16 Paul buys a number of large sacks of fertiliser costing $x each. He spends $27. (a) Write down, in terms of x, an expression for the number of large sacks which Paul buys. Answer(a) 27 (b) Rula buys a number of small sacks of fertiliser. Each small sack costs $2 less than a large sack. Rula spends $25. Write down, in terms of x, an expression for the number of small sacks which Rula buys. Answer (b) 25 X-2 (c) Rula buys 4 more sacks than Paul. Write down an equation in x and show that it simplifies to 2x²-3x-27= 0. ​

Answer:

(-1.29, 8.29) or (9.29,-2.29)

Step-by-step explanation:

Find what y equals in terms of x for both equations.

Set those two equations equal to each other (Because y=y)

Solve for x (you may need to use the quadratic formula for this step)

Plug the value(s) of x into the original equation to find y.

Answer:

  147.6 m²

Step-by-step explanation:

You want the surface area of a triangular pyramid with base sides of length 8 m and a slant height of 10 m. (The altitude of the base triangle is approximately 6.9 m.)

The surface area is the sum of the areas of the triangular faces.

  SA = 1/2(8 m)(6.9 m) + 3 × 1/2(8 m)(10 m)

  = 1/2(8 m)(6.9 m + 3×10 m) = (4)(36.9) m² = 147.6 m²

The surface area of the pyramid is 147.6 square meters.

Answer:

Step-by-step explanation:

In a parallelogram, opposite sides are parallel and equal.

     MA // HT  & AH is transversal.

        ∠MAG ≅ ∠GHT        {Alternate interior angles are equal}    --Angle

     MA = HT   {MATH is a parallelogram}   ---> Side

    MA // HT & MT is transversal.

         ∠AMG ≅ ∠HTG    {Alternate interior angles area equal} ---->Angle

ΔMGA ≅ ΔTGH   { Angle side angle congruent}

 AG = HG           {CPCT}

MT bisects AH

Answer: 1/7

Step-by-step explanation:

d d x ( ∑ n = 0 ∞ c n ( x − a ) n ) = c 1 + 2 c 2 ( a − a ) + 3 c 3 ( a − a ) 2 + ⋯ = c 1/7

The graph of y = sec(x) will have vertical asymptotes at these three values of x.

Vertical asymptotes are vertical lines on a graph where a function approaches either positive or negative infinity as the independent variable approaches a certain value.

The secant function, y = sec(x), is defined as the reciprocal of the cosine function:

sec(x) = 1/cos(x)

The cosine function has vertical asymptotes at x = (2n + 1)π/2 for any integer n, since cosine is undefined at these points (division by zero). Therefore, the secant function will have vertical asymptotes at the same points, since dividing 1 by a very small number (close to zero) results in a very large number (tending towards infinity).

So, the values of x in the interval -2π ≤ x ≤ 2π that will cause the secant function to have vertical asymptotes are:

x = (2n + 1)π/2, where n is an integer

For n = -2, we have x = -5π/2, which is outside the interval, so we can ignore it.

For n = -1, we have x = -π/2

For n = 0, we have x = π/2

For n = 1, we have x = 3π/2

For n = 2, we have x = 5π/2, which is also outside the interval.

Therefore, the values of x in the interval -2π ≤ x ≤ 2π that will cause the secant function to have vertical asymptotes are:

x = -π/2, π/2, 3π/2

So, the graph of y = sec(x) will have vertical asymptotes at these three values of x.

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Answer: To find the points where the two skaters might collide, we need to solve the system of equations:

y = -2x + 14 (Equation 1)

y = -2x^2 + 14x (Equation 2)

We can substitute Equation 1 into Equation 2 to eliminate y and get:

-2x + 14 = -2x^2 + 14x

Simplifying, we get:

2x^2 - 16x + 14 = 0

Dividing both sides by 2, we get:

x^2 - 8x + 7 = 0

This quadratic equation factors as:

(x - 1)(x - 7) = 0

So the possible values of x where the skaters might collide are x = 1 and x = 7.

