CEDAR
1
SOULANICTA
ARANTEN
In the figure below, mZ1 = (x-6)° and m/2=5x°.
Find the angle measures.
2

The evaluation of P10–Focus 1 indicates that we get by using the equation for the volume of a cuboid and by calculus;

(a) [tex]L = 4\times x + 4\times 2\cdot x + 4\times \frac{81}{2\cdot x} =12\cdot x + \frac{162}{x^2}[/tex]

(b) [tex]L_{min}[/tex] = 54 cm

(c) L'' at x = 3, (where [tex]L_{min}[/tex] = 54 cm) is positive, therefore, x = 3 is a minimum point

P10-Focus2

The area = [tex]2\times x \times y+\frac{\pi\cdot x^2}{4} =4[/tex], therefore, [tex]y = \frac{16- \pi \cdot x^2}{8 \cdot x}[/tex]

The minimum point, which is an extremum point is a point where the derivative of the function is 0.

(a) The length of the cuboid = 2·x

The width of the cuboid = x

The volume of the cuboid = 81 cubic centimeters

The thickness of the cuboid is therefore;

Thickness = 81/(2·x × x) = 81/(2·x²)

The sum of the length of the twelve edges, L, is therefore;

[tex]L=x + x + x + x + x + 2\cdot x + 2 \cdot x + 2 \cdot x + 2\cdot x + \frac{81}{2 \cdot x^2} + \frac{81}{2 \cdot x^2} + \frac{81}{2 \cdot x^2} + \frac{81}{2 \cdot x^2}[/tex]

Therefore;

[tex]L=4\times x + 4\times 2\cdot x + 4\times \frac{81}{2 \cdot x^2} =12\cdot x + \frac{162}{x^2}[/tex]

The sum of the lengths of the twelve edges, L, is [tex]L = 12\cdot x + \frac{162}{x^2}[/tex]

(b) The minimum value of L can be found using calculus by finding the point at which dL/dx = 0 as follows;

[tex]\frac{dL}{dx} = 12 - \frac{324}{x^3} =0[/tex]

Therefore;

[tex]12-\frac{324}{x^3}=0[/tex]

[tex]12 = \frac{324}{x^3}[/tex]

x³ = 324 ÷ 12 = 27

x = ∛(27) = 3

x = 3

The minimum value of L can be obtained when x = 3 as follows;

[tex]L = 12\cdot x + \frac{162}{x^2}[/tex]

[tex]L_{min} = 12\times 3 + \frac{162}{3^2} = 54[/tex]

The [tex]L_{min}[/tex] = 54

The minimum value of L is 54 centimeters

(c) The value of the minimum value of L can be confirmed by further differentiation using the second derivative, L'' as follows;

At the x-value of a minimum point of a function, the value of the further derivative is larger than zero

The further derivative of the specified function is found as follows;

[tex]L'' = \frac{dL'}{dx} =\frac{972}{x^4}[/tex]

At the critical number, x = 3, from the first derivative, we get;

[tex]L'' = \frac{972}{3^4} = 12 > 0[/tex]

The value of the further derivative of the function at the point x = 3, is L'' = 12 > 0, therefore the point x = 3 is a relative minimum.

(b) The radius of the quarter circle = x

The area of the figure = 4 m²

The sum of the area of the composite figures are;

[tex]A = 2 \times x \times y + \frac{\pi \cdot x^2}{4} = 4[/tex]

Therefore;

[tex]y =\frac{\left(4 - \pi \cdot \frac{x^2}{4}\right)}{2\cdot x} = 16 - \frac{\pi \cdot x^2}{8\cdot x}[/tex]

Rearranging the right hand side expression as a fraction, we get;

[tex]y = \frac{16-\pi \cdot x^2}{8\cdot x}[/tex]

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Answer: 1/3

Step-by-step explanation:

Applying reciprocal property the equation 36 / 5 = 8 / a is solved to find the value of a as 10/9

The definition of reciprocal in mathematics is the inverse of a value or a number.

If n is a real number, then 1/n will be n's reciprocal.

Therefore, we must change the number to its upside-down form. For instance, 1 divided by 9 is the reciprocal of 9, or 1/9.

The result of multiplying a number by its reciprocal now equals one. multiplicative Inverse is another name for it.

