In this warm-up activity, you will use your knowledge from the previous lesson on compound angle formulas to derive expressions for the double angle formulas.
Derive a general expression for sin(2θ) and cos(2θ). Hint: sin(2θ) = sin(θ + θ), and use the compound angle formula that was introduced in the previous lesson. Be sure to do this for both sin(2θ) and cos (2θ).

(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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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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The volume of the prism with height h is h times the volume of the prism with height 1 unit.

If the original prism has a volume of V and a height of 1 unit, then the volume of the prism with height h is given by:

Bh = (V)(h)

where B is the area of the quadrilateral base.

The volume of the prism with height h is h times the volume of the original prism. So, we can compare the volume of the two prisms by looking at their volume ratios:

Volume ratio = Volume of prism with height h / Volume of original prism

= Bh / (V)(1)

= Bh / V

= h

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Answer:

16 litres = 100% of the tank capacity

x litres = 75% of the tank capacity (since 3/4 is equivalent to 75%)

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

16 * 75 = 100x

1200 = 100x

x = 12

Therefore, 3/4 of the tank capacity when the tank holds 16 litres is 12 litres.

Answer:

= 12 liters.

Step-by-step explanation:

16*3/4 =48/4.

= 12

The probability of selecting a black checker from a bag of 6 black and 4 red checkers, replacing it, and selecting another black is 9/25.

To find the probability follow these-

Determine the probability of selecting a black checker on the first draw.

There are 6 black checkers and a total of 10 checkers (6 black + 4 red).

So, the probability of selecting a black checker on the first draw is 6/10,

which can be simplified to 3/5.

Since the black checker is replaced, there are still 6 black checkers and

a total of 10 checkers in the bag. Therefore, the probability of selecting a

black checker on the second draw is also 3/5.

To find the probability of both events occurring, multiply the probabilities together: (3/5) × (3/5) = 9/25.

So, the probability of selecting a black checker from a bag of 6 black and

4 red checkers, replacing it, and selecting another black is 9/25.

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Answer:

x = 0 y = 7/5

The gradient of the line is 200, hence that is the answer.

What does the word "point" actually mean?

A point is a dot on a sheet of paper or in a plane. There are no lengths, widths, or heights in a point. It establishes a plane's position or location. To make a point, mark a dot on a piece of paper with an A or another capital letter.

The slope-intercept version of the equations y = 200x + 500 is y = mx + b, wherein m is the line's gradient and b is its y-intercept.

The value of x, which really is 200, acts as the gradient (or gradient) in this equation. As a result, the line's gradient is 200.

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[tex]y^(3)z^(2)[/tex] is the greatest common factor for the given set of monomials.

The highest power of each variable that is shared by all of the given monomials must be identified in order to calculate the greatest common factor (GCF) for the given list of monomials. To put it another way, we must first determine the prime factors of each monomial before determining the factors that are shared by all the monomials.

Prime factorization for each monomial:

[tex]x^(3)y^(5)z^(3) = (x * x * x) * (y * y * y * y * y) * (z * z * z)y^(3)z^(3) = (y * y * y) * (z * z * z)xy^(3)z^(2) = (x) * (y * y * y) * (z * z)[/tex]

Factors are common to all monomials are [tex]y^(3), z^(2)[/tex]. The GCF is [tex]y^(3)z^(2)[/tex].

This is appropriate GCF by dividing each monomial by [tex]y^(3)z^(2)[/tex] and verify that result is whole number. For eg:

[tex]x^(3)y^(5)z^(3) / (y^(3)z^(2)) = x^(3)y^(2)zy^(3)z^(3) / (y^(3)z^(2)) = yzxy^(3)z^(2) / (y^(3)z^(2)) = x/y[/tex]

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Answer:

2

Step-by-step explanation:

PEMDAS

(-3+4)2

(1)2

2x1=2

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

[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$1200\\ r=rate\to 0.2\%\to \frac{0.2}{100}\dotfill &0.002\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{weekly, thus fifty two} \end{array}\dotfill &52\\ t=years\dotfill &10 \end{cases} \\\\\\ A = 1200\left(1+\frac{0.002}{52}\right)^{52\cdot 10}\implies A \approx 1224.24[/tex]

Answer: $1248 in compound interest for 10 years

Step-by-step explanation:

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 correct solution of the expression 5a² -25bˣ is 5(a² - 5bˣ)

From the question, we have the following parameters that can be used in our computation:

5a² -25bˣ

The expression is already in its simplest form in terms of combining like terms.