To find the corresponding y-values, we can plug each value of x into either Equation 1 or Equation 2. For x = 1:

y = -2(1) + 14 = 12 (using Equation 1)

y = -2(1)^2 + 14(1) = 12 (using Equation 2)

So the skaters might collide at the point (1, 12).

For x = 7:

y = -2(7) + 14 = 0 (using Equation 1)

y = -2(7)^2 + 14(7) = 0 (using Equation 2)

So the skaters might collide at the point (7, 0).

Therefore, the two skaters might collide at points (1, 12) and (7, 0).

Step-by-step explanation:

That sounds like a smart plan! Before investing his savings, Jose should do some research to determine the best investment strategy for his goals.

What is research?

He will need to consider factors such as his risk tolerance, the potential return on investment, and the length of time he has to invest. Once he has identified an investment strategy that aligns with his goals, he should start investing his savings.

Over time, as his investments generate returns, he can use these returns to fund his business venture. It is important for Jose to regularly monitor his investments and adjust his strategy as needed to ensure that he is on track to achieve his goals. With careful planning and a well-executed investment strategy, Jose can achieve his goal of retiring early and starting his own coaching classes.

What is an investment ?

An investment refers to the purchase of an asset with the goal of generating income, capital appreciation, or both. The asset could be anything of value, such as stocks, bonds, real estate, commodities, or mutual funds. Investors typically make an investment with the expectation of earning a return on their investment over a period of time.

The return could be in the form of interest, dividends, or capital gains, depending on the type of investment. Investors may choose to invest in different types of assets based on their investment goals, risk tolerance, and investment horizon. The process of investing involves conducting research, analyzing market trends, and making informed decisions based on the available information.

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Complete question is: Jose is a middle-aged professor who wants to retire early and start his own coaching classes. He decides to invest his savings and use the returns to fund his business venture. Before investing his savings, Jose should do some research to determine the best investment strategy for his goals.

When the probability of an event is high, we can say that the likelihood of that event occurring is also high. Conversely, when the probability of an event is low, the likelihood of that event occurring is also low.

What is probability?

In mathematics, probability is a measure of the likelihood that an event will occur. It is a way of quantifying the uncertainty or randomness associated with an event or set of events. The probability of an event is expressed as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain to occur. For example, the probability of flipping a fair coin and getting heads is 0.5, or 50%, since there are two equally likely outcomes (heads or tails) and getting heads is one of them. Probability theory is a fundamental branch of mathematics that has applications in many areas, including statistics, finance, engineering, and computer science, among others.

Here,

The likelihood of selecting a green ribbon from Emilie's drawer is high, since the probability of randomly selecting a green ribbon is 84%. This means that out of all the ribbons in the drawer, 84% of them are green.

Another way to interpret this is that if Emilie were to randomly select a ribbon from her drawer without looking, there is an 84% chance that the ribbon she selects will be green.

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Correct question is "Emilie has a drawer full of color for ribbons the probability of randomly selecting a green ribbon is 84% which describes the likelihood by the example of selecting a green ribbon".

Answer:

a. To complete the table:

# of times floss per week Frequency Relative Frequency Cumulative Frequency

0 6 0.0857 6

1 7 0.1 13

2 4 0.0571 17

3 11 0.1571 28

4 15 0.2143 43

5 11 0.1571 54

6 6 0.0857 60

7 7 0.1 67

Step-by-step explanation:

b. To find the cumulative relative frequency for flossing 3 times per week, we need to add up the relative frequencies for all values of flossing less than or equal to 3.

Cumulative relative frequency for flossing 3 times per week:

= relative frequency for flossing 0 times per week + relative frequency for flossing 1 time per week + relative frequency for flossing 2 times per week + relative frequency for flossing 3 times per week

= 0.0857 + 0.1 + 0.0571 + 0.1571

= 0.4

Therefore, the cumulative relative frequency for flossing 3 times per week is 40%.