Similarly,  36 / 5 = 8 / a

the reciprocal is 5 / 36 = a / 8

cross multiplying

5 * 8 = 36 * a

36a = 40

a = 40 / 36

a = 10/9

The value of a is 10/9

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The question is based on the calculus and the derivative for given equation [tex]t^{4} - 24t^{2} \\ \\[/tex] is t = 0.

Calculus is a discipline of mathematics that focuses on rates of change, to put it simply. For instance, you could wish to determine how quickly an automobile will slow down when it approaches a stop sign. Calculus can assist you in calculating that change. Calculus introduces movement into math, that's correct!

Calculus is a tool used by engineers to calculate things like rates of change and areas; in fact, calculus is the mathematical "backbone" for dealing with problems where variables change with time or some other reference variable. Calculus is concerned with two basic operations: differentiation and integration.

We are given

x(t) = [tex]t^{4} - 24t^{2} \\ \\[/tex]

we need to find its derivative only then we can comment on its increasing decreasing nature

x'(t) = [tex]4t^3 -48t\\\\[/tex]

we will check where derivative is zero

[tex]4t^3 - 48t = 0\\\\4t^2(t-4) = 0\\\\\\t = 0 \\\\t = 4\\[/tex]

hence increasing for all x E R except interval (0,4)

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Part a: y = -6x + 5 is : Not proportional.

Part b: y = 7/8x proportional:  proportionality constant = 7/8.

For the stated question;

Part a: y = -6x + 5

To check the proportional relation, put value of 'x' and find the corresponding value of 'y'.

The ratio = y/x must be same.

For x = 1, y = -1

y/x = -1

And, for x = 2, y = 7

y/x = 7/2

Thus, ratio is not equal, Not proportional.

Part b:  y = 7/8x

For x = 1, y = 7/8

y/x = 7/8

Now, For x = 2, y = 14/8

y/x = 7/8, both ratios are equal.

Thus,  y = 7/8x proportional:  proportionality constant = 7/8.

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The algebraic expression of the phrase Two-tenths of a number is 2x/10. The simplest form of the expression is x/5.

What is a phrase?

A mathematical expression is an expression that contains numbers, variables, and/or operations on those numbers and variables. A mathematical phrase is a verbal phrase that contains words and/or figures that can be turned into a mathematical expression.

Given phrase is Two-tenths of a number.

The value of two tenths is 2/10.

Assume that x be the number.

Two-tenths of x is 2/10 × x = 2x/10

Divide the denominator and numerator by 2:

= x/5

The expression is x/5.

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The cost of the pyramid shape sign is $240.

A pyramid is a 3D figure that has a base made of a polygon and faces that are all triangular and connected. The conventional shape of a pyramid is created by joining each vertex of the base to a single common tip or apex. In this section, let's study more about pyramids.

We know the volume of a pyramid is (1/3)×length of the bese×width of the base×height.

∴ The volume of the pyramid with base length of 6feet, base width of 3feet and height 5 feet is,

= (1/3)×6×3×5 cubic feet.

= 30 cubic feet.

Given, The aluminum costs $8 per cubic foot.

∴ The total cost of the pyramid is = (30×8) = $240.

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0.954 represents population of the town increases by 0.954 each year.

Two or more expressions with an Equal sign is called as Equation.

Given,

The population of a heard of buffalo after t

years is represented by the function

[tex]f(x)=2186(0.954)^{t}[/tex]

We need to find the what does the value 0.954 represent in this

situation.

In the given function 2186 represent the initial population.

t represents the time in years

and 0.954 represents population of the town increases by 0.954 each year.

Hence, 0.954 represents population of the town increases by 0.954 each year.

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Answer:

[tex] slope=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} [/tex]

[tex] slope=\frac{8-1}{7-1} [/tex]

[tex] slope=\frac{7}{6} [/tex]

slope = 7/6

$150 is the least amount of money you can spend if you want to increase the computer performance by 25%, assuming that you can improve one or all the components in the computer.

RAM is a type of volatile memory, which means that data is only stored there while the chip is powered and is completely erased when the computer is turned off. In gigabytes, RAM memory is measured (GB).

Random access memory is referred to as RAM. Because the data may be swiftly read and updated in any sequence, it is termed random access. Compared to previous storage devices like CD-RWs, which access data in a slower, predetermined order, this is faster.

