The roots of the quadratic function describing the relationship between number of products produced and overall profit margin are x=0 and 28. The vertex is a minimum at (14,−40).

[tex]\rm \dfrac{ x^2 }{100} + \dfrac{ y^2}{36} = 1[/tex] is the equation of the ellipse.

The equation of an ellipse is given by

[tex]\rm \dfrac{ (x-h)^2 }{a^2} + \dfrac{ (y-k)^2}{b^2} = 1[/tex]

here

(h,k) is the center

c = is the distance between center to focus

c = [tex]\rm \sqrt{a ^2 - b^2}[/tex]

as the a = 10 , component length

and c = 8

8 = [tex]\rm \sqrt{10 ^2 - b^2}[/tex]

8² = 10² -b²

b² = 36

b = 6

h = 0 ,k = 0

Substituting the values given

[tex]\rm \dfrac{ (x-0)^2 }{10^2} + \dfrac{ (y-0)^2}{6^2} = 1[/tex]

[tex]\rm \dfrac{ x^2 }{100} + \dfrac{ y^2}{36} = 1[/tex]

Therefore the equation of the ellipse has been determined.

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Answer:

12 and 14

Step-by-step explanation:

The first number can be assigned x, the second will be assigned x+2 so that it will be even.

3(x)+5(x+2) = 106

3x+5x+10 = 106

8x = 96

x = 12

x+2 = 14

The two consecutive even numbers such that the sum of three times the smaller and five times the larger is 106 are 12 and 14, computed using the linear equation in one variable 3x + 5(x + 2) = 106.

We are given two consecutive even numbers, thus the larger number will be two more than the smaller one, as consecutive even numbers differ by two.

We assume the smaller number to be x.

Thus, the larger number = x + 2.

We have been given that three times the smaller number summed with five times the larger number results in 106.

This can be shown by the linear equation in one variable:

3x + 5(x + 2) = 106.

To find the numbers, we need to solve this linear equation in one variable as follows:

3x + 5(x + 2) = 106,

or 3x + 5x + 10 = 106 {Simplifying},

or, 8x + 10 = 106 {Simplifying},

or, 8x + 10 - 10 = 106 - 10 {Subtracting 10 from both sides of the equation},

or, 8x = 96 {Simplifying},

or, 8x/8 = 96/8 {Dividing both sides of the equation by 8},

or, x = 12 {Simplifying}.

Thus the smaller number, x = 12.

The larger number, x + 2 = 12 + 2 = 14.

Thus, the two consecutive even numbers such that the sum of three times the smaller and five times the larger is 106 are 12 and 14, computed using the linear equation in one variable 3x + 5(x + 2) = 106.

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Answer:

Angle Q is acute.

Step-by-step explanation:

In an obtuse triangle, there is one obtuse angle and the other two angles are acute.

Answer:

SSS similarity

Step-by-step explanation:

From the smaller triangle to the larger triangle, the ratio of each pair of corresponding sides is 2.

Answer:

Option A

Step-by-step explanation:

   [tex]\sf Slope=\dfrac{Change \ in \ y}{Change \ in \ x}=\dfrac{rise}{run}[/tex]

We can find the equation of line, by comparing the slope of line segment AC and slope of AE.

Slope of AC = Slope of AE

           [tex]\sf \dfrac{4}{3}=\dfrac{y +2}{x}[/tex]

Cross multiply

4*x = 3*(y +2)

4x   = 3y + 3*2

4x  = 3y + 6

Equation of the line in  standard form:

         4x - 3y - 6= 0

Answer:

See below

Step-by-step explanation:

Diameter  of the semi circles = 15 - 3-3 = 9mm

   radius = 4.5 mm

area of rectangle tha goes through the semicircles = 5 x 15 = 75 mm^2

area of two semicircles =

                 area of circle with radius 4.5 = pi r^2 = 63.62 mm^2

    total area = 75 + 63.62 = `38.62 mm^2

Perimeter   Two small side rectangles = 2   * (3+5+3) =26 mm

   perimeter of cicle with diameter 9   =  pi * d = 9pi = 28.27 mm

           total perimeter = 54.27 mm

(check my math !)