Answer: To use Excel's NORM.INV() function to solve this problem, we can follow these steps:

Determine the z-score corresponding to the top 10% of the distribution. We can use the NORM.INV() function to find this value. Since we want the top 10%, we'll use a probability of 0.9 and a mean of 330lbs and a standard deviation of 15lbs. In Excel, we can use the formula:

=NORM.INV(0.9,330,15,TRUE)

This gives us a z-score of 1.281552.

Once we have the z-score, we can use the formula for a normal distribution to find the corresponding weight. The formula is:

z = (x - μ) / σ

where z is the z-score, x is the weight we're trying to find, μ is the mean weight (330lbs), and σ is the standard deviation (15lbs).

Rearranging this formula to solve for x, we get:

x = z * σ + μ

Substituting in the values we know, we get:

x = 1.281552 * 15 + 330

This gives us a weight of approximately 349lbs.

Therefore, the minimum weight of the sumo wrestlers at the highest weight of the league is about 349lbs, rounded to the nearest whole number.

Step-by-step explanation:

To solve the integral [tex]\rm\int_0^7 e^x \sin(x) dx\\[/tex], we can use integration by parts. Let u = sin(x) and dv = [tex]\rm e^x[/tex] dx, then we have:

[tex]\begin{align} \rm\int \rm e^x \sin(x) dx &= \rm-e^x \cos(x) + \rm\int e^x \cos(x) dx \\&= \rm -e^x \cos(x) + e^x \sin(x) - \int e^x \sin(x) dx\end{align}[/tex]

Rearranging, we get:

[tex] \begin{align}2 \rm \int e^x \sin(x) dx &= \rm e^x (\sin(x) - \cos(x)) \bigg|^7_0 \\& \rm= e^7 (\sin(7) - \cos(7)) - 1\end{align}[/tex]

Dividing both sides by 2, we get:

[tex] \rm\int_0^7 e^x \sin(x) dx = \frac{e^7 (\sin(7) - \cos(7)) - 1}{2} \\ [/tex]

Therefore, the value of the integral is

[tex] \rm \boxed{ \rm\frac{e^7 (\sin(7) - \cos(7)) - 1}{2}}[/tex]

Answer:

45.32

Step-by-step explanation:

We can conclude that m∠MNP has a measure of 0 degrees in parallelogram MNPQ.

A parallelogram is a four-sided geometric shape in which opposite sides are parallel and congruent (equal in length) to each other. It is a special case of a quadrilateral, which is any four-sided polygon.

In the given question,

Since opposite angles in a parallelogram are equal, we know that:

m∠MNP = m∠QMN

We also know that the sum of the interior angles of a triangle is 180 degrees. Therefore, we can find the measure of angle MQN as follows:

m∠MQN = 180 - m∠PQM = 180 - 135 = 45 degrees

Now, we can use the fact that the opposite angles in a parallelogram are equal to find the measure of angle QMN:

m∠QMN = m∠PQM = 135 degrees

Finally, we can use the fact that the angles in a triangle add up to 180 degrees to find the measure of angle MNP:

m∠MNP = 180 - m∠MQN - m∠QMN

= 180 - 45 - 135

= 180 - 180

= 0 degrees

Therefore, we can conclude that m∠MNP has a measure of 0 degrees. This means that the sides MN and PQ are parallel and do not intersect, so angle MNP does not exist.

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La hipotenusa mide 137 cm.

Podemos utilizar el teorema de Pitágoras para resolver este problema, que establece que en un triángulo rectángulo, el cuadrado de la hipotenusa es igual a la suma de los cuadrados de los catetos.

Sea "a" la medida del otro cateto, entonces:

a² + 77² = (a+49)²

Desarrollando los cuadrados y simplificando, tenemos:

a² + 5929 = a²+ 98a + 2401

Restando a ambos lados a², obtenemos:

5929 = 98a + 2401

Restando 2401 a ambos lados, obtenemos:

3528 = 98a

Dividiendo por 98, obtenemos:

a = 36

Por lo tanto, el otro cateto mide 36 cm, y la hipotenusa se puede calcular utilizando el teorema de Pitágoras:

h² = a² + b²

h2 = 36² + 77²

h² = 12996 + 5929

h² = 18925

Tomando la raíz cuadrada en ambos lados, obtenemos:

h = 137

Por lo tanto, la hipotenusa mide 137 cm.

