3.8m² of concrete is made into the shape of a cylinder.
The base has radius 0.5 metres.
Work out the height of the cylinder.
0.5 m
(2 Marks)

The slope of one of the dotted lines is equal to 1.

The slope of the line of reflection is equal to -1.

In Mathematics, the slope of any straight line can be determined by using this mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Substituting the given data points into the slope formula, the slope of one of the dotted lines include the following;

Slope, m = (4 - 1)/(-1 - (-4))

Slope, m = 3/(-1 + 4)

Slope, m = 3/3.

Slope, m = 1.

For the line of reflection, the slope is given by:

Slope, m = (2 - 4)/(1 - (-1))

Slope, m = -2/(1 + 1)

Slope, m = -2/2.

Slope, m = -1.

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(A) The global optimal solution is x = -π/2 and the local optimal solutions are x = (2n + 1)π/2

(B) The global optimal solution is x = (2n + 1)π/2

(C) the global optimal solution is x = (2n + 1)π/2

A function is a mathematical concept that assigns a unique output value for each input value. It is a rule that takes in an input or set of inputs and maps it to a specific output. Functions are represented by symbols, usually denoted by a letter, such as "f".

In mathematical notation, a function is usually expressed as "f(x)" where "x" is the input and "f(x)" is the corresponding output. For example, a function could be defined as f(x) = x^2, which means that for any value of x, the function will calculate and return the square of that value.

For the first problem:

(a) f(x) = sin x and X = [0, 3π]

The objective function f(x) = sin x has a global minimum at x = -π/2, where sin x = -1. The function also has a local minimum at x = (2n + 1)π/2, where n is an integer and sin x = -1.

The feasible set X = [0, 3π] contains all the local and global minima of f(x) = sin x, so the global optimal solution is x = -π/2 and the local optimal solutions are x = (2n + 1)π/2, where n is an integer.

For the second problem:

(b) f(x) = sin x and X = [1, 4π]

The objective function f(x) = sin x has a global minimum at x = -π/2, where sin x = -1. The function also has a local minimum at x = (2n + 1)π/2, where n is an integer and sin x = -1.

The feasible set X = [1, 4π] only contains some of the local minima of f(x) = sin x, so the global optimal solution is x = (2n + 1)π/2 where (2n + 1)π/2 is in the feasible set X and sin x is at its minimum.

For the third problem:

(c) f(x) = sin x and X = [-π/4, (11/4)π]

The objective function f(x) = sin x has a global minimum at x = -π/2, where sin x = -1. The function also has a local minimum at x = (2n + 1)π/2, where n is an integer and sin x = -1.

The feasible set X = [-π/4, (11/4)π] only contains some of the local minima of f(x) = sin x, so the global optimal solution is x = (2n + 1)π/2 where (2n + 1)π/2 is in the feasible set X and sin x is at its minimum.

For the fourth problem:

(d) f(x) = max(0, a,2-1) and X = (-00,00); and X = 10, (2/3) ; 0, If x = 0;

The objective function f(x) = max(0, a,2-1) has a global minimum at x = 0, where the function is 0. The function also has a local minimum at x = (2/3) where the function is (2/3)^2-1.

The feasible set X = (-00,00); and X = 10, (2/3) ; 0, If x = 0; only contains some of the local minima of f(x) = max(0, a,2-1), so the global optimal solution is x = (2/3) if (2/3) is in the feasible set X and the function is at its minimum, or x = 0 if 0 is in the feasible set X and the function is at its minimum.

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By answering the above question, we may state that The entire probability  sum of your raffle wins is represented by the random variable X.

Mathematicians use probability theory to determine the chance that an event will occur or a statement is true. A probability of about 0 reflects how probable an event appears to occur, whereas a risk is a value between 0 and 1, where 1 suggests certainty. Possibility is a mathematical representation of the likelihood that a specific event will take place. Other ways to express probabilities are as integers around 0 or 1 or as percentages between 0% and 100%. the ratio of events among equally likely options that lead to a certain event to all other outcomes.

a. As a raffle participant, you are curious about your odds of winning the prize and the potential value of your winnings.

b. The entire sum of your raffle wins is represented by the random variable X.

c. The potential values for X are 0 (in the event that you do not win the raffle), $150 (in the event that you do win the raffle and receive the two Broadway show passes), or -$20 (in the event that you buy four raffle tickets for $20 but do not receive a reward).

