An accounting firm agrees to purchase a computer for $120,000 (cash on delivery) and the delivery date is in 270 days. How much do the owners need to deposit in an account paying 0. 65% compounded quarterly so that they will have $120,000 in 270 days?

The new monthly salary after two increases of 6.9% and 15% is 2867.

The new monthly salary after two increases is calculated as follows:

First Increase:

The initial salary of 2332 increases by 6.9%. This is calculated by multiplying the salary by the percentage increase, i.e. 2332 x 0.069 = 160.68. The new salary after the first increase is 2332 + 160.68 = 2492.68.

Second Increase:

The salary after the first increase of 2492.68 is further increased by 15%. This is calculated by multiplying the salary by the percentage increase, i.e. 2492.68 x 0.15 = 373.90. The new salary after both increases is 2492.68 + 373.90 = 2866.58.

Rounded to the nearest dollar, the new monthly salary after both increases is 2867.

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We can calculate the percentages according to the amounts provided in the prompt, and then make a number line to illustrate each percentage as follows:

a) The price of the book decreased by 20%.

$15.00             $12.00

  |-----------------|------->

      100%            80%

b) The percentage decrease in the number of students who take the bus to school is 25%.

200               150

  |-----------------|------->

     100%            75%

Calculating percentages involves finding a portion of a whole expressed as a fraction of 100. To calculate a percentage, divide the part by the whole and then multiply the result by 100.

For example, if 20 out of 100 students are absent, the percentage of absent students is:

(20/100) x 100 = 20%.

With that in mind, we can conclude that we have correctly answered this question.

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We want to find the number of yards Adrian ran during the last game, which is represented by the variable a. Adrian ran 90 yards during the last football game.

Let's define the variables we will use in this problem:

r: the number of yards Reggie ran during his previous personal best.

m: the number of yards Monte ran during the last game.

a: the number of yards Adrian ran during the last game.

From the problem statement, we know that:

Reggie ran for 180 yards during the last game, which is 40 more yards than his previous personal best: r + 40 = 180.

Monte ran 50 more yards than Adrian during the same game: m = a + 50.

Monte ran the same amount of yards Reggie ran in one game for his previous personal best: m = r.

We want to find the number of yards Adrian ran during the last game, which is represented by the variable a. To solve for a, we need to use the information we have to create an equation that relates a to the other variables.

From the equation r + 40 = 180, we can solve for r by subtracting 40 from both sides: r = 180 - 40 = 140.

Substituting m = r into the equation m = a + 50, we get: r = a + 50.

Now we can substitute the value of r we found earlier into this equation: 140 = a + 50.

Solving for a, we subtract 50 from both sides: a = 90.

Therefore, Adrian ran 90 yards during the last football game.

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Based on the information in the table, the graph shows a diagonal going down from left to right.

To graph in a Cartesian plane we must be clear that it is a tool to graph the coordinates. The first number of the coordinate corresponds to the x-axis (horizontal) and the second number corresponds to the y-coordinate (vertical). According to the above, we can infer the graph would look like this (image attached). This graph corresponds to a function because it is a constant, in this case it is a diagonal line that descends from left to right.

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Answer:200

Step-by-step explanation:

A+B=25

A^2+B^2= 225

we use short multiplication formulas:

(A+B)^2=A^2+B^2+2AB

and solve the equation

25^2=225+2AB

625=225+2AB

625-225=2AB

400=2AB

AB=200

1) In the first line item - the Sales Price, Percent Saved and Percent Paid are given.

To work out the Original Price, we can state that: if 20% off the value of x = 800 what is x, (where x is the Original Price)?
If 20% is taken off the value of x, we need to find the original value of x before the discount was applied. We can set up the following equation:

x - 20% of x = 800

Simplifying by factoring out x on the left side, we get:

x * (1 - 20%) = 800

Converting 20% to a decimal, we get:

x * (1 - 0.20) = 800

Simplifying, we get:

x * 0.80 = 800

Dividing both sides by 0.80, we get:

x = 1000

Therefore, the original value of x before the 20% discount was applied is $1000.

Since the Original price is $1000, and the Sales price is $800, that means the Amount of Discount ($) is:

Original Price - Sales Price

= $1,000 - $800
= $200

Thus, the discount value for the Television is $200and the original price is $1000.


2) With regard to the sneakers, Original Price was $80 and the percent saved is 25%.The percentage saved is 75%.