There are much more methods through which we can improve computer performance but by spending the least amount of money we can do it by upgrading  your RAM. We can upgrade the RAM of the system to 8 GB and 16 GB. If we want to upgrade the RAM to 8 GB then it can be done for around $75 and If we want to upgrade the RAM to 16 GB then it can be done for around $150.

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The complete question is as follows:

What is the least amount of money you can spend if you want to increase the computer performance by 25%, assuming that you can improve one or all the components in the computer?

Answer:

a.) x=6 b.) y=-7 c.) a=0.25

Step-by-step explanation:

a.) 2(3x+2)=2x+28

6x+4=2x+28

subtract 4 on both sides

6x=2x+24

subtract 2x on both sides

4x=24

divide both sides by 4

x=6

b.) 5y+13=-43-3y

add 43 on both sides

5y+56=3y

subtract 5y on both sides

56=-8y

divide both sides by 8

-7=y or y=-7

c.) 4(2a+2)=8(2-3a)

8a+8=16-24a

add 24a on both sides

32a+8=16

subtract 8 on both sides

32a=8

divide both sides by 32

a=0.25

(hope you understood how I solved for each equation..)

The number will be decreased by a percentage of 32%.

If a number N is increased by a percentage X, then the new value is:

N' = N*(1 + X/100%)

If instead the number is decreased, then the new value is:

N' = N*(1 - X/100%)

So if we apply a decrease by 60% followed by an increase of 80%, we will have:

N' = N*(1 - 60%/100%)*(1 + 80%/100%)

N' = N*(1 - 0.6)*(1 + 0.8)

N' = N*(0.4)*(1.8)

N' = N*0.72

We can rewrite that as:

N' = N*(1 - 0.32)

0.32 = 32%/100%

So that is equivalent to a decrease by 32%

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Josue has 8 pennies and 5 Nickels and N+P=13 and 0.01P + 0.05N ≥ 0.33 are system of equations that could be used to determine the number of pennies and the number of nickels that Josue has

Two or more expressions with an Equal sign is called as Equation.

Given,

Josue has $0.33 worth of pennies and nickels.

total of 13 pennies and nickels altogether.

Let N be number of Nickels and P be number of Pennies

P + N =13..(1)

Therefore

0.01P + 0.05N ≥ 0.33..(2)

From 1,

P=13-N

0.01(13-N)+0.05N ≥ 0.33

0.13-0.01N+0.05N ≥ 0.33

0.13+0.04N ≥ 0.33

0.04N≥ 0.33-0.13

0.04N≥ 0.33-0.13

0.04N≥ 0.2

Divide both sides by 0.04

N=5

From 1, N+P=13

5+P=13

Subtract 5 from both sides

P=13-5

P=8

Hence, Josue has 8 pennies and 5 Nickels and N+P=13 and 0.01P + 0.05N ≥ 0.33 are system of equations that could be used to determine the number of pennies and the number of nickels that Josue has

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The p-value of the given hypothesis is; 0.9999

The formula for the z-score of proportions is;

z = (p^ - p)/√(p(1 - p)/n)

where;

p^ is sample proportion

p is population proportion

n is sample size

We are given;

p^ = 15% = 0.15

p = 20% = 0.2

n = 5000

Thus;

z = (0.15 - 0.2)/√(0.2(1 - 0.2)/5000)

z = -8.8388

From p-value from z-score calculator, we have;

P(Z < -8.8388) = 1 - 0.0001 = 0.9999

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Answer:

Step-by-step explanation:

1. Lin, Tyler, and Jada's information is missing.

2. 9 1/8 = 73/8

1 2/8 = 10/8

73/8 - 10/8 = 63/8.

His shoe length was 63/8 inches.

The final equation for the temperature of the coffee as a function of time is [tex]T = 80 + 100*e^{(-4t)[/tex]. There are no unknown constants in this equation.

The temperature of a substance will naturally cool down over time due to heat transfer to the surrounding environment. The rate at which the substance cools down is described by Newton's law of cooling, which states that the rate of change of the temperature of a substance is proportional to the difference in temperature between the substance and the surroundings. Mathematically, this can be expressed as:

dT/dt = k(T - Ts)

Where T is the temperature of the substance, Ts is the temperature of the surroundings, t is time, and k is the cooling coefficient, which is a constant that depends on the properties of the substance and the surroundings.