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A. The firm's short-run total cost (TC) is given by TC = rK + wL, where r is the interest rate for capital and w is the wage rate for labor.

B. The short-run total cost at 25 units of output: TC = $100 + 4((25/2)^2) = $100 + 100 = $200

The short-run total cost at 200 units of output: TC = $100 + 4((200/2)^2) = $100 + 1600 = $1700

Please let's go ahead with the explanation to fully understand the solution,

A. To calculate the firm's short-run total cost curve and average cost curve function, we need to know the amount of labor (L) the firm uses at each level of output (Q). We can use the production function to find the optimal combination of L and Q given the fixed amount of capital (K = 100).

The firm's short-run total cost (TC) is given by TC = rK + wL, where r is the interest rate for capital and w is the wage rate for labor.

The short-run average cost (SAC) can be calculated by dividing the short-run total cost by the level of output (Q): SAC = TC/Q

B. The short-run marginal cost (SMC) is the change in the total cost per unit change in the level of output. The SMC can be calculated as the derivative of the short-run total cost function with respect to Q.

At a level of output of 25, we have: TC = $100 + 4L, and Q = 2√L. So we can substitute L = (Q/2)^2 into the short-run total cost equation to find the short-run total cost at 25 units of output: TC = $100 + 4((25/2)^2) = $100 + 100 = $200

At a level of output of 200, we have: TC = $100 + 4L, and Q = 2√L. So we can substitute L = (Q/2)^2 into the short-run total cost equation to find the short-run total cost at 200 units of output: TC = $100 + 4((200/2)^2) = $100 + 1600 = $1700

The short-run average cost at 25 units of output is SAC = TC/Q = $200/25 = $8, and the short-run average cost at 200 units of output is SAC = TC/Q = $1700/200 = $8.5

The short-run marginal cost at 25 units of output is the derivative of the short-run total cost with respect to Q evaluated at 25 units of output. Since the short-run total cost is linear in Q, the short-run marginal cost is constant and equal to the wage rate, which is $4. The short-run marginal cost at 200 units of output is also constant and equal to $4.

C. A graph of the short-run total cost, short-run average cost, and short-run marginal cost functions can be plotted on the same graph, with Q on the x-axis and the corresponding cost on the y-axis.

D. The short-run average cost curve intersects the short-run marginal cost curve at the minimum point of the short-run average cost curve. This is because the short-run average cost is the derivative of the short-run total cost with respect to Q, and the short-run marginal cost is the derivative of the short-run total cost with respect to Q. At the minimum point of the short-run average cost curve, the first derivative (the short-run marginal cost) is equal to zero, and the second derivative (the short-run average cost) is negative.

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Answer:

y = - [tex]\frac{1}{3}[/tex] x + 3

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = 3x + 2 ← is in slope- intercept form

with slope m = 3

given a line with slope m then the slope of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{3}[/tex] , then

y = - [tex]\frac{1}{3}[/tex] x + c ← is the partial equation

to find c substitute (- 9, 6 ) into the partial equation

6 = - [tex]\frac{1}{3}[/tex] (- 9) + c = 3 + c ( subtract 3 from both sides )

3 = c

y = - [tex]\frac{1}{3}[/tex] x + 3 ← equation of perpendicular line

The elimination method involves adding or subtracting the two equations to cancel out one of the variables and obtain a single equation in one variable. We can then solve for that variable, and then use the original equations to find the other variable.

What does "system of equations" mean?

simultaneous equations, system of equations Two or more equations in algebra must be solved jointly (i.e., the solution must satisfy all the equations in the system). The number of equations must match the number of unknowns for a system to have a singular solution.

