The value of a is 12. Therefore, the equation of the function is f(x) = 12|x|.
What is a function's absolute value?In algebra, an objective function function is one where the variables is inside the bars denoting absolute values. The absolute value is also referred to as the modulus function, and its most popular representation is f(x) = |x|, wherein x is a real number.
How can I solve a function using absolute values?Finding x values that turn the equation inside the exact amount positive or negative into a constant is how absolute value equations are solved. Plot two for the negative and positive cases that intersect at the expression's 0 to graph exact value functions.
To find the value of a, we can use the point (1, 12) to set up the equation:
a = 12/|1| = 12
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Opportunity cost and production possibilities
Shen is a talented artist who sells hand-crafted goods on his website. Shen currently crafts and sells both crochet hats and baskets. He spends 8 hours
a day working on crafts. The following table gives different daily output scenarios depending on how much of his time is spent on each good.
Answer:
I'm not understanding can you say it in a similar way?
Answer:
Step-by-step explanation:
Opportunity cost and production possibilities
Shen is a talented artist who sells hand-crafted goods on his website. Shen currently crafts and sells both crochet hats and baskets. He spends 8 hours
a day working on crafts. The following table gives different daily output scenarios depending on how much of his time is spent on each good
To test if a regression coefficient is less than a critical value, C
, we conduct a one-sided test on the _________ tail of a ___________ distribution.
we conduct a one-sided test on the statistic one tail of a critical area of distribution.
The critical value for a two-tailed test of h0: β1 = 0 at α = .05 in a simple regression with 22 observations is 4.35.
What is a critical value?
A critical value can be calculated for different types of hypothesis tests. The critical value of a particular test can be interpreted from the distribution of the test statistic and the significance level.
Test of significance of linear regression between x and y.
Null hypothesis: There is no significant linear relationship between x and y.
Alternative hypothesis: There is a significant linear relationship between x and y.
Test statistic:
F0 = Mean square due to regression/ mean squared error ~ F(1, n-2)
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How to find the area of the major sector
The area of the major sector in the given diagram is 115.5 sq.unit.
What is a Major sector?The region having a greater area is known as Major Sector while the one which has a smaller area is the minor sector.
A sector with a central angle greater than 180° is called a major sector and the formula is given below:
/360° ₓ r²
The radius is 7m and is 270°
Therefore 270/360 × 3.142 × 7² = 115.5sq.unit.
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three students, a, b, and c are accused of creating a contradiction. three facts are known: if a is not guilty, then b and c are both guilty. either a is not guilty, or b is guilty. either b is not guilty, or c is not guilty. which one of the following statements correctly identifies the guilty students? [hint: review the end of the lecture for section 2.2.] group of answer choices either a or b is guilty, but not both, and c is guilty. neither a nor c is guilty, but b is guilty. only one of the three is not guilty and it is not a and not b. exactly two of the students are guilty.
Exactly two of the students are guilty. This can be deduced by looking at the facts given and understanding that if A is not guilty, then both B and C are guilty, while either B is not guilty or C is not guilty. This implies that exactly two of the students are guilty.
Given the three facts, we can determine that either A or B is guilty, but not both, and C is guilty. This means that exactly two of the students are guilty.
Since the statement either A is not guilty, or B is guilty, and the statement either B is not guilty, or C is not guilty, both cannot be true. This means that exactly two of the students are guilty.
Given the three facts, we can determine that either A or B is guilty, but not both, and C is guilty. This means that exactly two of the students are guilty. Since the statement either A is not guilty, or B is guilty, and the statement either B is not guilty, or C is not guilty, both cannot be true. This means that only one of the three is not guilty and it is not A or B. Therefore, exactly two of the students are guilty. This can be deduced by looking at the facts given and understanding that if A is not guilty, then both B and C are guilty, while either B is not guilty or C is not guilty. This implies that exactly two of the students are guilty.
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Write a function in a factored form that could represent the situation below:
A Polynomial function has zeros at 1, -2 and 5 (all multiplicity 2).
The polynomial equation with the given zeros is something like:
p(x) = a*(x - 1)²*(x + 2)²*(x - 5)²
How to find the function that can represent the situation?Remember that for a polynomial of degree a with the zeros {x₁, x₂, ...} the polynomial can be written as:
p(x) = a*(x - x₁)*(x - x₂)*...
