Annie made 4 glasses of juice,her 5 friends drank 1/2 glass each.what fraction of juice was left? how to solve it

A. The value of the function at x = 8 will be 4. Then the correct option is B.

B. The average rate of change of the function in the interval of [2, 6]  will be 24. Then the correct option is D.

C. The total mass of 5,000 of these microorganisms will be 0.012 g. Then the correct option is C.

A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.

A. The function is f(x) = 8 / (x - 6), then the value of the function at x = 8 is given as,

f(8) = 8 / (8 - 6)

f(8) = 4

Then the correct option is B.

B.  The function is f(x) = 3x² - 5, then the average rate of change of the function in the interval of [2, 6] is given as,

Average rate = [3(6² - 4²)] / (6 - 2)

Average rate = 24

Thus, the correct option is D.

C.  The average mass of a certain type of microorganism is 2.4 × 10⁻⁶ grams. Then the total mass of 5,000 of these microorganisms is given as,

⇒ 2.4 × 10⁻⁶ × 5,000

⇒ 0.012 grams

Thus, the correct option is C.

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A graph of the equation under this condition {(x, y): x + y = 5)} is shown in the image attached below.

In Mathematics, an ordered pair simply refers to a pair of two (2) elements or data points that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph.

Next, we would determine the ordered pairs that satisfies the given linear function in standard form.

When the value of x = 1, the output value of this linear function is given by;

y = 5 - x

y = 5 - 1

y = 4 ⇒ (1, 4)

When the value of x = 0, the output value of this linear function is given by;

y = 5 - x

y = 5 - 0

y = 5 ⇒ (0, 5).

When the value of x = 2, the output value of this linear function is given by;

y = 5 - x

y = 5 - 2

y = 0 ⇒ (2, 3).

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Suppose that the functions q and r are defined as q(x) = 2x+2 and r(x) = x² +1, the function  (r * q)(4) =(q * r)(4) and they are equal to 170.

Which is the commutative law?

commutative law, in mathematics, is either of two laws relating to number operations of addition and multiplication that are stated symbolically as a + b = b + a and ab = ba. From these laws, it follows that any finite sum or product is unaltered by reordering its terms or factors.

Given the following functions:

q(x) = 2x+2

r(x) = x² +1

Taking the product

q(x)*r(x) = (2x + 2)(x² +1)

Note that according to the communatative law, (r * q)(x) = (q * r)(x)

(r * q)(4) =(q * r)(4)  = (2(4)+2)((4)² +1)

(r * q)(4) =(q * r)(4)  = (10)(16+1)

(r * q)(4) =(q * r)(4)= 10(17)

(r * q)(4) =(q * r)(4) = 170

Hence, the function  (r * q)(4) =(q * r)(4) and they are equal to 170.

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Since the domain has 4 elements and the range has 2 elements, each element in the domain can be mapped to one of the two elements in the range. Thus, there are a total of [tex]$2^4 = 16$[/tex] functions from the domain to the range.

What is functions ?

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the codomain or range) with the property that each input is related to exactly one output. Functions can be represented in many ways, including algebraic expressions, graphs, tables, or verbal descriptions.

Functions are widely used in various fields of mathematics, science, and engineering to model and analyse real-world phenomena, and to make predictions or decisions based on data.

There are 4 elements in the domain and an empty set in the codomain.

Using two-line notation, the functions are:

There is only one possible image for each element in the domain, which means that there is only one function.

Since there is only one function, it is both surjective and injective. Therefore, there is only one bijective function.

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The lengths of the sides of the pentagon are;

12 Inches

13 inches

14 inches

15 inches

16 inches

The formula for the perimeter of a pentagon is expressed with the formula;

P = 5a

Given that the parameter are;

Since the sides are consecutive integers, then;

60 = 5(a)

expand the bracket

60= 5a

Make 'a' the subject

a = 12In

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Answer: See the attached image below for the proof table.

Explanation:

We start with what we're given. Simply repeat the given information word for word. This is statement 1 of the proof.

Statement 2 mentions AB is parallel to CD. This is valid due to the definition of a parallelogram. The name means "two sets of opposite parallel sides. Statement 3 is a similar situation.

Statement 4 relies on statement 2. The red angles in the diagram are congruent alternate interior angles. They are only congruent because of AB parallel to CD.