Here's the work for the given system of equations:

7x - 2y = 35

4x + 5y = -23

Multiplying the second equation by 2:

8x + 10y = -46

Adding the two equations to eliminate x:

15y = -11

Dividing both sides by 15 to solve for y:

y = -11/15

Using the first equation to find x:

7x - 2(-11/15) = 35

Expanding and solving for x:

7x + 22/15 = 35

Subtracting 22/15 from both sides:

7x = 35 - 22/15

Simplifying the right side:

7x = 35 - 22/15

7x = 77/15

Dividing both sides by 7 to solve for x:

x = 11/15

The solution to the system is (x,y) = (11/15, -11/15).

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Answer:

Step-by-step explanation:

You want the number of bracelets (x) and the number of necklaces (y) made from 120 grams of gold when a total of 14 were made with 4 grams in each bracelet and 20 grams in each necklace.

The equations describing the numbers can be written as ...

  4x +20y = 120 . . . . . . amount of gold used

  x + y = 14 . . . . . . . . . number of items made

The first equation can be graphed by plotting its x-intercept at (30, 0) and its y-intercept at (0, 6). The line between these points is the graph.

The graph of the second equation is the line between the two intercepts at 14: (14, 0) for the x-intercept, and (0, 14) for the y-intercept.

The lines cross at (x, y) = (10, 4).

The number of bracelets made was 10; the number of necklaces, 4.

When a singular die is rolled once, the results are represented by the probability distribution P(X = x) = 1/6, 1,2,3,4,5,6 for random variable X.

Probability is the potential outcome of any random occurrence. This expression relates to estimating the chance that any certain event will take place. How likely are we to get a head, for example, if we flip the coin into the air? The solution to this question is based on the potential number of occurrences.

When one die is rolled, there are only six possible outcomes: 1, 2, 3, 4, and 5. Due to the equal likelihood of each scenario, its sp risk is 1/6. The following is true for the required posterior distribution of the random variable x:

P(X = x) = 1/6, 1,2,3,4,5,6

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The table showing Luthando's income based on the number of bicycles he sells:

No. of bicycle. Income

0 R100

1. R115

2. R130

3. R145

4. R160

Amount Luthando receives per day= R100

Commission on each bicycle = R15

Number of bicycles sold = x

The equation:

100 + 15x

When x = 0

100 + 15x

= 100 + 15(0)

= 100 + 0

= R100

When x = 1

100 + 15x

100 + 15(1)

= 100 + 25

= R115

When x = 2

100 + 15x

= 100 + 15(2)

= 100 + 30

= R130

When x = 3

100 + 15x

= 100 + 15(3)

= 100 + 45

= R145

When x = 4

100 + 15x

= 100 + 15(4)

= 100 + 60

= R160

Therefore, Luthando earned the following income R100, R115, R130, R145, R160

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If you travel for three hours at a speed of 30 km/hr, you will go a distance of 90 km.

Speed is the rate of change of an object position relative to a given frame of reference. It is a scalar quantity meaning it has both magnitude and direction. Speed is usually measured in meters per second.

The distance travelled can be calculated by multiplying the speed of 30 km/hr by the number of hours travelled, which is 3. Therefore, the distance travelled is 90 km. To summarize, if you travel for three hours at a speed of 30 km/hr, you will go a distance of 90 km.

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Answer:

[tex]y=1-Ae^{\frac{1}{2}x^2-x}[/tex]

Step-by-step explanation:

Given equation:

[tex]\dfrac{\text{d}y}{\text{d}x}=(1-x)(1-y)[/tex]

Separate the variables by moving all the y terms to one side of the equation and all the x terms to the other side:

[tex]\implies \dfrac{1}{(1-y)}\;\text{d}y=(1-x)\;\text{d}x[/tex]

Integrate both sides of the equation separately:

[tex]\implies \displaystyle \int \dfrac{1}{(1-y)}\;\text{d}y= \int (1-x)\;\text{d}x[/tex]

[tex]\implies -\ln |1-y|+C=x-\dfrac{1}{2}x^2+D[/tex]

