equation (1) -
x + y = 8
x + 2 = 8
x = 8 - 2
x = 6
Now, to check equate the values of x and y in equation (2) -
x + 2y = 10
6 + 2(2) = 10
6 + 4 = 10
10 = 10
The graph is also plotted and it can be seen that the equations intersect at point (6,2).
Therefore, Morgan bought x = 6 peaches and y = 2 mangoes.
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x f(x)=2x−1 g(x)=12x −2 −34 −1 −1 −12 −12 0 0 0 1 1 12 2 3 1
The value of the composite function (f + g)(x) = 14x - 3
How to determine the composite functionFrom the question, we have the following parameters that can be used in our computation:
f(x) = 2x - 1
g(x) = 12x - 2
The composite function (f + g)(x) is calculated as
(f + g)(x) = f(x) + g(x)
Substitute the known values in the above equation, so, we have the following representation
(f + g)(x) = 2x - 1 + 12x - 2
Evaluate the like terms
(f + g)(x) = 14x - 3
Hence, the composite function is (f + g)(x) = 14x - 3
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Complete question
Consider the functions f(x)=2x−1 g(x)=12x −2
Calculate (f + g)(x)
As the price of tickets rises from $200 to $250, the price elasticity of demand for business travelers is , and the price elasticity of demand for vacationers is , using the midpoint method. Therefore, the demand for airline tickets in this price range is elastic for vacationers because business travelers are sensitive to changes in price.
Answer:
The price elasticity of demand measures how responsive the quantity demanded is to a change in price. It can be calculated using the midpoint method, which is (change in quantity demanded / average quantity demanded) / (change in price / average price).
Using the midpoint method to calculate the price elasticity of demand for business travelers and vacationers, you would need more information such as the change in quantity demanded and the average quantity demanded for each group at different prices.
It is not possible to say if the demand is elastic or not for either group based on the information provided.
Luke stacked seven pieces of wood on top of each other. IF each piece was ten over 12 of a foot tall, how tall was his pile?
The length of the pile of the piece of woods put together by Luke would be = 5.8ft
What is a pile?A pile of objects such as wood is defined as the combination of two or more different parts of the object in an orderly form.
The number of woods stacked by Like = 7
The length of each wood = 10/12ft
Therefore the length of the whole pile of wood ;
= 7×10/12
= 70/12
= 5.8ft
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Consider the following two algorithms which both are meant to print all multiples of 11 from 1 up to a user input positive integer value - upper. Which statement correctly compares the efficiency of these two algorithms?
Algorithm 1:
for (int i = 1; i <= upper; i++)
{
if (i % 11 == 0)
{
System.out.println(i + " ");
}
}
Algorithm 2:
for (int i = 1; i <= upper / 11; i++)
{
System.out.println(i * 11 + " ");
}
Algorithm 2 is more efficient than Algorithm 1 because it only needs to loop through the numbers up to upper/11 instead of upper. This means that it will take fewer iterations to reach the result, making it more efficient than Algorithm 1.
Algorithm 1 will make upper/11 more checks than Algorithm 2, resulting in a slower runtime. Additionally, Algorithm 2 avoids the need to check the modulus of each number, further reducing the computation time. Algorithm 2 only needs to loop through the multiples of 11, which are always 11 apart from each other, so it can skip over the other numbers between them.
This makes Algorithm 2 much faster than Algorithm 1 because it is only looping through a fraction of the numbers.
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a subscript n baseline equals area of the shaded region in term n. a subscript 1 baseline equals 4 drag the numbers and symbols to the line to create an expression that represents the recursive definition of a subscript n baseline.
The value of a sub n, we must start with the previous value of a sub n minus 1, and then add four to it.
a_n = a_n-1 + 4
This formula is used to represent the recursive definition of a subscript n baseline. The formula is read as "a sub n is equal to a sub n minus 1 plus 4". This means that in order to find the value of a sub n, we must start with the previous value of a sub n minus 1, and then add four to it.
For example, let's say we want to find the value of a sub 5. We can start by looking at the previous value of a sub 4, which is 12. We then add four to 12, giving us 16. Therefore, a sub 5 is equal to 16.
To calculate this in a more general way, we can say that a sub n is equal to 4 times n minus 1, plus the initial value of a sub 1. That is, a sub n is equal to 4 (n-1) + 4. For example, a sub 5 is equal to 4 (5-1) + 4, which is equal to 16.
