2.8.AP-1 Where Does The Line Cross The Y-axis? Please Help (2024)

The function f(x) = [tex]-3x^2[/tex] + 18x + 3 has the maximum value of 30 at x = 3

The quadratic function is

f(x) = [tex]-3x^2[/tex] + 18x + 3

The general form of the quadratic function is

y = [tex]ax^2[/tex] + bx + c

If a < 0, the quadratic function has maximum value

Here the value of a = -3

Therefore the function has maximum value.

Maximum value will be at x = -b / 2a

= -18 / 2(-3)

= -18 / -6

= 3

The maximum value of quadratic function will be (4ac - [tex]b^2[/tex]) /4a

Substitute the value in the equation

(4ac - [tex]b^2[/tex]) /4a = (4×-3×3 - 18×18) / 4×-3

= (-36 - 324 / -12

= -360 / -12

= 30

Hence, the function f(x) = [tex]-3x^2[/tex] + 18x + 3 has the maximum value of 30 at x = 3

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Its true that major difference simple random samples and systematic samples is that the latter are not selected randomly.

Simple random samples are those samples which are selected from the population totally randomly

here , each data point have equal probability of being chosen.

The data point are independent to each other. This sampling technique requires the reach throughout the total scope of the population.

That's why for sample random sample population size should be small for effective result

whereas , Systematic samples are those samples which are selected in a proper systematic order .

Here , only the starting point is being selected randomly and then next data points are selected as pre-defined order.

So , The major difference between simple random samples and systematic samples is that the latter are not selected randomly.

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The probability that both balls drawn have the same color if the first ball is replaced before the second is drawn will be 0.5165 that is option E.

What is probability?

The area of mathematics known as probability explores potential outcomes of events as well as their relative probabilities and distributions. Simply put, probability is the likelihood that something will occur. When we don't know how an event will turn out, we might discuss the likelihood or likelihood of several outcomes. Statistics is the study of occurrences that follow a probability distribution.

Here,

=9/22*9/22+13/22*13/22

=0.5165

If the first ball is replaced before the second is drawn, chance that both balls drawn will be the same color is 0.5165.

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The value of x for the point will be equal to 6.

What is a slope?

Slope or the gradient is the number or the ratio which determines the direction or the steepness of the line. Th slope is also defined as the ratio of the rise to the run of the line on a coordinate plane.

Given that the points are (x,-4) and (2,8) and the slope is m = -3. The slope formula can be written as,

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

Slope = ( 8 + 4 ) / ( 2 - x)

-3 = ( 8 + 4 ) / ( 2 - x)

-6 + 3x = 12

3x = 18

x = 18 / 3

x = 6

Therefore, the value of x for the point will be equal to 6.

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The solution of the inequality is found as x > 16.2.

Explain the term inequality of the number?Inequalities specify the relation between two values which aren't equal. Equal does not imply inequality. Typically, we use the "not equal sign (≠)" to indicate that two values are also not equal. But several inequalities are utilized to compare the numbers, if it is lower than or higher than.

As per the given inequality.

4x - 28. 2 > 36. 6

Add 28.2 on both sides

4x - 28. 2 + 28.2 > 36. 6 + 28.2

4x > 64.8

Divide both sides by 4

x > 64.8/4

x > 16.2

Thus, the solution of the inequality is found as x > 16.2.

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Question 1

(12-3) ÷ (6-1) ? 4 • (20-3) - 4 • 39 ÷ 5 ? 4 • 17 - 4 • 31.8 ? 4 • (17 - 3)1.8 ? 4 • 141.8 < 56

Question 2

(a) 0.8n = 12(b) n = 12/0.8 = 15

Question 3

(a) c + 2.47 = 9.98(b) c = 9.98 - 2.47 = 7.51(c) c = 7.51 is the initial cost of the toy

Answer:

y - 4 = 2/3(x - 6)

OR

y = 2/3x

Step-by-step explanation:

1. Use point slope form:

y - y1 = m(x- x1)

y1 = y coordinate in point

x1 = x coordinate in point

m = Slope

2. Substitute:

y - 4 = 2/3(x - 6)