[tex]\implies \ln |1-y|-C=\dfrac{1}{2}x^2-x-D[/tex]

Write the two constants as one (a = -D + C):

[tex]\implies \ln |1-y|=\dfrac{1}{2}x^2-x+a[/tex]

Take exponents of both sides:

[tex]\implies e^{\ln |1-y|}=e^{\frac{1}{2}x^2-x++a}[/tex]

[tex]\textsf{As }\; e^{\ln y}=y:[/tex]

[tex]\implies 1-y=e^{\frac{1}{2}x^2-x+a}[/tex]

[tex]\textsf{Apply exponent rule} \quad a^{b+c}=a^b \cdot a^c:[/tex]

[tex]\implies 1-y=e^{\frac{1}{2}x^2-x}e^{a}[/tex]

As [tex]e^{a}[/tex] is just a constant, replace it with A:

[tex]\implies 1-y=Ae^{\frac{1}{2}x^2-x}[/tex]

Rearrange to make y the subject:

[tex]\implies y=1-Ae^{\frac{1}{2}x^2-x}[/tex]

He have 20.5 miles to his destination after 68 minutes of driving

Since Hong has 54 miles to his destination after 38 minutes of driving, and he has 44.4 miles to his destination after 50 minutes of driving.

We will consider the two given data values as our points 1 and 2 respectively. And that is what we are going to use to create a linear function for pressure

Let the y and x represent the distance and time respectively

point 1 (54, 38) and point 2 (44.4, 50)

slope(m) = (y2-y1) / (x2-x1)

where  x1 =54, y1 = 38 and x2 = 44.4, y2 = 50

slope(m) = (50-38) / (44.4-54) = -5/4 = -1.25

Linear function are of the form y-y1 = m(x-x1). Thus:

y-38 = -1.25 (x-54)

y-38 = -1.25x + 67.5

y = -1.25x + 67.5 + 38

y =  -1.25x + 105.5

After 68 minutes of driving, the number of miles will be:

y =  -1.25x + 105.5

put x = 68

y =  -1.25(68) + 105.5

y = 20.5 miles

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Answer:

3x-2y=-6

Step-by-step explanation:

I recommend using desmos graphing calculator to find the answer

If a particle is located a distance x meters from the origin, a force of cos (pi(x)/3 newtons acts on it then the work is done in moving the particle from x = 1 to x = 2 is 0 J.

Work is the energy transferred to or from an object via the application of force along a displacement.

Given that  a particle is located a distance x meters from the origin

Force acting on it is cos (pi(x)/3 newtons.

We need to find the work.

W=∫F dx

Now replacing the values ​​with the information that was given and the equation, we have:

[tex]W=\int\limits^2_1 {cos(\pi x/3)} \, dx[/tex]

w=3/π(sinπx/3)

Applying the points where X is equal to 2 and 1 we find:

W=3/π(√3/2-√3/2)

W=0

Hence, if a particle is located a distance x meters from the origin, a force of cos (pi(x)/3 newtons acts on it then the work is done in moving the particle from x = 1 to x = 2 is 0 J.

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Answer:13 x

Step-by-step explanation: bc i did the test

Answer:

x≅101.5

Step-by-step explanation:

(88×cot60° + x)tan 30°=88

(88/√3 + x)1/√3 = 88

88/3 + x/√3= 88

x/√3=176/3

x= 176×1.73/3

x= 101.49

x≅101.5

the value of x will be 101.61 units.

A triangle is said to be right-angled if one of its angles is exactly 90 degrees. The total of the other two angles is 90 degrees. Perpendicular and the triangle's base are the sides that make up the right angle. The longest of the three sides, the third side is known as the hypotenuse.

Given two right-angle triangles with the same height of 88 units

Let the adjacent side of a triangle with a leg angle of 60 be "h"

From the general formula of a tangent:

Tan A = opposite side/ adjacent side

For the triangle with a leg angle of 60

tan 60 = 88/h

√3 = 88/h

h = 88/√3

For the triangle with a leg angle of 30

tan 30  = 88/(x +h)

1/ √3 = 88(x + h)

x +h = 88 * √3

x = 88*√3 - 88/√3

x = 101.613

therefore, the value of x will be 101.61 units.