The length of extra cable that is required to connect the two pieces of existing cable is equal to 182 meters.

In order to determine the length of extra cable that is required to connect the two pieces of existing cable, we would apply the law of cosine as follows:

B² = A² + C² - 2(A)(C)cosB

Substituting the given parameters into the formula, we have;

B² = 325.0² + 430.0² - 2(325.0)(430.0)cos23

B² = 105,625 + 184,900 - 279,500(0.9205)

B² = 290,525 - 257,279.75

B² = 33,245.25

B = √33,245.25

B = 182.33 ≈ 182 meters.

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Answer: 6.8

Step-by-step explanation:

By the Law of Cosines,

[tex]c^2 = 5^2 + 9^2 - 2(5)(9)\cos 48^{\circ}\\\\c=\sqrt{5^2 + 9^2 - 2(5)(9)\cos 48^{\circ}}\\\\c \approx 6.8[/tex]

Answer:

[tex]\frac{\sqrt{61} }{\sqrt{97} }[/tex]

Step-by-step explanation:

Sin(y) = opposite over hypotenuse

opposite side is [tex]\sqrt{61}[/tex]

hypotenuse is [tex]\sqrt{97}[/tex]

Answer:

36 total students in the class

Step-by-step explanation:

There are 24 boys in the class.

Since they sit in pairs, and exactly 1/4 of the boys sit with a girl, then one-quarter of the 24 boys sits with a girl.

1/4 * 24 = 6, so 6 of the boys sit with a girl.  Reflexively, 6 girls sit with a boy.

Since the problem states that exactly half of the girls sit with a boy, then half of the total number of girls is 6 (since we know that exactly 6 girls were sitting with a boy).

1/2 * x = 6

Multiplying both sides by 2...

x=12

So there are 12 girls total in the class, half (6) of whom sit with a boy.  

There are 24 boys total in the class, one-quarter (6) of whom sit with a girl.  The rest of the boys (18) sit in pairs with another boy (9 pairs), and the rest of the girls (6) sit with another girl (3 pairs).

Since there are 12 girls total and 24 boys total in the class, the total number of students is the sum of 12 and 24

12+24 =36 total students in the class

The sample size required is 1725.

(A) Sample size calculation

p=0.15

For 98%, z=2.326

d=0.02

Sample size = (z^2*p*(1-p))/d^2

= (2.326^2*0.15*0.85)/0.02^2

=1724.5

The sample size required= 1725

(B) Sample size calculation

p=0.5

For 98%, z=2.326

d=0.02

Sample size = (z^2*p*(1-p))/d^2

= (2.326^2*0.5*0.5)/0.02^2

= 3381.4

The sample size required= 3382

In records, the pattern size is the degree of the variety of character samples utilized in a test. For example, if we're testing 50 samples of folks that watch television in a town, then the pattern length is 50.

Sample size willpower is the act of selecting the range of observations or replicates to consist of in a statistical pattern. The sample size is a crucial function of any empirical look at in which the intention is to make inferences about a populace from a sample.

Your question is incomplete. Please find the complete question below.

A researcher for the U.S. Department of the Treasury wishes to estimate the percentage of Americans who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with98% confidence if

(a) Does he use a 2006 estimate of 15% obtained from a Coinstar National Currency Poll?

(b) he does not use any prior estimate?

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Answer:

45units^2

Step-by-step explanation:

Break the polygon into two shapes. A rectangle and a triangle.

The area formula for a rectangle is A=L•W or

Area = Length x Width. Counting the units for length and width is 9 x 2. Which would equal 18units.

Now for the triangle, the formula is A=1/2 (b•h)

The base is 9 and Height is 6.

9 • 6 = 54

Since a triangle is half the area of a rectangle you divide 54 by 2. Area for triangle is 27.

Now add the rectangle area and triangle area.

18+27=45

Area=45 units ^2

Answer:

Step-by-step explanation:

Discussion.

Not directly. But the quadratic formula can do it. But that's not your question.

the factors you get must contain the factors for 1 which are 1 and - 1

These factors must add to - 5. There's no way that will happen with 1 and - 1 and you would be creative math if you tried to say that you could make one of the factors (x -1-1-1-1-1-1). That creates a whole new question.