Here we don't know the leading coefficient, and we know that the zeros are:
{1, 1, -2, -2, 5, 5}
These appear twice because all have a multiplicity of 2.
Then the equation is:
p(x) = a*(x - 1)*(x - 1)*(x + 2)*(x + 2)*(x - 5)*(x - 5)
p(x) = a*(x - 1)²*(x + 2)²*(x - 5)²
Where a can be any real number different than zero.
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Sets
A, B, C
are subsets of the universal set
U
.
These sets are defined as follows.
U = f, g, h, p, q, r, x, y, z
A = f, g, p, q
B = g, h, q, x, y
C = p, q, x, y, z
Find
C ∩ (B’ ∪ A) =
.
What is roster form
∅
.
The required roster form of the given sets are C ∩ (B’ ∪ A) = {p, q}.
What is the roster form of a set?The roster form of a set is defined as a set that can be represented mathematically in a precise way using a roster notation. In a roster notation, if the set has more than one element, a comma is used to denote the separation between every pair of elements.
The given sets are defined as follows:
U = {f, g, h, p, q, r, x, y, z}
A = {f, g, p, q}
B = {g, h, q, x, y}
C = {p, q, x, y, z}
The complement of set B is the set of all elements in the universal set U that are not in set B.
U = {f, g, h, p, q, r, x, y, z}
B = {g, h, q, x, y}
B’ = {f, p, r, z}
The union of set B' and A is the set of all elements that are in set B' or in set A.
A = {f, g, p, q}
B’ = {f, p, r, z}
B’ ∪ A = {f, g, p, q, r, z}
The intersection of set C and (B’ ∪ A) is the set of all elements that are in both set C and set (B’ ∪ A).
C = {p, q, x, y, z}
B’ ∪ A = {f, g, p, q, r, z}
C ∩ (B’ ∪ A) = {p, q}
So, C ∩ (B’ ∪ A) = {p, q}
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I. Express each ratio 25 2 II. Express each num 1.
Each one of the given ratios should be expressed in its simplest form as follows:
50:150 = 1:3.25:40 = 5:8.33:99 = 1:3.What is a ratio?In Mathematics, a ratio can be defined as a mathematical expression that's used to denote the proportion of two (2) or more quantities with respect to one another and the total quantities.
In this exercise, you are required to express each of the given ratios in its simplest form. Therefore, we would divide both sides of the ratio by highest common factor (HCF) as follows;
50:150 = 50/150 = 1/5 = 1:5. (divide both sides by 50).
25:40 = 25/40 = 5:8. (divide both sides by 5).
33:99 = 1:3. (divide both sides by 33).
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Complete Question:
Express each one of the following ratios in its simplest form.
(i) 50: 150
(ii) 25:40
(iii) 33 : 99
If f(-8) = 1, then the point ( ____ , ____ ) is on the graph of f.
If f(-8) = 1, then the point (-8, 1) is on the graph of f.
How to obtain the point on the graph of the function?The numeric value of the function at x = -8 is given as follows:
f(-8) = 1.
The format of each point on the coordinate plane is given as follows:
(x,y).
In which y is the numeric value of the function at x, meaning that the point can also be written as follows:
(x, f(x)).
Considering that when x = -8, f(x) = 1, the point is given as follows:
(-8,1).
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a random sample of 1,175 people in a certain country were asked whether they thought climate change was a problem. the sample proportion of those who think climate change is a problem was calculated, and a 95 percent confidence interval was constructed as
The interpretation of the confidence interval is that "we can be 95% confident that the true population proportion of people who think climate change lies between 0.146 and 0.214".
Given :
A random sample of 1,175 people in a certain country was asked whether they thought climate change was a problem.
The sample proportion of those who think climate change is a problem was calculated, and a 95 percent confidence interval was constructed as (0.146, 0.214).
95% confidence interval means a person is 95% confident that true population lies in it.
95% confidence interval is given by (0.146,0.214).
So, the interpretation of the confidence interval is that "we can be 95% confident that the true population proportion of people who think climate change lies between 0.146 and 0.214".