Statement 5 relies on statement 3 for similar reasoning as the previous paragraph. Focus on the blue angles.

Statement 6 is hopefully self explanatory so there's not much for me to mention here.

Statement 7 has us connect statements 4, 5, and 6 to prove triangle ADB is congruent to triangle CBD. We use the ASA (angle side angle) theorem here.

Statements 8 and 9 finish up the proof. CPCTC stands for "Corresponding parts of congruent triangles are congruent". If two triangles are congruent, then their corresponding pieces must be congruent as well. It would be like saying two houses are identical if and only if their front doors are identical.

The theoretical probability that a sum greater than 10 is obtained is given as follows:

p = 1/12.

Out of 108 trials, the expected number of trials with a sum greater than 10 is given as follows:

9.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The 36 total outcomes for the pair of dices is given as follows:

Out of these outcomes, three of them, (5,6), (6,5) and (6,6) have a sum greater than 10, hence the probability is given as follows:

p = 3/36

p = 1/12.

Then the expected number is given as follows:

E(X) = 108 x 1/12

E(X) = 9.

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Answer:

9

Step-by-step explanation:

experiment, Kim rolled a fair die twice. The theoretical probability of both rolls equaling a sum greater than 10 is Predict how many times.

Therefore the experement is 9

The graph of the quadratic function is on the image at the end.

Here we want to graph the quadratic function:

f(x) = -x² - 2

To do so, we can complete the table of values by evaluating the function.

if x = ±2 then:

f(±2) = -(±2)² - 2 = -4 - 2 = -6

if x =  ±1

f(±1) = -(±1)² - 2 = -1 - 2 = -3

if x = 0

f(0) = -(0)² - 2 =  - 2 = -2

Then we have the points:

(-2, -6), (-1, -3), (0, -2), (1, -3), (2, -6)

Now just graph these points and connect them with a curve. The graph of the parabola is below.

The volume of the candle is 6740 cubic cm

From the question, we have the following parameters that can be used in our computation:

Radius, r = 9 cm

Height, h = 26.5 cm

The volume of a cylinder is calculated as

Volume = πr²h

substitute the known values in the above equation, so, we have the following representation

Volume = 3.14 * 9² *  26.5

Evaluate the products and exponents

Volume = 6740 cubic cm

HEnce, the volume is 6740 cubic cm

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The ratio of the number of sixth-graders who visited the science museum to the number of sixth-graders who visited the history museum is

10:39.

What is meant by the ratio of two quantities?

A ratio displays the multiplicity of two numbers. Mathematicians use the term "ratio" to compare two or more numbers. It serves as a comparison tool to show how big or tiny an amount is in relation to another. Two quantities are compared using division in a ratio. In this case, the dividend is referred to as the "antecedent" and the divisor as the "consequent." In general, a: b, which can be interpreted as "a is to b," is used to denote a ratio between two quantities, let's say "a" and "b."

Given,

  Number of students who picked the science museum = 10

Number of students who picked the history museum = 39

The ratio of students who visited the science museum to students who visited the history museum = 10:39.

Therefore the ratio of the number of sixth-graders who visited the science museum to the number of sixth-graders who visited the history museum is 10:39.

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Answer:

16x(1+2x^2)^3

Step-by-step explanation:

f(x)=(1+2x^2)^4

Use chain rule

4(1+2x^2)^3(4x)

Answer:

It depends on the amount of slices.

Step-by-step explanation:

The diameter of the pie is 10 inches. Multiply this by π and we get approximately 31.4. Basically just divide it by the number of slices. For example, if there were six slices, 31.4/6=5.23, so that's the arc length of one slice.

Answer:

5(3-2)

Step-by-step explanation:

15-10

5×3-5×2

The total surface area of cylinder formed is   924 cm²              

We are given a rectangular sheet of paper of dimensions 44 cm* 14 cm is rolled along its length

So, height of formed cylinder = Length of rectangular sheet = 14 cm

Circumference of base of cylinder = Breadth = 44 cm

Circumference of circle = 2πr

So,

2 × (22/7) × r =44

r = (44×7)/(22×2)

Radius of cylinder is 7 cm

Height = 14 cm

Total surface area of cylinder = 2πr(h+r)

                                                =    2  × 22/7 × 7 (14 +7 )                                      

                                                =     924 cm²                  

Hence the total surface area of cylinder formed is   924 cm²  .  