The total number of lemon candies is 1, solved using the linear equation in one variable 3l + 12 = 15, where l is the number of lemon candies.

We assume the number of chocolate candies to be c.

We assume the number of vanilla candies to be v.

We assume the number of peppermint candies to be p.

We assume the number of lemon candies to be l.

The total number of candies is given as 15.

This can be represented by the linear equation: c + v + p + l = 15 ... (i).

The number of peppermint candies and lemon candies together is given to be twice the number of chocolate and vanilla candies together.

This can be represented by the linear equation: p + l = 2(c + v) ... (ii).

There are eight more peppermint candies than lemon candies.

This can be represented by the linear equation: p = l + 8.

Substituting p = l + 8 in (ii), we get,

l + 8 + l = 2(c + v),

or, 2(l + 4) = 2(c + v),

or, c + v = l + 4.

Substituting c + v = l + 4, and p = l + 8, in (i), we get:

l + 4 + l + 8 + l = 15,

or, 3l + 12 = 15,

or, 3l = 15 - 12 = 3,

or, l = 3/3 = 1.

Therefore, the total number of lemon candies is 1, solved using the linear equation in one variable 3l + 12 = 15, where l is the number of lemon candies.

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Step-by-step explanation:

36 eggs in 3 cartons.

now we double the cartons, so, we need to double the eggs too, because there is a direct relationship between cartons and the number of eggs.

so. in 2×3 cartons are then 2×36 = 72 eggs.

The figures have corresponding sides that are proportional, therefore, they are similar figures.

When two figures are similar to each other, the corresponding sides are proportional to each other.

The two figures given have two opposite sides that are parallel and congruent to each other. Therefore:

DA/VS = 21/7 = 3

AB/ST = 15/5 = 3

Thus, we can conclude that since their corresponding sides are proportional, therefore, they are similar to each other.

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Answer:

Step-by-step explanation:

The daughter's age in 6 years time is 20

Let their current age be x and y

So, we have:

x = 3y

7 years ago, we have:

x - 7 : y - 7 = 5 : 1

Express as fraction

[tex]\frac{x - 7}{y - 7} = \frac 51[/tex]

Cross multiply

x - 7 = 5y - 35

This gives

x = 5y - 28

Substitute x = 3y in x = 5y - 28

3y = 5y - 28

Evaluate the like terms

2y = 28

Divide by 2

y = 14

In 6 years time, we have:

y + 6 = 20

Hence, the daughter's age in 6 years time is 20

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⊰________________________________________________________⊱

Answer:

x=5

Step-by-step explanation:

Apply the distributive property,

2(x+6)-10=12

2x+12-10=12

Collect a few like terms

2x+2=12

Subtract 2 from both sides

2x=10

Divide the equation by 2.

x=5

Done!!

⊱______________________________________________________⊰

2(x+6) - 10 = 12
2(x+6) = 22
x+6 = 11
x = 11 - 6 = 5

x = 5

Answer:

-14

Step-by-step explanation:

Given -x+5y=7

Now we substitute x+5y into -2(-x+5y)

-2(-x+5y) = (-2)(7)

= - 14

There were 15 blue marbles in the bag before removing some

Let B=blue marbles and Y= yellow marbles

The ratio of B to Y is

[tex]\frac{B}{Y} =\frac{8}{5}[/tex]

If we remove 12 blue marbles and add 12 yellow marbles,

the ratio will be

[tex]\frac{B-12}{Y+21} =\frac{1}{3}[/tex]

[tex]3(B-12)=1(Y+21)[/tex]

[tex]3B-36=Y+21\\[/tex]

[tex]3(\frac{8}{5}Y)-36=Y+21[/tex]

[tex]\frac{24}{5}Y-Y=21+36[/tex]

[tex]\frac{19}{5}Y=57[/tex]

[tex]Y=\frac{57*5}{19}\\[/tex]

[tex]Y=15[/tex]

[tex]B=\frac{8}{5} (15)\\B=24[/tex]

Hence, there were 15 blue marbles in the bag before removing some

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The probability of getting 3 or more who were involved in a car accident last year is 0.126.

Given 9% of the drivers were involved in a car accident last year.