Since we know that the initial temperature of the coffee is 180 and the temperature of the surroundings is 80, and the temperature of the coffee after 10 minutes is 130, we can write the following equation:

(180 - 130)/10 = k(180 - 80)

Solving for k, we get:

k = -4

Substituting this value back into the equation for the rate of change of temperature, we get:

dT/dt = -4(T - 80)

This is the equation for the temperature of the coffee as a function of time. To find the temperature of the coffee at any time t, we can integrate this equation with respect to time. The resulting equation will be:

[tex]T = 80 + C*e^{(-4t)[/tex]

Where C is an unknown constant of integration. Since we are given that the initial temperature of the coffee is 180, we can solve for C by substituting t = 0 and T = 180 into the equation:

[tex]180 = 80 + C*e^0[/tex]

Solving for C, we get:

C = 100

Substituting this value back into the equation for T, we get the final equation for the temperature of the coffee as a function of time:

[tex]T = 80 + 100*e^{(-4t)[/tex]

This is the final, simplified equation for the temperature of the coffee as a function of time. There are no unknown constants in this equation.

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The probability she makes at least 65 is 0.0139.

What is probability?

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a proposition is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

[tex]\begin{aligned}P(x \geq 65) & =1-P(x < 65) \\& =1-P\left(\frac{x-\mu}{n} < \frac{65-56}{4.0988}\right) \\& =1-P(z < 2.1958) \\& =1-0.9861 \\& =0.0139\end{aligned}[/tex]

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The correct name for the set of numbers {-1, √11, 21.6555..., 16} include the following;

A. Real numbers

B. Integers

C. Irrational numbers

E. rational numbers

In Mathematics, there are six (6) common types of numbers and these include the following:

In Mathematics, an irrational number can be defined a type of number which comprises non-terminating or non-repeating decimals such as the square root of 11 or √11.

In conclusion, when the square root of 11 or √11 is taken, the output is neither a terminating decimal nor a repeating decimal. Therefore, it would be classified as an irrational number.

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Complete Question:

Choose the correct name for the set of numbers. {-√11, 21.6555..., √16, 1/9}.

real numbers

integers

irrational numbers

natural numbers

rational numbers

whole numbers

Answer:

Step-by-step explanation:

the graph of g(x) is shifted 7 units to the right and 3 units below the graph of f(x).

The graph of g(x) is shifted 7 units to the left and 3 units below the graph of f(x).

The graph of g(x) is shifted

7 units to the right and 3 units above the graph of f(x).

The graph of g(x) is shifted 7 units to the left and 3 units above the graph of f(x)

Answer:

This would not form a triangle

Step-by-step explanation:

due to the triangle inequality theorem, in order to find out if three numbers can form a triangle, you have to have each pair of numbers add up to more than the remaining number. The first combination added is 5+10 which is 15, and that is not greater than 17, so the answer is no, this is not a triangle.

hope this helps! :3

The length of side AB is 8 units. Option B is the correct option.

What is right angle triangle?

A triangle is a area that bounded by three sides. The length of the sides may not be equal. A right triangle is a triangle with 90 degree angle. The rest two angles are acute angle.

Given that △ABC is a right triangle with right angle at B.

The length of side AC is 10 units. The length of side CB is 6 units.

Pythagoras theorem: The square of hypotenuse is equal to the sum of square of legs.

The legs of △ABC is AB and BC. The hypotenuse of triangle ABC is AC.

Therefore,

AB² + BC² = AC²

AB² + 6² = 10²

AB² + 36 = 100

Subtract 36 from both sides:

AB² = 100 - 36

AB² = 64

Take square root on both sides:

AB = √64

AB = 8

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Let "c" represent the amount of cashews they each bought.

Let "p" represent the amount of peanuts they each bought.

Mike: 2c + 1.5p = 26

Jim: 2.5c + 1 p = 29

That's the system you need to solve.  We can do this with substitution.

Start with Jim:

    2.5c + p = 29

               p = 29 - 2.5c

Substitute that into Mike and solve for c:

                  2c + 1.5p = 26    

   2c + 1.5(29 - 2.5c) = 26    

    2c + 43.5 - 3.75c = 26

           -1.75c + 43.5 = 26

                      -1.75c  = - 17.5

                             c  = 10

Now substitute that back into Jim's last equation:

               p = 29 - 2.5c

               p = 29 - 2.5(10)

               p = 29 - 25

               p = 4

So they each bought 10 pounds of cashews and 4 pounds of peanuts.