However, we can still look into factoring the expression

To do this, we can factor out the greatest common factor of 5 and rewrite the expression as follows:

5a² -25bˣ = 5(a² - 5bˣ)

This is the factored form of the expression, where the greatest common factor of 5 has been factored out.

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Answer:

[tex]130m^{2}[/tex] , [tex]65.000 rs[/tex]

Step-by-step explanation:

Given Length = 15m
Given Breadth = 12m

Since there is a 1m gap between the sides,

Actual Length = 15 - 2 = 13m
Actual Breadth = 12 - 2 = 10m

From here find the Area:

L * B = AR of a cuboid , so

13 x 10 = [tex]130m^{2}[/tex] is the Area

for every [tex]m^{2}[/tex] it is 500 rupees, so for [tex]130m^{2}[/tex] ,

130 * 500 = 65,000 rupees.

y varies directly as x means

[tex]y=kx[/tex]

y varies inversely with z means

[tex]y=k\div z[/tex]

therefore

[tex]y=(kx^2)\div z[/tex]

k=constant of variation

[tex]6=(k9^2)\div 27[/tex]

[tex]6=(k81)\div27[/tex]

[tex]6=3k[/tex]

divide both sides by 3

[tex]2=k[/tex]

the constant of variation is 2

If the quantity y varies directly with the square of x and inversely with z, then we can write the following equation:

What is equation ?

A assertion that two mathematical expressions are equal is known as an equation. It typically consists of variables, constants, and mathematical operations, and it may also include functions, exponents, and other mathematical symbols. Equations are used to represent relationships between quantities in mathematics and to solve problems in various fields such as physics, engineering, and economics.

y = k * (x^2) / z

where k is the constant of variation.

We are given that when x is 9, z is 27, and y is 6. Substituting these values into the equation, we get:

6 = k * (9^2) / 27

Simplifying, we get:

6 = k * 3

k = 6 / 3

k = 2

Therefore, the constant of variation is 2.
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Answer:

To find the pressure that the box exerts on the ground, we need to divide the force by the area of the base:

pressure = force / area

We are given that the force is 4.5 kN and the area is 1.1 m². However, we need to convert the force from kilonewtons to newtons before we can divide by the area:

force = 4.5 kN = 4,500 N

Now we can plug in the values:

pressure = 4,500 N / 1.1 m² = 4,090.91 N/m²

To express the pressure in kilonewtons per square meter, we need to divide by 1,000:

pressure = 4.09091 kN/m²

Rounding to 2 decimal places, we get:

pressure ≈ 4.09 kN/m²

Therefore, the pressure exerted by the box on the ground is approximately 4.09 kN/m².

Answer:

The volume of the container is 1872 cubic cm. The volume of the water is 672 cubic cm.

Therefore, 672/1872 = 0.36 or 36%

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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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 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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Based on the information in the image, we can infer that the correct four-digit code is: 4321

To find the four-digit secret code we must observe the image carefully, especially the intertwined lines. What we must do is follow the direction of the lines in the order from A to D to identify which number the line of each letter matches.

According to the above we can infer that the lines of each letter coincide in the following way with the numbers:

So according to the image, the four-digit secret code would be 4321.

Note: This question is incomplete. Here is the complete information:

Attached image

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The range of values of x that satisfy the inequality 0 ≤ x < 6 is the set of all real numbers between 0 (inclusive) and 6 (exclusive), i.e., the interval [0, 6).

The inequality 0 ≤ x < 6 can be interpreted as follows: x can take on any value between 0 and 6, including 0 but not including 6. This means that x can be any number greater than or equal to 0 and less than 6. In other words, the range of values for x that satisfy this inequality is the set of all real numbers in the interval [0, 6), where the square bracket indicates that 0 is included in the interval (i.e., 0 is a valid value of x) and the round bracket indicates that 6 is excluded from the interval (i.e., 6 is not a valid value of x). This interval can be visualized as a number line with a closed circle at 0 and an open circle at 6, indicating that 0 is included in the range of values for x but 6 is not.

Therefore, any value of x between 0 and 6 (excluding 6) satisfies the inequality 0 ≤ x < 6.

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The number οf histοry bοοks is 180 that sοld in a week. The number οf chemistry textbοοks is 108 sοld in a week.

There are many different ways tο define an equatiοn. The definitiοn οf an equatiοn in algebra is a mathematical declaratiοn that illustrates the equality οf twο mathematical expressiοns.