Thus, the Sales Price = Original Price - (Original Price * Percent Saved)
Sales Price = 80 - (80 * 25%)
Sales Price = 80 - 20
Sales Price = $60

Since $20 was removed form the Original Price, that means the Amount of Discount ($) is $20.

3)

For the Video Game, we know the Sales Price and the Percent Paid, which is 90%.
That means $54 = 90% of x, where x is the Original Price.

To find x,
54 = 0.9x
x = 54/0.9
x (Original Price) = $60
Thus, the Original Price of the Video Game is $60.

Since Original Price = $60; and
Sales Price = $54
Amount discounted = Original Price - Sales Price
Amount discounted = 60-54
Amount discounted = $6
As a percentage of the Original Price $6 is: 6/60 * 100

= 10%


4) For the MP3 player, we know that that the Sales Price = $51.60 which is 60% (Percent Paid) of the Original Price. We also know that the Percent saved is 40% of the Original price.

Thus, if $51.6 is 60% of the original price, we can use the following proportion to find the original price:

60% = $51.6 / original price

To solve for the original price, we can isolate it by multiplying both sides by the reciprocal of 60%, which is 100% / 60%, or 5 / 3:

original price = $51.6 / (60%)

original price = $51.6 / (0.60)

original price = $86

Therefore, the original price before the 60% reduction was $86.

Amount of Discount = Original Price - Sales Price
= 86 - 51.6
= $34.4

Thus, Original Price and Amount of Discount for the MP3 Player are $86.00 and $34.40 respectively.


5)

For the book, we know that the Amount of Discount is $2.60 which is 80% (Percent Paid) of the Original Price.
If $2.6 is 20% of the original price, we can use the following proportion to find the original price:

20% = $2.6 / original price

To solve for the original price, we can isolate it by multiplying both sides by the reciprocal of 20%, which is 100% / 20%, or 5:

original price = $2.6 / (20%)

original price = $2.6 / (0.20)

original price = $13

Therefore, the original price before the 20% reduction was $13.
Since Original price = $13 and Discount given is $2.60, Thus,

Sales Price = Original Price - Discount Given
Sales Price = 13- 2.60
Sales Price = $ 10.40

Note that the percentage saved (discount) =
2.6/13 * 100

= 20%

Thus, the Percentage Saved  = 20%.

6)
For the Snack Bar, we know that the Sales Price is $1.70 and the Amount of Discount is $0.30. This means that the Original Price  =

Sales Price + Discount Given = 1.7 + 0.3

Original Price = $2.00


To find what percent 0.3 is of 2, we can use the following proportion:

x/100 = 0.3/2

where x is the percent we want to find.

To solve for x, we can cross-multiply and simplify:

2x = 30

x = 15

Therefore, $0.3 is 15% of $2 which is the percent saved.

If the percent saved is 15%, then the percent paid is:

100 - 15
= 85%

Thus, the percent paid is 85% of the Original Price.


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Answer:

Decimal: 0.25

Fraction: 1/4

Step-by-step explanation:

2 / 3 * 3/8

2/3 X* 3/8

=  

2 × 3

3 × 8

=  

6/ 24

=  

6 ÷ 6

24 ÷ 6

=  

1/4

Please mark this answer brainliest if it helps!

The first mechanic wοrked fοr 10 hοurs, and the secοnd mechanic wοrked fοr 15 - 10 = 5 hοurs.

Let x be the number οf hοurs the first mechanic wοrked, then the number οf hοurs the secοnd mechanic wοrked wοuld be (15 - x).

The tοtal cοst οf the first mechanic wοuld be 65x, and the tοtal cοst οf the secοnd mechanic wοuld be 45(15 - x) = 675 - 45x.

Tοgether, their tοtal cοst wοuld be 65x + (675 - 45x) = 20x + 675.

We knοw that their tοtal cοst was $875, sο we can set up the equatiοn:

20x + 675 = 875

Subtracting 675 frοm bοth sides, we get: 20x = 200

Dividing bοth sides by 20, we get: x = 10

Therefοre, the first mechanic wοrked fοr 10 hοurs, and the secοnd mechanic wοrked fοr 15 - 10 = 5 hοurs .

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The explicit formula for the population in the United States between 1930 and 1975 is [tex]P(t) = 0.00004t^1.9 + 1[/tex], where t is the number of years after 1930.

To calculate the population for any year between 1930 and 1975, we can simply substitute the corresponding year for t in the equation. For example, if we want to calculate the population in 1950, we can substitute t = 1950 in the equation.