3. Solve for y (if necessary)

y = 2/3x - 4 + 4

y = 2/3x

The length of the inner rectangle is 2 foot

Consider the length of the inner rectangle = x

The area of the inner rectangle = x × 1

= x unit square

The dimensions of the middle rectangle = (x+2) and 3

The area of the middle rectangle = 3 × (x+2)

= 3x+ 6

The area of the shaded region = The total area of the middle rectangle - The area of the inner rectangle

= 3x+6 - x

= 2x + 6

The dimensions of the outer rectangle = (x+4) and 5

The area of the outer rectangle = 5(x+4)

= 5x + 20

The area of the shaded region of outer rectangle = 5x + 20 - (3x+6)

= 5x+ 20 - 3x -6

= 2x + 14

The areas of the three differently colored regions form an arithmetic progression

Then the they have common difference

2x+6 - x = 2x+14 - (2x + 6)

x + 6 = 8
x = 8 -6

x = 2

Hence, the length of the inner rectangle is 2 foot

The complete question is

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is 1 foot wide on all four sides. What is the length in feet of the inner rectangle

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The radius of soap bubble is 5.702 mm and diameter of soap bubble is 11.404 mm.

What is the formula for spherical volume?The volume will be calculated by multiplying the surface area by the diameter, with every point on its surface being equidistant from its centre. The following formula is used to compute the volume of a sphere mathematically: A sphere's volume is equal to 4/3(π)r³, where r is the radius of the sphere.

Given: The volume of a soap bubble is 775. 3 mm³

We need to find the radius and diameter of the soap bubble.

We know that,

Volume of soap bubble = 4/3(π)r³

775.3 = 4/3(3.14)(r³)

r³= 185.47

r = [tex]\sqrt[3]{185.47}[/tex]

r = 5.702 mm

Radius of soap bubble = 5.702 mm

Diameter of soap bubble = 2(5.702)

= 11.404 mm

Therefore, the radius of soap bubble is 5.702 mm and diameter of soap bubble is 11.404 mm.

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Probability that a salesperson makes a bonus given that the salesperson attended the sales training program will be 2/5.

What is probability?

Simply put, probability is the likelihood that something will occur. When we don't know how an event will turn out, we might discuss the likelihood or likelihood of several outcomes. Statistics is the study of occurrences that follow a probability distribution. Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events. Information about the possibility that something will happen is provided by probability. For instance, meteorologists utilize weather patterns to forecast the likelihood of rain. Probability theory is utilized in epidemiology to comprehend the connection between exposures and the risk of health outcomes.

Here,

Total percentage of people in team=100%
the salespeople attended the training program=40%

Probability that a salesperson makes a bonus given that the salesperson attended the sales training program,

=40/100

=2/5

Given that the salesperson completed the sales training session, there is a 2/5 chance that they will receive a bonus.

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The final price after the sale tax is $137.8.

Sale tax:

Sales tax is an amount of money, calculated as a percentage, that is added to the cost of a product or service when purchased by a consumer at a retail location.

Here we have to find the final price after the sale tax.

The amount before the sale tax was $130.

Percentage of sale tax = 6%.

The sale tax = 130 × 6/100

= $7.8

The final price = 130 + 7.8

= $137.8

Therefore the final amount is $137.8.

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The two different names of k are constant and coefficient of x

What are coefficient and constant?

Constants are the terms in an expression that includes only numbers, the value of which does not change. They are mostly represented by 'k' in an expression. for example in variation 'k' is represent by the proportionality constant

Coefficient of a variable is the number written along with the variable in the term.

In the expression k stands as the coefficient of x, if x and y are two different variables and at the same time k is a constant if y varies directly as x

Therefore k stands for both constant and coefficient

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Over the 30 year period, the values of Minneapolis and St. Paul's values will be: Mean 1.580666667 1.769333333 and Standard Deviation 0.890741799 1.123161779

Using the R software or any other graphing software and typing in the, we get the graphs.

The commands were

qqnorm(c(0.32,0.52,0.77,0.81,0.96,1.2,1.31,1.43,1.62,1.87,1.95,2.1,2.48,3,3.37)) and qqnorm(c(0.47,0.59,0.81,0.9,1.18,1.2,1.35,1.51,1.74,1.89,2.05,2.2,2.81,3.09,4.75))

We see there's a slight difference between the two graphs.