Given that the tοtal number οf bοοks that sοld in a week is 288.

The number οf chemistry textbοοks sοld was 72 less than the number οf histοry textbοοks sοld.

Assume that the number οf histοry textbοοks sοld was x.

The number οf chemistry textbοοks sοld was (x-72)

The tοtal number οf bοοks sοld was x + (x-72) = 2x - 72.

Accοrding tο the questiοn:

2x - 72 = 288

Add 72 frοm bοth sides:

2x - 72 + 72 = 288 + 72

2x = 360

Divide bοth sides by 2:

x = 360/2

x = 180

The number οf histοry textbοοks sοld was 180.

The number οf chemistry textbοοks sοld was (180-72) = 108.

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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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The nurse will have to administer 3 milliliters with each dose.

When the prescriber orders medication f 150 mcg po q.12h, the pharmacy sends a bottle labeled:

medication f 0.05 mg/ml.

Milliliters will the nurse administer with each dose :

The conversion factor from mcg to mg is 1/1000.

Therefore, 150 mcg = 150/1000 = 0.15 mg.

The nurse has to administer 0.15 mg of medication f per dose.

The medication f available with the pharmacy is 0.05 mg/ml.

So, the nurse has to administer 0.15 mg in a single dose, which is equal to three times the quantity of medication f available in the bottle (0.05 mg/ml).

Therefore, the nurse has to administer 3 ml (0.05 mg/ml × 3 ml) of medication f with each dose.

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By using the scale diagram, the height of this person in meters is equal to 1.6 meter.

In Geometry, the scale factor of a geometric figure can be calculated by dividing the dimension of the image (new figure) by the dimension of the pre-image (original figure):

Scale factor = Dimension of image (new figure)/Dimension of pre-image(actual figure)

Based on the information provided about the lorry, the lorry is approximately 11.2 m in length and 14 cm on a scale diagram. Therefore, the height of the person in meters can be calculated as follows;

1 centimeter = 11.2/14

1 centimeter = 0.8 meter.

Height of the person in meters = 0.8 meter × 2

Height of the person in meters = 1.6 meter.

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The number of students who passed the examination is 152.

Given: Marks distribution in Mathematics

To Find: Plot an ogive, the median, the number of students who scored distinction marks (75% and above), and the number of students who passed the examination if the pass marks are 35%.

Solution:

Marks Number of students

Less than 10 7

Less than 20 28

Less than 30 54

Less than 40 71

Less than 50 84

Less than 60 105

Less than 70 147

Less than 80 180

Ogive:

An ogive is a graph that represents the cumulative frequency distribution of a set of data. To plot an ogive, we plot the upper limits of each class on the x-axis and the corresponding cumulative frequency on the y-axis. We then join the points with a free-hand smooth curve to obtain the ogive.

Median:

To find the median from the graph, we draw a line from the value of 50 on the y-axis to the ogive curve and then drop it down to the x-axis. The point where the line intersects the x-axis gives us the median. In this case, the approximate median value from the graph is 52.

The number of students who scored distinction marks:

To find the number of students who scored distinction marks (75% and above), we first calculate 75% of the total number of students, which is 0.75 x 180 = 135.

From the ogive graph, we see that the number of students who scored less than 75% marks is 105.

Therefore, the number of students who scored 75% or more marks is 180 - 105 = 75.

The number of students who passed the examination:

To find the number of students who passed the examination if the pass marks are 35%, we first calculate 35% of the total number of students, which is 0.35 x 180 = 63.

From the ogive graph, we see that the number of students who scored less than 35% marks is 28.

Therefore, the number of students who passed the examination is 180 - 28 = 152.

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Answer: SA=5183.62788

Step-by-step explanation:

A=2πrh+2πr2=2·π·15·40+2·π·152≈ 5183.62788

~Find the radius- 30 divided by 2= 15 so radius =15

Option (a) is correct Mr. Reynolds walks 2000 feet per minute for 8 minute. Here it is the condition of speed.

Speed is define as the distance covered by a person in a given interval or time. It is generally denotes in Km/h or m/sec.

[tex]Speed\ (S)= Distance\ (d) * Time\ (t)[/tex]

Distance, It is an area cover by a person to complete his journey.

Distance is generally denoted by km, m etc,

Time, It is an interval in which generally person complete it task.

It is generally denoted by hour (h), Minute (min) or Second (Sec).

The correct is (a) Mr. Reynolds walks 2000 feet per minute for 8 minute. Here it is the condition of speed.

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