Therefore, the correct option is C) We are 95 percent confident that the proportion of people in the sample who think climate change is a problem is between 0.146 and 0.214.
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If 15 workers can build a wall in 48 hours, how many workers will be
required to do the same work in 30 hours?
Answer:
24 workers
Step-by-step explanation:
solution
A wall can be built in 48 hrs by = 15 workers
A wall can be built in 1 hr by = 15× 48 workers
A wall can be built in 30 hrs by= 15 × 48/ 30 workers
= 24 workers
select the correct prefix to complete the equality. drag the appropriate labels to their respective targets. note: not all labels will be used.
The answers for the given units are 1 g = 0.001 kg; 1 m = 1000 mm; 1 cm = 10 mm; 1 dL = 100 mL; 1 kg = 1000 g; 1 pg = 1 × 10⁻¹² g; 1 g = 1 × 10¹² pg; and 1 mL = 1 cc.
A unit is a name given to a known quantity that is used to measure something. A gram is one-thousandth of a kilogram used to measure weight. This is written as 1 g = 0.001 kg or 10⁻³ kg and kg to g is written as 1 kg = 1000 g. In terms of their relationship, 1 m and 0.001 mm are equivalent. This is written as 1 m = 1000 mm or 10³ mm. The relationship between millimeters and centimeters is written as 1 cm = 10 mm.
Over a milliliter, a deciliter is greater. This is written as 1 dL = 100 mL. The relationship between picogram and gram is written as 1 pg = 1 × 10⁻¹² g
and 1 g = 1 × 10¹² pg. The relationship between a milliliter and a cubic centimeter is written as 1 mL = 1 cc.
The required answers are
1 g = 0.001 kg
1 m = 1000 mm
1 cm = 10 mm
1 dL = 100 mL
1 kg = 1000 g
1 pg = 1 × 10⁻¹² g
1 g = 1 × 10¹² pg
1 mL = 1 cc
The complete question is -
Select the correct prefix to complete the equality. Drag the appropriate labels to their respective targets. Note: not all labels will be used.
1 g = -----kg
1 m = -----mm
1 cm = ----- mm
1 dL = ----- mL
1 kg = ----- g
1 pg = ----- g
1 g = ----- pg
1 mL = ----- cc
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Given the figure below shows the regular hexagon ABCDEF, we can draw BF and CE to create two isosceles triangles.
Consider your answer to the previous question to help find the measures of these TWO angles. Show all work and provide all calculations:
The angle measures for this problem are given as follows:
m < AFB = 60º.m < BFE = 60º.How to obtain the angle measures?The sum of the measures of the internal angles of a polygon with n sides is given as follows:
S(n) = 180 x (n - 2).
Hence, for an hexagon, with six sides, the sum is given as follows:
S(6) = 180 x (6 - 2)
S(6) = 720º.
A regular hexagon has all angle measures equal, hence the measure of each angle is given as follows:
720/6 = 120º.
The segment FB is a bisection of the angle F of the hexagon, dividing it into two equal angles, hence the angle measures are given as follows:
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symbol for sample deviation is
The symbol for sample deviation or standard deviation is "σ"(sigma).
What is the standard deviation?The standard deviation is a metric that reveals how much variance from the mean there is, including spread, dispersion, and spread. The standard deviation shows a "typical" variation from the mean. Because it uses the data set's original units of measurement, it is a well-liked measure of variability.
Because it is the square root of the means of the squared deviations from the arithmetic mean, the standard deviation is often referred to as root-mean-square deviation or sample deviation, sample standard deviation.
the standard deviation of sample deviation is denoted by σ(sigma).
Hence the symbol is "σ"(sigma).
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The mortgage is $158,750. Interest is 6.25% monthly payment is $1,168. How much will the principal be after the third payment?
How do I know what formula to use to figure the correct answer?
These are the choices:
A. $344.74
B. $342.95
C. $341.18
D. $826.82
Answer:
Step-by-step explanation:
B. $342.95 is your answer
(50 POINTS AND BRAINLYEST)
The height of the cylinder is 30 units. What is the volume of the cylinder? Ignore the prism.