Therefore, [tex]P(1950) = 0.00004(1950)^1.9 + 1 = 0.0035 + 1 = 1.0035[/tex].

This is the population in 1950. In conclusion, the explicit formula for the population in the United States between 1930 and 1975 is [tex]P(t) = 0.00004t^1.9 + 1[/tex], where t is the number of years after 1930.

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The area οf οnly the frame is apprοximately 50.27 square inches.

The area οf a circle is the amοunt οf surface inside the bοundary οf a circle. It is a measure οf hοw much space the circle takes up in twο-dimensiοnal space. The fοrmula tο calculate the area οf a circle is:

[tex]A = \pi r^2[/tex]

where A is the area οf the circle and r is the radius οf the circle.

The diameter οf the mirrοr with the frame is 17 inches, and the diameter οf οnly the mirrοr (withοut the frame) is 15 inches.

Therefοre, the width οf the frame is:

17 inches (diameter οf mirrοr with frame) - 15 inches (diameter οf οnly mirrοr) = 2 inches

Sο the radius οf the mirrοr withοut the frame is:

r₁= (15 inches / 2) = 7.5 inches

The area οf the mirrοr withοut the frame is:

A₁= π(7.5)² square inches ≈ 176.71 square inches

The radius οf the mirrοr with the frame is:

r₂= (17 inches / 2) = 8.5 inches

The area οf the mirrοr with the frame is:

A₂= π(8.5)² square inches ≈ 226.98 square inches

The area οf οnly the frame is the difference between these twο areas:

=A₂-A₁

=π(8.5)²- π(7.5)² square inches

≈ 50.27 square inches

Therefοre, the area οf οnly the frame is apprοximately 50.27 square inches.

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For the given figure, the area of the shaded region is obtained as 100 m².

What is area?

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.

The area of shaded region can be obtained as -

Area of shaded region =  Area of rectangular concrete path - Area of rectangular lawn

The formula for area of rectangle is -

Area of rectangle  =  Length × width

The area of rectangular concrete path is -

Length  =  (30+2) m  =  32 m

width  =  (18+2) m  =  20 m

Area of rectangular concrete path  =  32 × 20

Area of rectangular concrete path  =  640 m²------(1)

The area of rectangular lawn is -

Length  =  30 m and width  =  18 m

Area of rectangular lawn  =  30 × 10

Area of rectangular lawn  =  540 m²------(2)

To find the are of shaded region subtract equation (2) from (1) =

Area of shaded region = (1) - (2)

Area of shaded region =  (640 – 540) m²

Area of shaded region =  100 m²

Therefore, the total area of concrete is 100 m².

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A rectangular lawn 18 m by 30 m is surrounded by a concrete path 1 m wide. Draw a diagram of the situation and find the total area of concrete.

Answer:

x=120 and y=60

Step-by-step explanation:

As we know, a straight line is 180 degrees.

To find x, you would do 180= 60 + x to get 120.

I forget how but I know y= 60 because its the same angle as 60 degrees.

Step-by-step explanation:

The correct order of the functions in decreasing order of their periods is:

y = -10sin(pi/5 - 2pi)

y = 2/3cot(pi/4)+6

y = 5csc(3x)+6

y = 1/2tan(5pi/6 + pi)

y = -3cos(x+2pi)

Note: The periods of trigonometric functions are determined by the coefficient of the independent variable (x in this case). The period of y = asin(bx + c) or y = acos(bx + c) is 2pi/b, and the period of y = atan(bx + c) or y = acot(bx + c) is pi/b. The period of y = acsc(bx + c) or y = asec(bx + c) is 2pi/|b|.

The criteria that we could use to prove the congruency that ΔABC ≅ ΔDEC is;

Vertical Angles are Congruent. We could then use AAS to prove that ΔABC ≅ ΔDEC

There are different triangle congruency postulates namely:

SSS: Side Side Side Congruency Postulate

SAS: Side Angle Side Congruency Postulate

ASA: Angle Side Angle Congruency Postulate

AAS: Angle Angle Side Congruency Postulate

HL: Hypotenuse Leg Congruency Postulate

We want to prove that ΔABC ≅ ΔDEC

Now, from the given image, we see that we are given 2 congruent sides.

However, angle ECD is congruent to angle DCA because they are vertically opposite angles and vertical angles are congruent.

Thus, the triangles are congruent by AAS Congruency postulate.