For Minneapolis, the curve goes upwards to 0.4

and the for St. Paul the curve goes downwards to 0.4

The term "mean" refers to the mathematical average of a set of two or more numbers. The arithmetic mean and the geometric mean are two types of means that can be computed. By adding all of the numbers in a set and dividing by the total number of numbers in the set, the arithmetic mean is determined. The standard deviation is a measurement of how widely distributed or how much a group of data differ. A low standard deviation indicates that the values typically tend to be close to the set mean, while a high standard deviation suggests that the values are dispersed over a wider range.

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a. If 1% of the shells are actually defective and we assume Independence, the probability of 0 defective shells in the sample is 82%,

b. if 1% of the shells are actually defective and we assume independence, the probability of 1 defective shell in the sample is 16% and

c. If 1% of the shells are actually defective and we assume Independence, the probability of more than 1 defective shell in the sample is 1.58%.

Given batch = 350 shells

sample = 20 shells

Given 1%10 f shells are defective = 0.01

That is 99% are non detective = 0.99

Using binomial theorem,

1. P(zero defective shells) = 20c0 (0.01)^0(0.99)^20

= 0.82

= 82%

2. P(One defective shells) = 20c1(0.01)(0.99)^19

= 0.16

= 16%

3. P(two defective shells) = 20c2(0.01)^2(0.99)^18

= 0.0158

= 1.58%

Hence the a. If 1% of the shells are actually defective and we assume Independence, the probability of 0 defective shells in the sample is 82%, b. if 1% of the shells are actually defective and we assume independence, the probability of 1 defective shell in the sample is 16% and c. If 1% of the shells are actually defective and we assume Independence, the probability of more than 1 defective shell in the sample is 1.58%.

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The correct statement is option (D) No, you forgot to add in 15% for what is deducted in taxes before you divided by 8.

The additional amount that you needed = $1140

The payment per hour = $8

Consider the number of hours that he worked = x

The tax percentage = 15%

$1140 is 85% of the amount that you needed to save

Then the total amount that you need to save is

x × (85/100) = 1140

x × 0.85 = 1140

x = 1140 / 0.85

x = $1341.18

Hence, the correct statement is option (D) No, you forgot to add in 15% for what is deducted in taxes before you divided by 8.

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a = solution at 20% acid

b = solution at 35% acid

c = solution at 80% acid

now, we know she wants to use two times as much of "c" as there's of "b", which is a mumble jumble for saying whatever "b" is, well "c" is twice that much, namely c = 2b. We also know the whole mixture will be 108 liters of 40% acid.

let's convert those percentages to decimals and let's see how much all those liters are

[tex]\begin{array}{lcccl} &\stackrel{liters}{quantity}&\stackrel{\textit{\% of }}{amount}&\stackrel{\textit{liters of }}{amount}\\ \cline{2-4}&\\ \textit{20\% sol'n}&a&0.20&0.2a\\ \textit{35\% sol'n}&b&0.35&0.35b\\ \boxed{\textit{80\% sol'n}}&c&0.80&0.8c\\ \textit{80\% sol'n}&2b&0.80&1.6b\\ \cline{2-4}&\\ mixture&108&0.40&43.2 \end{array}~\hfill \begin{cases} a+b+\stackrel{c}{2b}=108\\\\ 0.2a+0.35b+1.6b=43.2 \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{using the 1st equation}}{a+b+2b=108\implies} a+3b=108\implies 3b=108-a \\\\\\ b=\cfrac{108-a}{3}\implies \boxed{b=36-\cfrac{a}{3}} \\\\\\ \stackrel{\textit{substituting on the 2nd equation}}{0.2a+0.35b+1.6b=43.2}\implies 0.2a+1.95b=43.2[/tex]

[tex]0.2a+1.95\left( 36-\cfrac{a}{3} \right)=43.2\implies 0.2a+70.2-0.65a=43.2 \\\\\\ 0.2a-0.65a=-27\implies -0.45a=-27\implies a=\cfrac{-27}{-0.45}\implies {\Large \begin{array}{llll} a=60 \end{array}} \\\\\\ \stackrel{36-\frac{60}{3}}{{\Large \begin{array}{llll} b=16 \end{array}}} \hspace{5em} \stackrel{2(16)}{{\Large \begin{array}{llll} c=32 \end{array}}}[/tex]

The range of the relations (-8, -3), (-3, -8), (-1, 6) and (4, 2) is

-8 ≤ y ≤ 6

What is range?