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Since the triangle was rotated by 30 degrees, the angle AOA' is 30 degrees, as is the angle A'OC. Thus, m/AOC is equal to 60 degrees. To demonstrate that BD is perpendicular to CD, OB = OC, angles AOB and COA are both 60 degrees, and angles BOD and COD are both 60 degrees. BD and CD are both perpendicular to the diameter through A.

A triangle is said to be equilateral if all three of its sides are the same length and all three of its angles are exactly 60 degrees. It has three regular sides and is a polygon. When dividing an equilateral triangle into two congruent 30-60-90 triangles, the altitude, median, and angle bisector from any vertex are also the same line.

5. Let's identify the circle's intersection points as the equilateral triangle's A, B, and C. All angles in a triangle that is equilateral are 60 degrees. Point A will change locations when the triangle is rotated by 30 degrees; let's designate this new position A'. Since O is the centre of the circle, the angle AOC remains at 60 degrees. Since the triangle has been turned by 30 degrees, the angle AOA' is 30 degrees, as is the angle A'OC. Thus, m/AOC is equal to 60 degrees.

6. The center of the circle is O, and D is the point where the diameter through B intersects the diameter through C. To demonstrate that BD is perpendicular to CD, the center of the circle, OB = OC, and angles AOB and COA are both 60 degrees, so angle BOC is 120 degrees. Angle BOD is a right angle, and angle OBD is half of angle BOC, so it is 60 degrees. Angle COD is also a right angle, and angle OCD is half of angle BOC, so it is also 60 degrees. Since angles OBD and OCD are both 60 degrees, and they are angles in a triangle, the third angle (angle BDC) must also be 60 degrees.

Therefore, triangle BDC is equilateral, and BD = CD. Since BD and CD are both perpendicular to the diameter through A, BD is perpendicular to CD.

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Step-by-step explanation:

Here , the eqn is given as,

Let 1 kg onion cost Rs. y and 1kg potato cost Rs. y then,

3x + 2y= 240....eqn i

The When rate of potato increases by 20 percent,

New rate of unit kg potato becomes,

[tex]x \: + \: 20\% \: of \: x[/tex]

i.e. Rs.6x/5

And when price of onion decreases by 10% ,

New unit price is ,

[tex]y - 10\% \: of \: y[/tex]

i.e. Rs.9y/10

Now the second eqn becomes ,

20x + 21y= 2060

By using elimination method we get x= Rs 40 and y= Rs.60

Finally, we have to find percent change i.e

{(cost of 1kg onion - cost of 1kg potato)/ cost of 1kg potato}x 100%

This gives the ans as 50%

Step-by-step explanation:

if we extend the line VT to reach also the opposite side of the circle, we see that this line cuts the circle in half.

the whole arc angle of a half-circle is 360/2 = 180°.

now we can use the exterior angle rule :

the vertex angle (V) = 1/2 × difference of the angles of the intersected arcs.

the intersected arcs are

shorter = angle at T : the arc from U to the circle intersection with the line VT.

longer : the arc from U to the circle intersection with the extended line VT.

shorter + longer = 180°

longer = 180 - shorter

30 = 1/2 × ((180 - shorter) - shorter) =

= 1/2 × (180 - 2×shorter) = 90 - shorter

-60 = -shorter

shorter = 60°

the angle at T = 60°.