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The median increases by 0.5 and the mean decreases by 4.375

What are median and mean?

Median and mean are measures of central tendency that are commonly used in statistics. The median is the middle value in a sorted list of numbers. To find the median, you need to arrange the numbers in order from lowest to highest (or highest to lowest) and then find the middle value. If there is an odd number of values, the median is the middle number. If there is an even number of values, the median is the average of the two middle numbers. The mean is the average of all the numbers in a list. To find the mean, you need to add up all the numbers in the list and then divide by the total number of values. It is calculated by summing up all the values and dividing the result by the total number of values. The mean is influenced by outliers and extreme values in the data.

(a) The median is the middle value in a sorted list of numbers. In the original list, the median is 547. If the number 605 is changed to 549, then the new list becomes:

381, 465, 496, 537, 547, 589, 604, 549

Now the median is the average of the two middle values, which are 547 and 549. Therefore, the median increases by:

(549 + 547)/2 - 547 = 0.5

So, the median increases by 0.5.

(b) The mean is the average of all the numbers in a list. In the original list, the mean is:

(381 + 465 + 496 + 537 + 547 + 589 + 604 + 605) / 8 = 527.5

If the number 605 is changed to 549, then the new list becomes:

381, 465, 496, 537, 547, 589, 604, 549

The mean of this new list is:

(381 + 465 + 496 + 537 + 547 + 589 + 604 + 549) / 8 = 523.125

Therefore, the mean decreases by:

527.5 - 523.125 = 4.375

So, the mean decreases by 4.375

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The equation that represents the variation is given by F = 42m/p

An equation is an expression that shows the relationship between numbers and variables using mathematical operators such as addition, subtraction, exponent, multiplication and division.

F varies directly as m and inversely with p, hence:

F ∝ m/p

Let k represent the constant of proportion. Therefore:

F= k*m/p

F = 36 when m = 6 and p = 7. Hence:

36 = k*6 / 7

k = 36 * 7 / 6

k = 42

The equation is F = 42m/p

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m= y2-y1/x2-x1

m= 5-5/2+6= 0

Answer: The slope is equal to zero

Step-by-step explanation:

Find the slope of (-6,5) and (2,5)

First subtract the x coordinates 2 and -6

which equals 8

Next subtract the y coordinates 5 and 5

which equals o

Then you get 8/0 and when you simplify you get 0, so the slope is zero.

a is correct

Question 1 (1 point )If y varies directly as x and y = 15 when x = 12, find y when x = 32. The value of y when x = 32 is 40.How to solve direct variation problems?A direct variation is a function that can be expressed in the form y = kx, where k is a nonzero constant. Two quantities x and y are said to be in direct variation if they satisfy this relationship. To solve direct variation problems, you can use the following formula: y = kx, where k is the constant of variation.The solution to the problem is given below:Given:y varies directly as xx = 12, y = 15To find:y when x = 32Formula:Since y varies directly as x, we have:y = kxLet's substitute x and y values in the above equation:15 = k × 1215/12 = kk = 5/4Thus, the equation of the direct variation is: y = (5/4) xTo find the value of y when x = 32, substitute 32 for x:y = (5/4) × 32y = 40Therefore, y = 40 when x = 32. Hence, the option (a) is correct.

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The probability that the battery lasts less than 25 hours is 0.3446.

The probability of P(Z > -1.5) is 0.9332. The value of -1.5 has to be positive for this computation because the normal distribution curve is symmetrical about the z = 0 line. The probability of being at z = -1.5 is the same as being at z = +1.5. Thus, the probability is found by referencing the z-table for the area under the curve to the right of the mean: 0.9332.

The probability of P(Z < 1.15) is 0.8749. The area below z = 1.15 is needed for this calculation. The probability that the standard normal random variable is less than 1.15 is 0.8749.iii) The probability of P(Z > 0.35) is 0.3632. The area under the curve to the right of the mean is needed. The probability that the standard normal random variable is greater than 0.35 is 0.3632.