Answer:
9 234
Step-by-step explanation:
base area multiplied by height is volume
I do not get this at all please help
Answer:I think it is 6
Step-by-step explanation:
Answer:
h(2) = 6
Step-by-step explanation:
This function is called h.
In function notation, it is h(x).
h(2) means the value of the y-coordinate of the point whose x-coordinate is 2.
Look up 2 on the y axis.
Now go up until you get to the blue line of the function.
What y value does that correspond to? Go left horizontally until you get to the y-axis. You intersect the y-axis at 6. That means that
h(2) = 6
The diagram below shows the dimensions of a can of tomatoes. HELP!! TYY
A can of tomatoes is shown. A dashed line across the top of the can is labeled nine centimeters. The height of the can is labeled twelve and four tenths centimeters.
How much tin was used to make the can rounded to the nearest square centimeter? Use 3.14 for π.
Enter the correct answer in the box.
_______cm^2
The amount of tin was used to make the can of cylindrical can is 478 cm²
What is an equation?An equation is an expression that shows the relationship between numbers and variables.
Area is the amount of space occupied by a figure. Area is usually calculated for a two dimensional shape while volume is calculated for a three dimensional shape.
The surface area of cylindrical can = 2π(radius * height + radius²)
From the image:
radius = 9 cm/2 = 4.5 cm, height = 12.4 cm
Hence, substituting:
The surface area of cylindrical can = 2(3.14)(4.5 * 12.4 + 4.5²) = 478 cm²
The surface area of cylindrical can is 478 cm²
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Solve the following equation for x. Round the answer to the nearest hundredth.
−1.5r = 4(c + 2.18)
A. 0.63
B. 3.49
C. -1.59
D. -0.40
The equation -1.5x = 4(x + 2.18) solved for x is -1.59. Option C
What are algebraic expression?Algebraic expressions are described as expressions that are mainly composed of variables, terms, constants, coefficients and factors.
These expressions are also composed of arithmetic operations, such as;
AdditionSubtractionDivisionMultiplicationParenthesesFloor divisionBracketFrom the information given, we have that;
-1.5x = 4(x + 2.18)
expand the bracket
-1. 5x = 4x + 8.72
Now, collect like terms, we get;
-1.5x - 4x = 8.72
subtract the like terms
-5.5x = 8.72
Divide both sides by the coefficient of x
x = -1.59
Hence, the correct option is Option C
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Complete question:
Solve the following equation for x. Round the answer to the nearest hundredth.
−1.5x= 4(x + 2.18)
A. 0.63
B. 3.49
C. -1.59
D. -0.40
50 POINTS HELP ASAP!
The table shows the distance a bicyclist travels over time.
Time (hours) 2 4 9 16
Distance (miles) 6 11 34 81
Based on the table, which statement is true?
Group of answer choices
The table shows a linear relationship and a proportional relationship.
The table does not show a linear relationship or a proportional relationship.
The table shows a linear relationship, but not a proportional relationship.
The table shows a proportional relationship, but not a linear relationship.
Based on the table, a statement which is true include the following: C. The table shows a linear relationship, but not a proportional relationship.
What is a proportional relationship?In Mathematics, a proportional relationship is a type of relationship that generates equivalent ratios and it can be modeled or represented by the following mathematical expression:
y = kx
Where:
y represents the distance.x represents the time.k is the constant of proportionality.Next, we would determine the constant of proportionality (k) for the data points on this graph as follows:
Constant of proportionality, k = y/x
Constant of proportionality, k = 6/2 ≠ 11/4 ≠ 34/9 ≠ 81/16
In this context, we can reasonably infer and logically deduce that the table shows a linear relationship because as the value of time (in hours) increases, the value of distance (miles) also increases. However, it does not show a proportional relationship.
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In a statistics course a linear regression equation was computed to predict the final exam score from the score
on the first test. The equation of the least-squares regression line was y=10+0.9x where y represents
the predicted final exam score and x is the score on the first exam.
As per the given equation of the least squares regression line y=10+0.9x the score of Carla's final exam is 95.5.
As given in the question,
Given equation of the least squares regression line is :
y = 10 + 0.9x
Where 'y' shows the predicted final exam score
And 'x' represents the score of the first exam.