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The most specific name for the quadrilateral is trapezoid, the correct option is C.

A quadrilateral is defined as a two-dimensional shape with four sides, four vertices, and four angles. There are two main types: concave and convex. There are also various subcategories of convex quadrilaterals, such as trapezoids, parallelograms, rectangles, rhombi, and squares.

Given;

The figure with 4 sides and two opposite are equal

Now,

In the figure two opposite sides are equal in length

But they all are not parallel only on pair of opposite sides are parallel.

Also, the figure the adjacent angle are equal.

Therefore, the quadrilateral will be trapezoid.

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Answer:

Rafael and Lucy's credit for retirement plan contributions is $570.

Step-by-step explanation:

The Saver's Credit is a tax credit for low and moderate-income taxpayers who make contributions to a retirement plan. The credit is worth between 10% and 50% of the taxpayer's contributions, up to a maximum credit of $1,000 ($2,000 for married taxpayers filing jointly).

To determine the credit amount, we need to calculate the taxpayers' adjusted gross income (AGI) and their credit rate.

For the tax year in question, the AGI is $40,500 and the contribution limit is $2,850 per person. The total contribution amount is $2,850 x 2 = $5,700.

The credit rate is determined by the taxpayer's AGI and filing status, as follows:

For AGI up to $19,750 for married taxpayers filing jointly, the credit rate is 50%.

For AGI between $19,750 and $32,000 for married taxpayers filing jointly, the credit rate is 20%.

For AGI between $32,000 and $40,000 for married taxpayers filing jointly, the credit rate is 10%.

For AGI above $40,000 for married taxpayers filing jointly, the credit is not available.

Since the taxpayers' AGI is $40,500, the credit rate is 10%. The credit amount is calculated as follows:

Credit amount = Credit rate x Contribution amount

Credit amount = 10% x $5,700 = $570

So, Rafael and Lucy's credit for retirement plan contributions is $570.

Answer:

The Empirical Rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% of the data falls within two standard deviations of the mean, and approximately 99.7% of the data falls within three standard deviations of the mean.

In this case, the mean speed is 70 mph and the standard deviation is 3 mph. So, we can estimate the percent of vehicles whose speeds are between 64 mph and 76 mph as follows:

64 mph = 70 mph - 3 mph * 1

76 mph = 70 mph + 3 mph * 1

Since the speeds are within one standard deviation of the mean, we can use the Empirical Rule to estimate that approximately 68% of the vehicles have speeds between 64 mph and 76 mph.

So, the answer is approximately 68%.

Jean purchased a snow thrower for $1,980 using an installment plan that required a 10% down payment of $198 and 18 monthly payments of $116. This allowed Jean to spread out the cost of the snow thrower over time, avoiding a large upfront cost.

Jean recently purchased a snow thrower for $1,980 using an installment plan. To start, Jean had to make a down payment of 10% of the total purchase price, which was $198. This down payment allowed Jean to spread the remaining cost of the snow thrower across 18 monthly payments of $116 each. After making all 18 payments, Jean will have paid the full purchase price of $1,980. This installment plan option is an excellent way for Jean to get the snow thrower he needs while avoiding a large upfront cost and spreading the payments out over a period of time. With this installment plan, Jean can pay off the purchase of the snow thrower while still having money left over each month to cover other expenses.

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The expression that represents the area of each section is 14z-9

The expression to denote the area of each section would be: 14z -9

Given that,

The expression for the area of plot  area 84z -54

The number of equal sections in which it is divided 6

To find,

The expression for each section = ?

The area of each section = (total area of the garden(/number of equal sections

= (84x-54)/6

= 6(14z-9)/6

=14z - 9

Thus,  is the expression that would represent every section's area.

Answer:

= (4x^2 + y^2)(2x + y)(2x - y).

Step-by-step explanation:

16x^4 - y^4

= (4x2)^2 - (y^2)^2

= (4x^2 +y^2)(4x^2 - y^2)

=(4x^2 + y^2)( (2x)^2 -y^2)

= (4x^2 + y^2)(2x + y)(2x - y).