For the given functions f(x) and g(x), the inverse functions of f and g are:

a. f(g(x)) = x; g(f(x))  = x

ƒ and g are inverses of each other.

b. (g(x)) = x; g(f(x))  = x

ƒ and g are inverses of each other

The inverse functions of the functions f(x) and g(x),are determined as follows;

a. f(x) = 2x, g(x) = x/2

f(g(x)) = f(x/2)

f(g(x)) = 2(x/2)

f(g(x)) = x

g(f(x)) = g(2x)

g(f(x))  = (2x)/2

g(f(x))  = x

Since f(g(x)) = g(f(x)) = x, ƒ and g are inverses of each other.

b. f(x) = 2x + 1, g(x) = (x -1)/2

f(g(x)) = f((x-1)/2)

f(g(x)) = 2((x-1)/2) + 1

f(g(x)) = x - 1 + 1

f(g(x)) = x

g(f(x)) = g(2x + 1)

g(f(x)) = ((2x + 1) - 1)/2

g(f(x)) = 2x/2

g(f(x)) = x

Since f(g(x)) = g(f(x)) = x, ƒ and g are inverses of each other.

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Using division, we can find that Marcus will take 2 hours or 120 minutes to read the book.

Repetitive subtraction is the process of division. It is the multiplication operation's opposite. It is described as the process of creating equitable groups. By dividing numbers, we divide a larger number down into smaller ones such that the larger number obtained will be equal to the multiplication of the smaller numbers.

Marcus is reading a book with 80 pages.

He reads first 20 pages in 30 mins.

So, to read 1 page time taken

= 30/20

= 3/2

= 1.5 mins

Now to read 80 pages, time taken will be:

80 × 1.5

= 120 mins.

Therefore, Marcus will take 2 hours or 120 minutes to read the book.

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a. f(x) = x³

b. g(x) = -f(x-2)

Answer:

  a) parent: f(x) = x³

  b) g(x) = -f(x -2)

Step-by-step explanation:

You want the parent function and the given function written in terms of the parent function for g(x) = -(x -2)³.

The parent function is the function that remains after scale factors and translations are removed. In general, a transformed function will look like ...

  g(x) = c·f((x-a)/b) +d

which scales the function f(x) horizontally by a factor of b, vertically by a factor of c, and translates the result by (a, d).

Here, we recognize that ...

  g(x) = -(x -2)³

has c = -1, a = 2, b = 1, d = 0

and the parent function is ...

  f(x) = x³

Using the values for a, b, c, d that we recognized above, we have ...

  g(x) = -f(x -2)

__

Additional comment

When scale factors are negative, they reflect the function over the relevant axis. Here, the negative vertical scale factor reflects the function vertically over the x-axis. The translation is 2 units to the right.

Answer:

Step-by-step explanation:

scale factor is -4

D(-1 , -1) and if you divide either the x coordinate or the y coordinate by

4 - you get negative four which is the scale factor.

Answer:

To find ƒ⁻¹(x), we need to solve for x in terms of ƒ⁻¹(x).

So, we start with the equation:

x = 2 (ƒ⁻¹(x) - 1)

We isolate ƒ⁻¹(x) by dividing both sides by 2 and adding 1 to both sides:

ƒ⁻¹(x) = (x/2) + 1

Therefore, the inverse of the given function f(x) is:

ƒ⁻¹(x) = (x/2) + 1

we have successfully simplified the expression x⁵-x³ into x³(x + 1)(x - 1). This can be useful for further algebraic manipulation or simplification in larger expressions.

How to simplify?

To simplify the expression x⁵-x³, we can factor it using the distributive property of multiplication. First, we can factor out x³ from both terms:

x⁵ - x³ = x³(x² - 1)

Now, we can further simplify by recognizing that x² - 1 is a difference of squares, which can be factored into (x + 1)(x - 1). So, we have:

x⁵ - x³ = x³(x² - 1) = x³(x + 1)(x - 1)

This is the fully simplified form of the expression. We can also expand the expression to verify that it is equivalent to the original expression:

x³(x + 1)(x - 1) = (x³ * x) + (x³ * -1) + (x³ * 1) + (x³ * -1) = x⁴ - x³ + x³ - x³ = x⁴ - x³

So, we have successfully simplified the expression x⁵-x³ into x³(x + 1)(x - 1). This can be useful for further algebraic manipulation or simplification in larger expressions.