Given that, the amount of time the battery lasts in a toy racing car is normally distributed with a mean of 26 hours and a standard deviation of 2.5 hours. Let X denote the time a battery lasts, which is a normally distributed random variable. Find the following probabilities:a) P(25 < X < 29)To calculate the probability,

we first standardize the values of 25 and 29 using the formula below:

z1 = (25 - 26) / 2.5 = -0.4 and z2 = (29 - 26) / 2.5 = 1.2 Now, we find the probability as shown:P(-0.4 < Z < 1.2) = P(Z < 1.2) - P(Z < -0.4) = 0.8849 - 0.3446 = 0.5403 Thus, the probability that the battery lasts between 25 and 29 hours is 0.5403.b) P(X > 29)

To find the probability, we first standardize the value of 29 using the formula below:z = (29 - 26) / 2.5 = 1.2 Now, we find the probability as shown:P(Z > 1.2) = 1 - P(Z < 1.2) = 1 - 0.8849 = 0.1151 Thus, the probability that the battery lasts longer than 29 hours is 0.1151.c) P(X < 25)To find the probability, we first standardize the value of 25 using the formula below:z = (25 - 26) / 2.5 = -0.4 Now, we find the probability as shown:P(Z < -0.4) = 0.3446

Thus, the probability that the battery lasts less than 25 hours is 0.3446.

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The equation with y as the subject is y = x^m / (4x^n - 2)

To solve for y as the subject of the equation, we need to isolate y on one side of the equation. Here's how we can do it,

First, we can rearrange the equation to get all the y terms on one side:

4x^n y - 2y = x^m

Next, we can factor out y from the left-hand side,

y(4x^n - 2) = x^m

Finally, we can divide both sides by (4x^n - 2) to isolate y,

y = x^m / (4x^n - 2)

Therefore, the equation with y as the subject is:

y = x^m / (4x^n - 2)

where m and n are the constants given in the original equation.

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The given question is incomplete, the complete question is:

Hence , write down the equation with y as the subject ( in terms of x ). The value of m and q must be indicated. equation 4x^ny -2y - x^m = 0, m and n are constant

By using the "acosd" function in MATLAB, you can easily calculate the angles of obtuse triangles in real-life objects.

The shapes of obtuse triangles occur in real-life objects such as kites and architectural designs. An obtuse triangle is a type of triangle that has one obtuse angle, which means that one of its angles is greater than 90 degrees.

These types of triangles can be seen in many real-life objects, such as kites, which often have an obtuse angle at the top where the two sides of the kite meet. Similarly, architectural designs often include obtuse angles in order to create unique and interesting shapes.

In terms of using MATLAB to calculate the angles of an obtuse triangle, you can use the "acosd" function to calculate the angle in degrees. For example, if you have a triangle with sides of length 3, 4, and 5, you can use the following code to calculate the angle between the sides of length 3 and 4:

angle = acosd((3^2 + 4^2 - 5^2)/(2*3*4));

This will give you an angle of 90 degrees, which is a right angle. However, if you change the lengths of the sides to create an obtuse triangle, such as a triangle with sides of length 3, 4, and 6, you can use the same code to calculate the obtuse angle:

angle = acosd((3^2 + 4^2 - 6^2)/(2*3*4));

This will give you an angle of approximately 98.13 degrees, which is an obtuse angle. By using the "acosd" function in MATLAB, you can easily calculate the angles of obtuse triangles in real-life objects.

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Answer:

x = 0 y = 7/5

A circular park of radius 14 m has a road of 7 m width all around on its outside then the  area of the road is approximately 153.943 square meters.

The circular park has a radius of 14 m. This means that the diameter of the park is 2 x 14 = 28 m. The road around the park has a width of 7 m. This means that the total width of the park and the road is 28 + 7 + 7 = 42 m. The area of the circular park can be found using the formula for the area of a circle: Area of park =[tex]π x r^2 = π x 14^2 = 615.752 m^2[/tex] (rounded to 3 decimal places)

The area of the park and the road can be found by calculating the area of the larger circle and subtracting the area of the park: Area of park and road =[tex]π x (r+7)^2 - π x r^2 = π x (14+7)^2 - π x 14^2 = π x 21^2 - π x 14^2 = π x (441 - 196) = π x 245 = 769.695 m^2[/tex] (rounded to 3 decimal places)

To find the area of just the road, we need to subtract the area of the park from the area of the park and road: Area of road = Area of park and road - Area of park = 769.695 - 615.752 =[tex]153.943 m^2[/tex] (rounded to 3 decimal places)

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Answer:

V = (4/3)π(4^3) = 256π/3 cubic units

Step-by-step explanation:

the acorn falls from the height of 25 feet above the ground, it means the initial time when it falls is t = 0. The time when it lands on the ground is t = 1.25

So the acorn was in the air for 1.25 seconds

Part 1: Triangles are similar by Side-Side-Side similarity.