We have to find the probability of getting 3 or more who were involved in a car accident last year if 14 were selected randomly.

We have to use binomial theorem which is as under:

n[tex]C_{r}p^{r} (1-p)^{n-r}[/tex]

where p is the probability an r is the number of trials.

Probability that 3 or more involved in a car accident last year if 14 are randomly selected=1-[P(X=0)+P(X=1)+P(X=2)]

=1-{[tex]14C_{0}0.9^{0} (0.1)^{14} +14C_{1} (0.9)^{1} (0.1)^{13} +14C_{2} (0.9)^{2} (0.1)^{12}[/tex]}

=1-{1*0.2670+14!/13!*0.9*0.29+14!/2!12!*0.0081*0.2358}

=1-{0.2670+0.3654+0.2358}

=1-0.8682

=0.1318

Among the options given the nearest is 0.126.

Hence the probability that 3 or more are involved in the accident is 0.126.

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Question:

A teacher puts students' names in a hat and chooses without looking to get a sample of 3 students. What type of sample is this?

Answer:

Simple random

The orthocentre for the triangle ABC is (1.85, 5.88).

Triangle ABC has vertices at A(2, 8), B(6, 2) and C(-3, 2).

We need to use analytical geometry to determine the coordinates of the orthocentre.

Steps to find the orthocentre:

Step 1: First, we will find the slopes of any two sides of the triangle (say AC and AB).

Step 2: Next, we can find the slopes of the corresponding altitudes. Remember that if two lines are perpendicular to each other, they satisfy the following equation.

Step 3: Next, we will use the slope-point form of the equation of a straight line to find the equations of the lines that are coincident with the altitudes BE and AD.

Step 4: Next, we can solve the equations of BE and CF simultaneously to find their solution, which gives us the coordinates of the orthocentre H.

Now, the slope of AC=2-8/-3-2=6/5 and the slope of BE=5/6.

The slope of AB=2-8/6-2=-6/4=-3/2 and the slope of CF=-2/3

The slope of BE=y-2/x-6=5/6

⇒6x-5y-26=0----(1)

The slope of CF=y-2/x+3=-2/3

⇒-2x-3y=0----(2)

By solving (1) and(2), we get

y=26/14=13/7=1.85 and x=5.88

Therefore, the orthocentre for the triangle ABC is (1.85, 5.88).

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Answer:

Its 31.0 as the the two triangles are similar and the scale factor is 2

[tex]in \: pounds \\ 12 \times 288 = 3456 \: pounds[/tex]

[tex]1 \: kilo = 2.2046 \: pounds[/tex]

[tex]x \: kilos = 3456 \: pounds[/tex]

[tex]x = \frac{3456}{2.2046} = 1567.61 \: kilos[/tex]

The ages of Lute,  Maya and Aran are respectively; 24, 8 and 6 years old.

Let Lute's age be x

Thus;

Maya's age = ¹/₃x

Aran's age = ¹/₄x

We are told the sum of their ages is 38. Thus;

x + ¹/₃x + ¹/₄x = 38

Multiply through by 12 to get;

12x + 4x + 3x = 456

19x = 456

x = 456/19

x = 24

Maya's age =  ¹/₃ * 24 = 8 years

Aran's age = ¹/₄ * 24 = 6 years

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It will take 29.4 years for the population of Africa to be two times what it is today. This reflects: doubling time.

Doubling time is the time it takes for a particular amount of size or value to double at a given growth rate. By applying the 70 rules, we can find the doubling time of the exponentially increasing population. To do this, divide 70 by the growth rate (r).

Rule of 70 is a simplified way to determine the doubling time using the equation doubling time = 70 / r. Where r is the percentage growth rate of the population. For example, if the population of 10 species increases by 2 individuals per year, the r value will be 20%.

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The system of inequalities is:

v + h ≤ 8

v < 2

h ≥ 1.5

Let's define the variables:

The maximum time that you can spend on both activities is 8 hours, then:

v + h ≤ 8

You want to spend less than 2 hours on video games, so:

v < 2

You want to spend at least, 1.5 hours on homework, so:

h ≥ 1.5

Then the system of inequalities is:

v + h ≤ 8

v < 2

h ≥ 1.5

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