Answer:

Step-by-step explanation:

1. Increase of 10% in price of shirt.
2. Decrease of 10% in price of shirt.
Hence- both amounts cancel out, therefore there is 0% change in price of shirt as both percentages cancelled out, or we can say nullified each other. Because they added the same amount to the item and then subtracted the same percent leading to no change.

The maximum number of spheres that can be made through ice will be 135.

To convert any unit into another is called a unit conversion.

In order to convert units, we need to care about their dimensions their dimension should not be changed.

As per the given,

Number of ice portions = 40

Weight of one ice portion = 3 ounce

Since, 1 lb = 16 ounce

3 ounce = 3/16 lb

Weight of 40 portion = 120/16 lb

The weight of one sphere of ice = 18 lb

Number of possible spheres = (120/16)/18

⇒ 120/16 x 18

⇒ 135 sphere

Hence "135 spheres are the most that can be produced by ice".

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If cos x= 2/3 and x is in quadrant I, then the exact value of sin(2x) is (4/3)sin x.

A branch of mathematics called trigonometry examines the connections between triangles' angles and sides. It is predicated on the idea of right triangles, where one angle is 90 degrees and two sides are perpendicular to one another. The six fundamental functions of trigonometry—sine, cosine, tangent, cotangent, secant,  cosecant. These values of functions can be used to address real-world angle and length issues, such as determining the height of a structure or the distance between two locations. Calculus, geometry, and other areas of mathematics are all strongly connected to trigonometry.Modeling actual phenomena like item motion, wave behaviour, and light dispersion is possible with these functions. Physics, engineering, and astronomy are just a few of the disciplines that use trigonometry extensively.

How to solve?

In quadrant I, cos x = 2/3 and sin x > 0.

sin(2x) = 2sin x cos x

sin(2x) = 2 * 2/3 * (sin x)

sin(2x) = (4/3)sin x

If sin x > 0, then sin(2x) = (4/3)sin x > 0.

The exact value of sin(2x) is (4/3)sin x

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A researcher obtains a negative value for the chi square test.1 is the value is negative .

What is chi-square value?

The difference between the frequencies of results of a collection of events or variables that were observed and those that were expected is measured by the chi-square (2) statistic.

                   Such variations in categorical variables can be examined using chi-square, particularly when they are nominal in nature.

We know that the chi-square is the sum of square of difference between observed and expected frequencies divided by the expected frequency. So, it is clear that the chi-square value can never be negative because sum of squares is always positive.

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the weight percentage of VOC content in the paint would be 4.05 pounds / 9 pounds x 100 = 45%.

A percent of a whole is referred to as a percentage. By dividing the portion by the entire and multiplying the result by 100, it is computed and denoted by the sign "%". For instance, if a book contains 100 pages and you have read 50 of them, you have read 50% of the book. In several disciplines, including finance, statistics, and mathematics, percentage is often utilized. To figure out interest rates, taxes, and discounts in finance, we employ percentages. In statistics, percentages are used to represent data that takes the form of proportions or ratios. In mathematics, questions involving proportions and ratios are solved using percentage.

How to solve?

the total weight of VOC in the paint would be 0.45 pounds x 9 pounds per gallon = 4.05 pounds.

the weight percentage of VOC in the paint would be 4.05 pounds / 9 pounds x 100 = 45%.

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Answer:

Step-by-step explanation:

To solve the system of equations 2y + 5x = 13 and 2y-3x = 5 using the substitution method, you can follow these steps:

Begin by solving one of the equations for one of the variables. For example, you could solve the first equation for y:

2y + 5x = 13

2y = 13 - 5x

y = (13 - 5x) / 2

Substitute the expression for y into the second equation and solve for x:

2((13 - 5x) / 2) - 3x = 5

13 - 5x - 3x = 5

-8x = -8

x = 1

Substitute the value of x into the expression for y to find the value of y:

y = (13 - 5(1)) / 2

y = (13 - 5) / 2

y = 8 / 2

y = 4

The solution to the system of equations is (1,4).

Therefore, the correct ordered pair is (1,4).

Answer: 300000000000000000

Step-by-step explanation:

50000x6000000000000 can be written as 3 times 10^17

So the answer is 300000000000000000