Part 2: Triangles are similar angle-Angle (AA) similarity.

Part 1: Ratios must be equal for triangle similarity.

Taking the ratios of the sides:

17/37.4  = 20/44 = 25/55

On solving

0.4545 = 0.4545 = 0.4545

As the ratios are equal, triangles are similar.by Side-Side-Side similarity.

Part 2:

∠F = ∠H (given)

∠EGF = ∠JGH (vertically opposite)

By Angle-Angle (AA) similarity.

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The correct question is 1 and 2.

The probability that at least nine flashlights will work is 0.3874 or 0.387 to three decimal places.

Given: A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages.

Suppose that 90% of all batteries from a certain supplier have acceptable voltages. Among ten randomly selected flashlights,

Let p be the probability that any single battery has an acceptable voltage.

Let X be the number of flashlights out of 10 that work.

Then X has a binomial distribution with n=10 and p=0.9.

The probability that at least nine of the ten flashlights work is P(X≥9).

We have

[tex]P(X=9)= 10C_9 \times 0.9⁹ \times 0.1 = 0.3874

P(X=10)= 10C_10 \times 0.9¹⁰ \times 0 = 0[/tex]

P(X≥9)  = P(X=9) + P(X=10)

            = 0.3874 + 0

            = 0.3874

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(d) The value of k for which the vertex of the parabola y=x2+kx+9 lies on the line x=−3 is 6. (e) The value of k for which the vertex of the parabola y=kx2+3x+4 lies on the line x=3 is -2.

(d) k=6 (e) k=-2

The vertex form of a quadratic function is given as;

f(x) = a(x-h)² + kf(x) = a(x−h)²+k, with vertex (h, k)

∴ for the quadratic equation y=x²+kx+9, the vertex is given by;

x = -b/2a = -k/2 ∴ k = -2x, using x=-3k = -2x = 6,

thus the vertex of the parabola y=x²+6x+9 lies on line x = -3

(e) Similarly, for the quadratic equation y=kx²+3x+4, the vertex is given by;

x = -b/2a = -3/2k ∴ k = -2x/3, using x=3k = -2x/3 = -2(3)/3 = -2,

thus the vertex of the parabola y=-2x²+3x+4 lies on the line x = 3

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The value of [tex]\( \ln \left[\frac{\left(x^{3}-2\right)^{5} \cdot \sqrt{x^{3}}}{2 \cdot \sqrt[3]{(4 x+1)^{7}}}\right] \)[/tex] is:
[tex]\ln \sqrt{x^{3}} - \ln \left(2 \cdot \sqrt[3]{(4 x+1)^{7}}\right)\][/tex]

To find the value of \[tex]( \ln \left[\frac{\left(x^{3}-2\right)^{5} \cdot \sqrt{x^{3}}}{2 \cdot \sqrt[3]{(4 x+1)^{7}}}\right] \)[/tex], we can use the following steps:

1. Factor out [tex]\( \left(x^{3}-2\right)^5 \)[/tex]:
[tex]\[ \ln \left[\frac{\left(x^{3}-2\right)^{5} \cdot \sqrt{x^{3}}}{2 \cdot \sqrt[3]{(4 x+1)^{7}}}\right][/tex] = [tex]ln \left[\frac{\left(x^{3}-2\right)^{5} \cdot \frac{\sqrt{x^{3}}}{2 \cdot \sqrt[3]{(4 x+1)^{7}}}}{1}\right]\][/tex]

2. Use the product rule for logs:
[tex]\[ \ln \left[\frac{\left(x^{3}-2\right)^{5} \cdot \frac{\sqrt{x^{3}}}{2 \cdot \sqrt[3]{(4 x+1)^{7}}}}{1}\right] = \ln \left(x^{3}-2\right)^5 + \ln \left(\frac{\sqrt{x^{3}}}{2 \cdot \sqrt[3]{(4 x+1)^{7}}}\right)\][/tex]

3. Evaluate each logarithm separately:
[tex]\[ \ln \left(x^{3}-2\right)^5 = 5\ln \left(x^{3}-2\right) \]\[ \ln \left(\frac{\sqrt{x^{3}}}{2 \cdot \sqrt[3]{(4 x+1)^{7}}}\right)[/tex] = [tex]\ln \sqrt{x^{3}} - \ln \left(2 \cdot \sqrt[3]{(4 x+1)^{7}}\right)\][/tex]

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