Range is dictated by the output variables which are plotted in the y coordinates or represented by y in ordered pairs

considering the given ordered pairs (-8, -3), (-3, -8), (-1, 6) and (4, 2) the range is the possible extents of the numbers

Examining the numbers the least number is -8 while the highest is 6, hence the range will be values from -8 to 6

This can be written as -8 ≤ y ≤ 6 or {-8, 6}

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The team's total salary decreased by 24.14%.

A percentage is a number that tells us how much something is out of 100, and it can be written as a decimal or a fraction.

Given that,

A team's total salary is $4,614,244.

The team wishes to reduce the total to $3.5 million = $3500000

We need to determine the percentage decrease in the team's total salary.

The percentage decrease/increase in any value is given by :

Percentage decrease = (original value - new value/original value) x 100

Here original value = $4,614,244 and new value = $3500000

= [(4614244 - 3500000)/4614244] x 100

= (1114224/4614244) x 100

= 0.2414 x 100

= 24.14%

Therefore, the percentage decrease in the team's total salary is 24.14%.

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The volume of the given solid that is bounded by the coordinate planes and the plane 9x + 5y + z = 45 is 45 units³.

What is volume?Volume is a measurement of three-dimensional space that is occupied. It is frequently expressed quantitatively using SI-derived units, as well as several imperial or US-standard units. Volume and the notion of length are connected.For instance, a cup's capacity is stated to be 100 ml if it can hold 100 ml of water in its brim. The quantity of space occupied by a three-dimensional object can also be used to describe volume.

So, we have: 9x + 5y + z = 45

Let us consider z = 0.

Then we have:

9x + 5y = 45

Now, we obtain:
x = 45/9 - y

OR
y = 45/5 - x

Hence,

45/9 × 45/5

2,025/45

45 units³

Therefore, the volume of the given solid that is bounded by the coordinate planes and the plane 9x + 5y + z = 45 is 45 units³.

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The relationship between the amounts loaned from each bank be, x + y = 9000

and the relationship between the interests earned for the first year be, 3x + 8y = 60000

Given, John is taking out student loans from two banks.

The first bank offers 3% apr compounded annually.

The second bank offers 8% apr compounded annually.

John took out a combined total of $9000 in loans and the total interest for the first year was $600.

Let x represent the amount of money loaned at 3%. let y represent the amount loaned at 8%. a.

(i) amount from first bank be, x

and the amount from second bank be, y

so, x + y = 9000

(ii) interest for the first year from first bank is,

3x/100

and the interest for the first year from second bank is,

8y/100

so, 3x/100 + 8y/100 = 600

3x + 8y = 60000

Hence, the relationship between the amounts loaned from each bank be, x + y = 9000

and the relationship between the interests earned for the first year be, 3x + 8y = 60000

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The results for the variable t in each linear equation are listed below:

t = - 1t = - 24t = - 3t = - 2t = - 10

How to solve linear equations by algebra properties

In this question we have linear equations with a single variable that must be cleared by using algebra properties. Now we proceed to solve each equation for the variable t:

Equation 1

16 · t + 4 · (5 - t) = 8

16 · t + 20 - 4 · t = 8

12 · t = - 12

t = - 1

Equation 2

18 = 2 · t - 3 · (t + 2)

18 = 2 · t - 3 · t - 6

24 = - t

t = - 24

Equation 3

- 10 · t - (t + 5) = 28

- 10 · t - t - 5 = 28

- 11 · t = 33

t = - 3

Equation 4

- 6 · (3 · t + 5) = 6

- 18 · t - 30 = 6

- 18 · t = 36

t = - 2

Equation 5

12 = 3 · (t + 14)