Carla's first exam score is equal to 95
Then predicted value 'y' of the Carla's final exam score is given by :
y = 10 + ( 0.9 ) × 95
⇒ y = 10 + 85.5
⇒ y = 95.5
Therefore, the predicted final score of Carla's as per the given least squares regression line is equal to 95.5.
The above question is incomplete, the complete question is :
In a statistics course a linear regression equation was computed to predict the final exam score from the score
on the first test. The equation of the least-squares regression line was y=10+0.9x where y represents the predicted final exam score and x is the score on the first exam.
Carla scored 95 on the first test. What is the predicted value of the Carla's score on the final exam?
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baseball coach buys frozen treats for his team following their game.
The coach buys a total of 15 treats, which are either ice cream sandwiches or frozen juice pops
• An ice cream sandwich costs $2.00 and a juice pop costs $2.50.
• The coach spends a total of $33.00 on the treats (no tax is charged).
How many ice cream sandwiches does the coach buy? How much does the coach spend on juice pops? Use the drop-down menus to select the answers that make the statement true.
The coach buys ~ ice cream sandwiches and spends
~ on frozen juice pops.
According to the information, the coach will buy 9 ice cream sandwches and 6 frozen juice pops.
How to calculate how many sandwiches and frozen juices did the coach buy?To calculate how many sandwiches and frozen juices the coach bought we have to perform the following mathematical operation:
First we have to built a equation incluiding all the values of the problem.
I + J = 15 I = The number of ice cream sandwiches J = The number of juice pops15 = Treats were purchased in totalNow, we have to include the price of each product
$2 * I + 2.50 * J = 33.00
Then we have to solve the equation.
2 I + 2.5 ( 15 - I) = 3320 I + 25 ( 15 - I ) = 330 20 I + 375 - 25 I = 330 375 - 5 I = 330 - 5 I = 330 - 375 - 5 I = - 45 I = 9
According to the above the number of sanwiches is 9, and the number of juices is 6.
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from a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. how many different committees are possible if ross, sheldon. first course in probability, a (p. 16). pearson education. kindle edition.
There are 16128 ways to form the committee when 2 of the men refuse to serve together, 23040 ways to form the committee when 2 of the women refuse to serve together, and 5040 ways to form the committee when 1 man and 1 woman refuse to serve together.
(a) When two of the men refuse to serve together, there are 6 men to choose from for the first 3 positions, 4 for the next, and 2 for the last. So there are 6 × 4 × 2 = 48 ways to choose the 3 men.
There are 8 women to choose from for the first 3 positions, so there are 8 × 7 × 6 = 336 ways to choose the 3 women. Thus there are 48 × 336 = 16128 ways to form the committee when 2 of the men refuse to serve together.
(b) When two of the women refuse to serve together, there are 8 women to choose from for the first position, 6 for the next, and 4 for the last. So there are 8 × 6 × 4 = 192 ways to choose the 3 women.
There are 6 men to choose from for the first 3 positions, so there are 6 × 5 × 4 = 120 ways to choose the 3 men. Thus there are 192 × 120 = 23040 ways to form the committee when 2 of the women refuse to serve together.
(c) When 1 man and 1 woman refuse to serve together, there are 7 women to choose from for the first 3 positions and 6 men to choose from for the last 3 positions.
So there are 7 × 6 × 5 × 4 × 3 = 5040 ways to form the committee when 1 man and 1 woman refuse to serve together.
The answer to each of the questions is the number of different committees that can be formed under the given conditions.
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--The given question is incomplete; the complete question is
"From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How many different committees are possible if (a) 2 of the men refuse to serve together? (b) 2 of the women refuse to serve together? (c) 1 man and 1 woman refuse to serve together?"--
select the correct answer. Which hyperbola has both vertices in the same quadrant?
Option (d) is the correct answer i.e. that hyperbola has both vertices in the same quadrant.
How does hyperbola work?A hyperbola is a conic section in analytical geometry that is formed when a plane intersects a double right circular cone at an angle that spans both cone halves.
Let's see each option one by one:
(a) We've a = 11 and b = 10
This hyperbola is centered at (-10,3) and has vertices as (1,3) and (-21,3).
So, both vertices are not in the same quadrant.
