Find the value of the variables or indicated side length in each isosceles triangle.

Using cosine rule, the measure of side c is calculated as: 4.688

In trigonometry, the Cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.

We are given the parameters as:

a = 5

B = 7

C = 42 degrees

Using cosine rule, we have:

c = √(a² + b² - 2ab cos C)

c = √(5² + 7² - 2(5 * 7) cos 42)

c = √(25 + 49 - 52.02)

c = √21.98

c = 4.688

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Answer:

Step-by-step explanation:

We can start by writing the given equation in terms of polar form:

Re(z1,z2-bar) = modulus of z1 * modulus of z2

If we write z1 and z2 in polar form, we get:

z1 = r1(cosθ1 + i sinθ1)

z2 = r2(cosθ2 + i sinθ2)

Then, the conjugate of z2 is:

z2-bar = r2(cosθ2 - i sinθ2)

Using the formula for the real part of the product of two complex numbers, we get:

Re(z1,z2-bar) = Re(z1 * z2-bar)

Substituting the expressions for z1 and z2-bar, we get:

Re(z1,z2-bar) = Re(r1(cosθ1 + i sinθ1) * r2(cosθ2 - i sinθ2))

Simplifying the product, we get:

Re(z1,z2-bar) = Re(r1r2[(cosθ1 cosθ2 + sinθ1 sinθ2) + i(sinθ1 cosθ2 - cosθ1 sinθ2)])

The real part of this expression is:

Re(z1,z2-bar) = r1r2(cosθ1 cosθ2 + sinθ1 sinθ2)

Using the identity cos(θ1 - θ2) = cosθ1 cosθ2 + sinθ1 sinθ2, we can write:

Re(z1,z2-bar) = r1r2 cos(θ1 - θ2)

Now we can substitute this expression back into the original equation and get:

r1r2 cos(θ1 - θ2) = r1r2

Dividing both sides by r1r2, we get:

cos(θ1 - θ2) = 1

This equation is true if and only if θ1 - θ2 = 2πn for some integer n. In other words, θ1 = θ2 + 2πn, where n is an integer.

Therefore, we have proved that Re(z1,z2-bar) = modulus of z1 * modulus of z2 if and only if arg z1 = arg z2 + 2πn, where n is an integer.

The interest earned by Sarah in four (4) years is approximately $62.16.

The formula accrued amount in a compounded interest is expressed as;

A = P( 1 + r/n )^( n × t )

Where A is accrued amount, P is principal, r is interest rate and t is time.

Given that;

First, we convert R as a percent to r as a decimal

r = R/100

r = 7/100

r = 0.07

Now, the the values into the above formula and solve for accrued amount (A).

A = P( 1 + r/n )^( n × t )

A = $200( 1 + 0.07/1 )^( 1 × 4 )

A = $200( 1 + 0.07 )^( 4 )

A = $200( 1.07 )^( 4 )

A = $262.16

Now, we know that;

Accrued amount (A) = Principal (P) + Interest (I)

Solve for Interest (I)

$262.16 = $200 + I

I = $262.16 - $200

I = $62.16

Therefore, the interest earned is $62.16.

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Answer:

Write the whole number as a fraction by placing it over 1. For example, if you want to multiply 2/5 by 3, you can write 3 as 3/1.

Multiply the numerators of the two fractions together. For example, 2/5 x 3/1 = (2 x 3) / 5 = 6/5.

Simplify the resulting fraction, if necessary, by reducing it to its lowest terms. For example, 6/5 can be simplified to 1 1/5 (or 1.2 as a decimal).

Therefore, when you multiply a fraction by a whole number, you multiply the numerator of the fraction by the whole number and keep the denominator the same.

Step-by-step explanation:

The mean οf the remaining claim is R6739.

The average οf a grοup οf variables is referred tο as the mean in mathematics and statistics. There are several methοds fοr calculating the mean, including simple arithmetic means (adding the numbers tοgether and dividing the result by the number οf οbservatiοns), geοmetric means, and harmοnic means.

Here We knοw that,

Mean = Sum οf amοunts οf claim  / Tοtal number οf claim

Accοrding tο the given,

Mean οf 50 claim οf insurance pοlicy = R 4500 then,

Sum οf amοunt οf 50 claim = 4500*50 = R 225000

Mean οf 15 οf these claim = R 2109 then,

Sum οf amοunt οf 15 claims = 2109*15 = R 31635

Mean οf 14 οther claim = R 3704 then

Sum οf 14 οther claim = 3704*14 = R 51856

Nοw remaining claim amοunt = 50-14-15 = 21

Sum οf 21 remaining claim = 225000-31635-51856 = R 141509

Now Mean = Sum of 21 amount of claim / Total number of claim

=> Mean = 141509/21 = R 6739

Hence the mean of remaining 21 sample is R 6739.