Answer:

Step-by-step explanation: [tex]16x^{4} -4y^{4} = (4x^{2} )^{2} -(2y^{2} )^{2}= (4x^{2} + 2y^{2} )(4x^{2} - 2y^{2})[/tex]

identity used =

[tex](a^{2} -b^{2} ) = (a+b)(a-b)[/tex]

A rectangle that has been divided into equal parts is referred to as a rectangular fraction. Depending on how many equal pieces you divide the whole into, the equal parts may be divided into halves, thirds, fourths, and so on.

1/2 of 2/3 is equal to 1/3

What is the equation?

The equation is defined as a mathematical statement with at least two terms that contain variables or values that are equal.

To represent the multiplication of 1/2 and 2/3, we can draw a rectangle and divide it into equal parts.

1) First, we draw a rectangle.

2) Then, we divide it into 2 equal parts horizontally.

3) Next, we divide each half vertically into 3 equal parts.

4) Now we can see that each small rectangle represents 1/6 of the whole rectangle.

So, 1/2 of 2/3 is equal to 1/2 of 1/3.

Therefore, 1/2 of 2/3 is equal to 1/3.

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The company’s revenue (R) from a particular distributor that buys x units is R(x) = 8999.93x

Algebra is the branch of mathematics that helps in the representation of problems or situations in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, subtraction, multiplication, and division to form a meaningful mathematical expression.

Given:

original price = $9000 per unit.

let the number units bought be x

then amount= 9000x.

If the price reduces the list price by seven cents times the number of units bought .

7cents= 0.07 dollar

Then,

R(x) = 9000x - 0.07x

R(x) = 8999.93x

Hence, R(x) = 8999.93x.

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The solution is, 12 (long division) 924 = 77.

Division is the process of splitting a number or an amount into equal parts.

Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication.

here, we have,

given that,

12 (long division) 924

so, we get,

12 | 924  | 77

     84

----------------

       84

       84

-----------------

       0

Hence, The solution is, 12 (long division) 924 = 77.

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The sentence that uses the word "obstinancy" incorrectly is: if we eat out too often, its result in obstinancy and poor health. The Option C is correct.

Obstinancy is a noun that refers to the quality of being stubborn or resolutely unyielding. It describes a person's refusal to change their views, opinions, or actions, even in the face of compelling evidence or persuasive arguments. Example of its use: despite the pleas of his family, the man's obstinancy led him to continue his dangerous lifestyle.

We often use the word "obstinancy" for the followings:

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1) Note that the altitude of the top of the hill is -136 feet above sea level.

2) Note that Checkpoint 1 is 3,558 feet higher than Checkpoint 5.

Altitude or height is a distance measurement between a reference datum and a point or object, generally in the vertical or "up" direction. The precise definition and reference datum vary depending on the circumstance.

1) Checkpoint 4 has an altitude of -202 feet above sea level. The top of the hill rises 66 feet above Checkpoint 4. Therefore, the altitude of the top of the hill is:

-202 + 66 = -136 feet

So, the altitude of the top of the hill is -136 feet above sea level.

2) To find the difference in altitude between Checkpoint 1 and Checkpoint 5, we need to subtract the altitude of Checkpoint 5 from the altitude of Checkpoint 1.

Altitude difference = Altitude of Checkpoint 1 - Altitude of Checkpoint 5

Substituting the values given in the table, we get:

Altitude difference (AD) = 3,459 - (-99)

AD = 3,459 + 99

Thus, AD = 3,558

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The point that lies in the solution to the system of inequalities is (5,-2). The correct answer would be an option (B).

An Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.

The system of inequalities is given in the question, as follows:

y ≤ x -5 and y ≥ -x-4

Since the required point is to satisfy the system of inequalities.

Then the graph attached below represents a solution to the system of inequalities.

Point A (-5,2) does not lie in the solution to the system of inequalities.

Point B (5,-2) lies in the solution to the system of inequalities.

Point C (-5,-2) does not lie in the solution to the system of inequalities.

Point D (5,2) does not lie in the solution to the system of inequalities.

Therefore, the point that lies in the solution to the system of inequalities is (5,-2).

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[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{25\% of 79600}}{\left( \cfrac{25}{100} \right)79600}\implies \text{\LARGE 19900}[/tex]