(b) We've a = 9 and b = 12
This hyperbola is centered at (-6,8) and has vertices as (-6,17) and (-6,-1).
So, both vertices are not in the same quadrant.
(c) We've a = 12 and b = 8
This hyperbola is centered at (1,11) and has vertices as (13,11) and (-11,11).
So, both vertices are not in the same quadrant.
(d) We've a = 9 and b = 11
This hyperbola is centered at (-1,10) and has vertices as (-1,19) and (-1,1).
So, both vertices in the same quadrant.
Hence, option (d) is the correct answer.
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What is the sequence of transformations that maps △ABC to △A′B′C′? Select from the drop-down menus to correctly identify each step. Step 1: Choose... Step 2: Choose... Two triangles on the coordinate plane. Triangle A B C has vertex A at negative 1 comma 2, vertex B at negative 2 comma 5, and vertex C at negative 5 comma 1. Triangle A prime B prime C prime has vertex A prime at 3 comma negative 2, vertex B prime at 4 comma negative 5, and vertex C prime at 7 comma negative 1.
The sequence of transformation which includes a reflection and a translation that maps triangle ΔABC to ΔA'B'C' are;
A reflection across the y-axisA translation of 2 units to the rightA reflection across the x-axisWhat is a reflection transformation?A reflection transformation is one in which the mirror image of the figure is produced, on the other side of a line known as the line of reflection.
The vertices of the triangle ΔABC are A(-1, 2), B(-2, 5), C(-5, 1), the vertices of triangle ΔA'B'C' are A'(3, -2), B'(4, -5), C'(7, -1)
A reflection of the triangle ΔABC across the y-axis, we get;
A''(1, 2), B''(2, 5), C''(5, 1)
A reflection of the triangle ΔA'B'C', across the x-axis produces;
A'''(3, 2), B'''(4, 5), C'''(7, 1)
The rule that maps the coordinates of the vertices of ΔA''B''C'' to the coordinates of the vertices of ΔA'''B'''C''' is a translation of triangle 2 units to the right,
Therefore, the sequence of transformation that maps ΔABC to ΔA'B'C' are;
A reflection across the y-axis, followed by ⇒A translation of two units to the right, then ⇒ A reflection across the x-axis.
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Provide reasons for the proof.
Given:22 24
and 22 and 23 are supplementary
Prove: 21 23
Answer:
1.22 24
2. m/2 = m/4
3. 22 and 23 are supplementary
Statement
4. m22+ m23 = 180°
5. 21 and 24 are supplementary
6. m21+ m24 = 180°
7. m21+ m24 = m/2 + m23
8. m21+ m24=m24+ m23
9. m/1 = m23
10. 21 23
T
1. Given
3. Given
V
7.
P
8.
3
2. Angle congruence postulate
10.
2
M
Reason
4. Definition of supplementary angles
6. Definition of supplementary angles
Here proof i.e. ∠1=∠3 is also given in the image for the given geometric shape.
What are geometric shapes?
In Mathematics, Geometric shapes are the figures which demonstrate the shape of the objects we see in our everyday life. In geometry, shapes are the forms of objects which have boundary lines, angles and surfaces. There are different types of 2d shapes and 3d shapes.
Shapes are also classified with respect to their regularity or uniformity. A regular shape is usually symmetrical such as a square, circle, etc. Irregular shapes are asymmetrical. They are also called freeform shapes or organic shapes. For example, the shape of a tree is irregular or organic.
In plane geometry, the two-dimensional shapes are flat shapes and closed figures such as circles, squares, rectangles, rhombus, etc. In solid geometry, the three-dimensional shapes are cube, cuboid, cone, sphere and cylinder. We can observe all these shapes in our daily existence also. For example books (cuboid shape), glasses (cylindrical shape), traffic cones (conical shape) and so on.
Now
To prove ∠1=∠3
Here as given ∠1+∠4=180°
and also ∠2+∠3=180° (supplementary angles)
therefore,
∠1+∠4=∠2+∠3
and also ∠2=∠4 ( complementary angles)
Hence,
∠1=∠3
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Find a function that represents the amount of air required to inflate the spherical balloon from a radius of r inches to a radius of r+1 inches.