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Answer:

To find the roots of 4x² - 12x + 9 = 0 graphically, we need to plot the equation on a graph and find the x-intercepts, which are the roots.

Here are the steps to graph the equation:

Plot the x-axis and y-axis on the graph

Plot the points (x, 4x² - 12x + 9) for several values of x within the range of -1 to 4.

Connect the points to form a smooth curve.

Find the x-intercepts, where the curve intersects the x-axis. These points represent the roots of the equation.

After finding the x-intercepts, we can use the x-value of each intercept to substitute back into the original equation to find the corresponding y-value. This confirms that the x-intercepts are indeed the roots of the equation.

Note: The above steps can also be done using a graphing calculator or software.

The area of the alphabet S that it covers on a square sheet is 12.375 in²

Area of a Square = Side × Side. Therefore, the area of square = Side2 square units. and the perimeter of a square = 4 × side units.

Given here: The letter S drawn on a square sheet with each square with length 1.5 inch.

Now the letter S includes 2 full squares and 7 half-squares.

Thus the total Area= 2×1.5²+7×1/2 ×1.5²

   A=12.375 inch²

Thus the area of the alphabet S that it covers on a square sheet is 12.375 in²

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Answer:

B. 63

Step-by-step explanation:

Let's first find angle M.

Angle M has 118 by it. The line below it is also a straight line. The angle of a straight line is 180. This means that 180-118 equal angle M.

Angle M = 180-118 = 62

Now the angles of a triangle also equal 180 and since 1 angle is 62, these other two angles must equal 118.

[tex](6x - 9) + (4x + 7) = 118[/tex]

Lets combine like terms. We combine the numbers. -9+7 equals -2. 6x + 4x = 10x. Putting those answers together

[tex]10x - 2 = 118[/tex]

If we add 2 to both sides, that gets rid of the 2 on the left side

[tex]10x = 120[/tex]

Lastly, we divide by 10

[tex]x = 12[/tex]

Now that we have X, we can plug it into angle L's equation!

[tex](6x - 9) = (6(12) - 9) = ((72) - 9) = 63[/tex]

To show it's correct, plugging it into the other equation makes it 55.

55 + 63 + 62 = 180

The below are the equations of exponential graphs

[tex]y=5^xy=(1/5)^x[/tex]

What is exponential growth function ?

A process called exponential growth sees a rise in quantity over time. When a quantity's derivative, or instantaneous rate of change with respect to time, is proportionate to the quantity itself, this phenomenon takes place.

A exponential decay is a function that, as the name implies, decays exponentially. So it decays fast at the beginning and slower as the value of the variable increases.

An exponential function will have constant as a base and exponent contains a variable

The below are the equations of exponential graphs

[tex]y=5^xy=(1/5)^x[/tex]

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Answer:

false

Step-by-step explanation:

Answer:

The equation has 3 solutions x(1) = -2 x(2) = 0 x(3) = 3

Step-by-step explanation:

1.When the product of the factor equals 0, atleast one of the factors is 0

x=0

x^2 - x - 6 = 0

2.solve the equation for x

x = 0

x = -2

x = 3

hope this helped :D

The probability that a randomly selected Atlantic cod has a length of 53.68 cm or less is given by the equation P ( x < 53.68 ) = 0.8439

The relationship between a value and the mean of a set of values is expressed numerically by a Z-score. The Z-score is computed using the standard deviations from the mean. A Z-score of zero indicates that the data point's score and the mean score are identical.