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no calculator is allowed for this question. show all of your work, even though the question may not explicitly remind you to do so. clearly label any functions, graphs, tables, or other objects that you use. justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. your work will be scored on the correctness and completeness of your methods as well as your answers. answers without supporting work will usually not receive credit. unless otherwise specified, answers (numeric or algebraic) need not be simplified. if your answer is given as a decimal approximation, it should be correct to three places after the decimal point. unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
The answers contain a. 5 b. [tex]y=-\frac{3}{4} x+\frac{1}{2}[/tex] c. 148.5 d. 1/7
a. f"(2)
f"(x) = df'(x)/dx = d(sin(πx) + x² +3)/dx = cos(πx) + 2x
f"(2)=cos(π × 2) + 2 × 2
f"(2)=cos(2π) + 4
f"(2)=1 + 4
f"(2)=5
b. Equation for the line tangent to the graph of y = 1/f(x) at x = 0
We first find f(x) by integrating f'(x)
f(x) = ∫f'(x)dx = ∫(sin(πx) + x² +3)dx = -cos(πx)/π + x³/3 +3x + C
f(0) = 2 so,
2 = -cos(π × 0)/π + 0³/3 +3 × 0 + C
2 = -cos(0)/π + 0 + 0 + C
2 = -1/π + C
C = 2 + 1/π
f(x) = -cos(πx)/π + x³/3 +3x + 2 + 1/π
f(x) = [1-cos(πx)]/π + x³/3 +3x + 2
y = 1/f(x) = 1/([1-cos(πx)]/π + x³/3 +3x + 2)
The tangent to y is thus dy/dx
dy/dx = d1/([1-cos(πx)]/π + x³/3 +3x + 2)/dx
dy/dx = -([1-cos(πx)]/π + x³/3 +3x + 2)⁻²(sin(πx) + x² +3)
at x = 0,
dy/dx = -([1-cos(π × 0)]/π + 0³/3 +3 × 0 + 2)⁻²(sin(π × 0) + 0² +3)
dy/dx = -([1-cos(0)]/π + 0 + 0 + 2)⁻²(sin(0) + 0 +3)
dy/dx = -([1 - 1]/π + 0 + 0 + 2)⁻²(0 + 0 +3)
dy/dx = -(0/π + 2)⁻²(3)
dy/dx = -(0 + 2)⁻²(3)
dy/dx = -(2)⁻²(3)
dy/dx = -3/4
At x= 0,
y = 1/([1-cos(π × 0)]/π + 0³/3 +3 × 0 + 2)
y = 1/([1-cos(0)]/π + 0 + 0 + 2)
y = 1/([1 - 1]/π + 2)
y = 1/(0/π + 2)
y = 1/(0 + 2)
y = 1/2
So, the equation of the tangent at (0, 1/2) is
[tex]\frac{y-\frac{1}{2} }{x-0} =-\frac{3}{4} \\\\y-\frac{1}{2} =-\frac{3}{4} x\\\\y=-\frac{3}{4} x+\frac{1}{2}[/tex]
c. If g(x) = f (√(3x² + 4). Find g'(2)
g(x) = f (√(3x² + 4) = [1-cos(π√(3x² + 4)]/π + √(3x² + 4)³/3 +3√(3x² + 4) + 2
g'(x) = [3xsinπ√(3x² + 4) + 18x(3x² + 4) + 9x]/√(3x² + 4)
g'(2) = [3(2)sinπ√(3(2)² + 4) + 18(2)(3(2)² + 4) + 9(2)]/√(3(2)² + 4)
g'(2) = [6sinπ√(12 + 4) + 36(12 + 4) + 18]/√12 + 4)
g'(2) = [6sinπ√(16) + 36(16) + 18]/√16)
g'(2) = [6sin4π + 576 + 18]/4)
g'(2) = [6 × 0 + 576 + 18]/4)
g'(2) = [0 + 576 + 18]/4)
g'(2) = 594/4
g'(2) = 148.5
d. If h is the inverse function of f. Find h' (2)
If h(x) = f⁻¹(x)
then h'(x) = 1/f'(x)
h'(x) = 1/(sin(πx) + x² +3)
h'(2) = 1/(sin(π2) + 2² +3)
h'(2) = 1/(sin(2π) + 4 +3)
h'(2) = 1/(0 + 4 +3)
h'(2) = 1/7
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one quart of water weighs about 2.1 pounds.there are 4quarts in a gallon. how much does a gallon of water weigh?
There are 4 quarts in a gallon, and a gallon weighs 8.34 pounds, so a quart weighs 2.085 pounds.
How much does a gallon of water weigh?The gallon is a volumetric unit that is used in both imperial and American customary measurements. In use right now are three different variations:
the imperial gallon (imp gal), defined as 4.54609 litres, which is or was used in the United Kingdom, Ireland, Canada, Australia, New Zealand, and some Caribbean countries; the US gallon (US gal), defined as 3.785411784 L,[1] (231 cubic inches), which is used in the US and some Latin American and Caribbean countries; and the US dry gallon ("usdrygal"), defined as 18 US bushel (exactly 4.40488377086 L).
gallon equals 4 quarts
2.1*4 =8.4
gallon of water weigh is 8.4
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Given the graph for one quadratic function and the
table of values for another, determine which has a
smaller minimum.
The Minimum for quadratic function f(x) is smaller than g(x)
and it is -5.
What is a quadratic function?
Quadratic function is also a second-degree polynomial function. The graph of a quadratic function is a parabola. The parabola opens upwards if a graph is made for the quadratic formula.
The point at which the function attains maximum or minimum value is the vertex of the quadratic function. When we say second degree, then the variable is raised to the second power like x^2.
A quadratic function is of the form f(x) = ax^2 + bx + c, where a, b, and c are the numbers with a not equal to zero.
Now,
As the we find minimum value by calculating the value for f(x) or g(x)
and from given values f(x) have minimum value of -5 and g(x) have minimum value of 3.
hence,
Minimum for quadratic function f(x) is smaller than g(x)
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