A function that represents the amount of air that is f(r)=4/3π[3r²+3r+1]
Since a function is a rule that applies a specific output to a specific input, we can define a function in a variety of ways. This could involve a change in variables, for example, or defining a function that represents a rate of change.
The amount or volume of air required to inflate the spherical balloon from a radius of r inches to a radius of r+1 inches.
The formula for the volume of a sphere is V = 4/3 πr³
where V = volume and r = radius.
V= 4/3π[(r+1)³−r³]
V=4/3π[r³+3r²+3r+1−r³]
V =4/3π[3r²+3r+1]
For r in inches and V(r) in inches cubed,
f(r)=4/3π[3r²+3r+1].
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Consider a function f defined as follows: f(x)=(x-3)^2, x belongs to [1,2]. Is f invertible in the domain [1,2]? If yes, find the inverse of the function.
(Looking for an complete explanation on how to prove invertible, probably by showing the function is one-to-one and then determining the inverse. )
The inverse of the function is f⁻¹(x)=3+√18+x, 3-√18+x
What is a function?A relation is a function if it has only One y-value for each x-value.
Let us consider the function f(x)=(x-3)²
Where x belongs to [1, 2]
f(x)=x²+9-6x
y=x²+9-6x
Interchange the variables.
x=y²+9-6y
y=3+√18+x
y=3-√18+x
Relace y with f⁻¹(x)
f⁻¹(x)=3+√18+x, 3-√18+x
Hence, the inverse of the function is f⁻¹(x)=3+√18+x, 3-√18+x
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4 (Simple Interest and Arithmetic Sequences MC)
After hip surgery, your physical therapist tells you to slowly return to walking. The therapist suggests walking for 5 minutes each day for the first week and increasing that time by 5 minutes per day each week thereafter. How many weeks will it be before you are up to 45 minutes of walking per day?
10
11
9
13
The number of weeks, it will take before it is up to 45 minutes of walking per day is, 9 weeks.
What is an arithmetic sequence?In the arithmetic sequence the common difference remains constant between any two consecutive terms. In this sequence we can get the next term by adding one fixed value which is known as a common difference. It is also known as arithmetic progression(AP).
Formula for any nth term in AP,
T = a + nd
d = T₂-T₁
Where, a is first term , n is nth term and d is common difference
Given that,
The therapist suggests walking for 5 minutes each day for the first week
Increasing that time by 5 minutes per day each week
Number of weeks will it be before you are up to 45 minutes of walking per day = ?
At the end of first week, he will walk for 5 minute
in the beginning of second he started increasing time 5 minutes per day
So, before he reach up to 45 minutes weeks passed,
Before up to 45 minutes,
number of weeks = ?
45 = 5 + 5n
n = 8
Total number of weeks of walking = 1 + 8
= 9
Hence, the number of weeks is 9
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Laura rents out small and large trucks. She has a total of 22 trucks. The number of large trucks is two more than triple the number of small trucks. how many of each type of truck does she have?
Answer:
see below
Step-by-step explanation:
S+L = 22
L=2+3S
so S+2+3S =22
4S=22-2 = 20
S =5
L =17
Answer:
Step-by-step explanation:
A rock is thrown upward with a velocity of 18 meters per second from the top of a 36 meter high cliff, and it misses the cliff on the way back down. when will the rock be 4 meters from the water, below? round your answer to two decimal places.
The time that is taken to fall is 3.1 s.
What is the time required?In this case, we have been told that A rock is thrown upward with a velocity of 18 meters per second from the top of a 36 meter high cliff, and it misses the cliff on the way back down.
We have to know that the maximum height that the rock has attained by the use of the formula;
v^2 = u^2 - 2gh
v = final velocity
u = initial velocity
g = acceleration
h= distance
v = 0 m/s at the maximum height
h = u^2/2g
h = (18)^2/2 * 9.8
h = 324/19.6
h = 16.5 m
The height from the cliff upwards is; 16.5 m + 36 m = 52.5 m
The distance that is covered when falling is; 52.5 m - 4 m = 48.5 m
If it is falling then it is dropped from a height and u = 0 m/s
h = 1/2gt^2
48.5 = 0.5 * 9.8 * t^2
t = √48.5/0.5 * 9.8
t = 3.1 s
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