The Z-score is calculated using the formula:

z = (x - μ)/σ

where z: standard score

x: observed value

μ: mean of the sample

σ: standard deviation of the sample

Given data ,

Let the probability that a randomly selected Atlantic cod has a length of 53.68 cm or less be represented as P

Now , the equation will be

The observed value of x = 53.68 cm

The mean of the sample μ = 49.9 cm

The standard deviation of the sample σ = 3.74 cm

Now , z-score is calculated using the formula:

z = (x - μ)/σ

Substituting the values in the equation , we get

z = ( 53.68 - 49.9 ) / 3.74

z = 1.0107

Now ,the p value from the z table is

P ( x < 53.68 ) = 0.84392

And , the probability is P ( x < 53.68 ) = 84.392 %

Hence , the probability is 84.392 %

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Answer:

$3.24

Step-by-step explanation:

Cost for the coffee + Cost for the bagel

$1.39 + $1.85

$3.24

The proposition "I come to class whenever there is going to be a quiz" can be written as "If there is going to be a quiz, then I come to class."

The converse of this proposition is "If I come to class, then there is going to be a quiz."

The contrapositive of this proposition is "If there is not going to be a quiz, then I do not come to class."

The inverse of this proposition is "If I do not come to class, then there is not going to be a quiz."

A mathematical proposition is a statement such as "3 is bigger than 4," "an infinite set exists," or "7 is prime." A proposition that is presumed to be true is referred to as an axiom. A proposition is an either true or incorrect statement. A mathematical assertion will be referred to as a theorem throughout our course. A theorem is a significant result. A lemma is a claim that serves primarily to establish a bigger theorem.

Here,

The proposition "I come to class whenever there is going to be a quiz" can be written as "If there is going to be a quiz, then I come to class."

The converse of this proposition is "If I come to class, then there is going to be a quiz."

The contrapositive of this proposition is "If there is not going to be a quiz, then I do not come to class."

The inverse of this proposition is "If I do not come to class, then there is not going to be a quiz."

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The variables are the number of candy bars and lollipops and their prices. The possible equality is   2x + 4y ≥ 180.

In mathematics, the variables are factors that might change. Due to this, in maths variables are often expressed using letters as their exact value is not known. Then the variables are:

Based on the previous information, the correct inequality would be:

2x + 4y ≥ 180

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The probability will be 1/17 i.e. A.

What exactly is probability?

Probability expresses the likelihood of an event. This mathematical branch of study is concerned with the occurrence of a random event. The value ranges from zero to one. Probability is used in mathematics to anticipate the possibility that events will occur.

Probability refers to the possibility of something happening. This fundamental idea in probability theory is also used by the probability distribution. Before determining the chance of a certain event occurring, we must first determine the total number of possibilities.

The probability that an event will occur P(E) = the number of positive outcomes divided by the total number of outcomes.

Now,

Total cards =52

No. of ways to select 2 cards=52C₂

=52*51*50!/50!*2!=51*26

Now to select two cards with same values we have no. of ways =13*6

so probability that two cards are chosen at random from a deck of 52 and they have the same value=13*6/26*51=3/51=1/17

Hence,

           The probability will be 1/17.

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Answer:

YStep-by-step explanation:

Answer:

3/4 servings

Step-by-step explanation:

Each serving = 1/2

There are 3/8 cookies.

How many times 1/2 "fits" into 3/8 = how many servings.

3/8 / 1/2

= 3/8 * 2

= 3/4 servings

Therefore, in this sort of triangle with angles 30-60-90, the value of y is 10√3  and the value of x is 10.

In algebra, a logo (typically a letter) used to represent an undetermined numerical score in a calculation or algebraic expressions. Simply put, a variable is an amount that may have been changed and is not fixed. Categorical and numeric elements are the two main types of variables. Then, discrete and continuous subcategories are made for each category for the categorical and numerical elements, respectively. An overview of various types is given in this section. a metric that has a wide range of values. Someone that can change; a variable's sign; an element of a laboratory test that could vary.

given

As a result, this is a "special triangle." According to the 30-60-90 triangle, if the short leg is x, the hypotenuse will be 2x, and the long leg will be  x√3.

Since we have the short leg, we can divide it by 2 to determine the hypotenuse, or y in this case:

20 /2 = 10

Next, divide the short leg by 3 to determine the long leg, or x in our case:

10 × √3 = 10√3

Therefore, in this sort of triangle with angles 30-60-90, the value of y is 10√3  and the value of x is 10.

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