12 = 3 · t + 42

3 · t = - 30

t = - 10

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Answer: 1.875 gallons a day

Step-by-step explanation:

Answer: The correct answer is 1.875 Gallons leaked in 1 day

Step-by-step explanation:

Rate = 1.25 cups of water per hour

Time = 24 hours (1 day)

Total = Rate x Time

Total = 1.25 x 24

Total = 30 cups

Convert to gallons:

16 cups = 1 gallon

Total in gallons = 30 / 16 = 1.875 Gallons

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Answer:

60 kilometers

Step-by-step explanation:

(7/3) × 140 = 60

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Answer: 60 km

Step-by-step explanation: To find how far the car travels on 1 liter of petrol, do 140 divided by 7 to get 20 km. But since it wants to know how far it goes on 3 liters, we can multiply 20 by 3 to get 60 km.

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Answer: $2321

Step-by-step explanation:

Sales tax means that the total price is the original price, plus the tax.

The tax is 5.5%, so to get the tax we multiply the sales price by 5.5% and add that to the original price.

To turn a percent into a decimal that we can use on a calculator, simply move the percentage's decimal point two spaces to the left.

5.5 --> 0.55 --> 0.055.

Now multiply 2200 by 0.055

2200 * 0.055 = 121

2200 + 121 = 2321

Using the given point and slope the equation is y = 11/3 x -7

To find slope

y= -3/11x+5/11

Compare this equation with y = mx +c.

slope = m

y-intercept = b

So, m=-3/11

If a line is perpendicular then,

m₁ * m₂ = -1

-3/11 * m₂ = -1

Slope, m₂ = 11/3

Now, one slope and one point formula:

(y - y₁) = m (x - x₁)

Given

point (3,4) and slope = 11/3

(y - 4) = 11/3 (x - 3)

y = 11/3 x - 11 + 4

y = 11/3 x -7

Hence, using the given point and slope the equation is y = 11/3 x -7

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The substitution property is used then x is 4 .The substituted property of equality states that one value may be substituted for another in an expression or equation and the result will be the same.

What is substitution property?

The substituted property of equality states that one value may be substituted for another in an expression or equation and the result will be the same.

If the values of x and y are equivalent, then x and y may be substituted for one another. The property's opposite is likewise true.

For exmaple,

If x=y for any real integer x and y, then y may be replaced for x in any statement.

4+2√3

x is 4 whereas y is 2

In the above instance, x = 4and hence can be substituted for x in the first expression given as:

4+2√3

4-2-√3

= x +y√3find x and y.

The _____ property is used here.

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Instead of attempting to remember each numeral individually, Wayne chooses to memorize the series of four ages. Wayne is employing a tactic known as "chunking."

Define the term called chunking?

In a "chunking" project, students rewrite the "portions" of a challenging material in their own words after breaking it down into more digestible chunks.

This method can be applied to difficult texts of every length. Chunking teaches students how to recognize important words and concepts, improves their paraphrasing skills, and makes it simpler for any of them to organize as well as synthesize material.Chunking is an illustration of a technique that aids pupils in breaking up challenging content into more manageable chunks. The ability to paraphrase is developed as a result of breaking up the content into smaller chunks, which also makes it simpler for students acquire organize and synthesize knowledge.

Thus, when instead of attempting to remember each numeral individually, Wayne chooses to memorize the series of four ages. Wayne is employing a tactic known as "chunking."

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The coordinates of the other endpoint, K is (-6, 31)

Given that, the midpoint of JK is point L at (–1, 8). One endpoint is J(4, –15).

What is the midpoint of a line segment?

A midpoint of a line segment is the point on a segment that bisects the segment into two congruent segments.

We know that, the formula to find the midpoint of line segment is (x, y) = [(x1+x2)/2, (y1+y2)/2].

Let the coordinates of the other endpoint be (x, y)

Now, (-1, 8) = [(4+x)/2, (-15+y)/2]

-1 = (4+x)/2 and 8 = (-15+y)/2

⇒ -2=4+x and 16= -15+y

⇒ x=-6 and y=31

Therefore, the coordinates of the other endpoint, K is (-6, 31)

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