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Camila's decision to return to work part-time instead of relying solely on her benefits reflects her personal investment in her human capital. By continuing to work, she is not only earning a salary but also investing in herself and her future career prospects.

How to solve the problem?

Camila's decision to return to work part-time after her accident instead of maintaining all her benefits without working shows that she values her human capital and is invested in it. Human capital refers to the skills, knowledge, and abilities that individuals possess, which are gained through education, training, and experience. By choosing to work, Camila is not only earning a salary, but she is also developing and enhancing her skills, knowledge, and abilities, thereby increasing her human capital.

Furthermore, Camila's decision to return to work part-time also shows that she has a strong work ethic and is committed to her career. She understands the importance of staying active in the workforce, even if it means working part-time, to maintain her professional network and stay up-to-date with industry developments. This commitment to her career and her willingness to work despite her physical limitations is a testament to her determination and resilience.

Overall, Camila's decision to return to work part-time instead of relying solely on her benefits reflects her personal investment in her human capital. By continuing to work, she is not only earning a salary but also investing in herself and her future career prospects.

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A binomial experiment has given number of trails n and the given success probability n=2, p=0.2 then P(0) = 1.

A binomial experiment is an experiment where there have a fixed number of independent trials with only have two outcomes either success or failure.

A binomial experiment has given number of trails n and the given success probability p.

n=2, p=0.2

The probability of obtaining x successes in n independent trials of a binomial experiment, where the probability of success is p, is given by binomial probability distribution   P(x)= ₙ c ₓ p ˣ  (1-p)ˣ

Where x = 0, 1, 2, ---- , n

so, P(0)= ₂C₀ (0.2)⁰ (1-0.2)⁰

             = 1×1×1

             =1

Hence P(0)= 1

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Complete question:

1. The 95% confident that the true population mean is 64.97 and 435.03 hours. 2. The 99% confident is between 0.407 and 0.500. 3. The 95% estimate of the mean is (45,424, 48,576). 4. Sample size is 91.27.

A fundamental idea in statistics known as the central limit theorem (CLT) argues that, under specific circumstances, the sampling distribution of the mean of a random sample from any population tends to resemble a normal distribution as the sample size rises.

1. For 95% confidence interval we have:

CI = x ± z*(σ/√n)

Substituting the values:

CI = 400 ± 1.96*(39/√30) = (364.97, 435.03)

2. For 99% confidence interval we have:

CI = p-cap ± z*(√(p - cap(1- p-cap)/n))

Substituting the values:

CI = 165/360 ± 2.58*(√((165/360)*(195/360)/360)) = (0.407, 0.500)

Hence, we can be 99% confident that the true population proportion of married men, 18 or older, is between 0.407 and 0.500.

3. 95% estimate of the mean:

CI = x ± t*(s/√n)

Substituting the values:

CI = 47,000 ± 2.064*(1,050/√26) = (45,424, 48,576)

4. For sample size:

CI = 47,000 ± 2.064*(1,050/√26) = (45,424, 48,576)

Substitute the values:

n = (1.96*39 / 20)^2 = 91.27

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Answer:

y = 4 meters

Step-by-step explanation:

to find a volume of a cube you do (y × y) y

so if Y equals 4 then it would look like this:

(4 × 4)4 which = 64

a) The amount of the insurance or the sum assured by Julia Carson on his brick home is $80,000.

b) The annual premium is $5,148.00.

The sum assured represents the fixed amount that the insurance company pays to Julio Carson (the policyholder) if the unpredictable event, such as fire, occurs and damages his brick home.

The purpose of the payment is to reimburse Julio not for the costs incurred but according to the risks undertaken and it is a fixed sum of money unlike the sum insured, which reimburses to cover the costs incurred.

Replacement value of the brick home = $100,000

Sum insured = 80%

= $80,000 ($100,000 x 80%)

Monthly premium = $429

Number of insured years = 10 years

Annual premium = $5,148 ($429 x 12)

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