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The 24 parking spοt length is 68 yards.

The same feature is expressed in a different unit οf measurement thrοugh a unit cοnversiοn. Time can be stated in minutes rather than hοurs, and distance can be expressed in kilοmetres rather than miles, οr in feet rather than any οther unit οf length.

Here the given width of One parking spot is eight and one-half feet.

Now  we know that 1 feet = [tex]\frac{1}{3}[/tex] yard,

Then , [tex]8\frac{1}{2}[/tex] feet = [tex]\frac{17}{2}[/tex] feet = [tex]\frac{17}{2}\times\frac{1}{3}[/tex] = [tex]2\frac{5}{6}[/tex] yd.

Now Parking lot have 24 parking spot, Then,

=> Total length of parking lοt = 24 [tex]\times[/tex] [tex]2\frac{5}{6}[/tex] = [tex]24\times\frac{17}{6}[/tex] = 68 yard.

Hence The length of parking lοt is 68 yard.

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Answer:

[tex]36 \: cm ^{2} [/tex]

Step-by-step explanation:

triangle area below

[tex] \frac{10 \times 5}{2} = 25[/tex]

EG segment

[tex] \sqrt{10 ^{2} + 5 ^{2} } = \sqrt{125} [/tex]

EF segment

[tex] \sqrt{ (\sqrt{125})^{2} - 2^{2} } = \sqrt{121} [/tex]

triangle area above

[tex] \frac{ \sqrt{121} \times 2}{2} = \sqrt{121} [/tex]

total area

[tex]25 + \sqrt{121} = 36[/tex]

Answer:

The Answer Is 6

Explanation: Divide y by x on each dot to get k

The product of -2³ + x - 5 and x³ -3x +4 is -5x³ + 16x - 28.

The given expression is:

-2³ + x - 5 x (x³ -3x +4)

We need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

First, let's simplify -2³, which means -2 x 2 x 2 = -8, so we have:

-8 + x - 5 x (x³ -3x +4)

Next, we need to distribute the -5 to the terms inside the parentheses:

-8 + x - 5x³ + 15x - 20

Now we can combine like terms:

-5x³ + 16x - 28

Therefore, the product of -2³ + x - 5 and x³ -3x +4 is -5x³ + 16x - 28.

Now, to answer the second part of the question, we need to check if:

-2³ + x - 5 x (x³ -3x +4) = (x³ -3x +4) * (-2³ + x - 5)

We can simplify both expressions first:

-8 + x - 5x³ + 15x - 20 = -8 + x - 5x³ + 15x - 20

We can see that both expressions are identical, which means that the product of -2³ + x - 5 and x³ -3x +4 is equal to the product of x³ -3x +4 and -2³ + x - 5, regardless of the order in which we multiply the factors.

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Answer:

1. 7/16 = 43.75%

2. 4/5 = 80%

3. 55% = 55/100 = 11/20

4. 248% = 2 48/100 = 2 12/25

5. 0.00031 = 0.031%

6. 6.005 = 600.5%

7. 12% = 12/100 = 3/25 (Gigi spent 3/25 of her birthday money on the new sunglasses)

8. The total bill for pizza and soft drinks was $27.50. With a 20% tip, the amount of tip left would be:

Tip = 20% of total bill

Tip = 20/100 * $27.50

Tip = $5.50

Therefore, Veronica and her friends left a $5.50 tip for their server.

Step-by-step explanation:

Answer:

1. 43.75%

2. 60%

3. 11/20

4. 2 and 12/25

5. 0.031%

6. 600.5%

7. 3/25

8. $5.50

Step-by-step explanation:

1. 7 ÷ 16 = 0.4375

0.4375 x 100 = 43.75%

2. 3 ÷ 5 = 0.6

0.6 x 100 = 60%

3. 55/100

11x25 / 20x5 = 11/20

4.248/100

62x4 / 25x4

62/25

62/25 = 2R12

2 12/25

5. 0.00031 x 100 = 0.031%

6. 6.005 x 100 = 600.5%

7. 12% = 12/100.

12/100 simplified = 3/25

8. $27.50 x 20% = $5.50

Answer:

0, -3.

Step-by-step explanation:

12x(x+3)=0

Either 12x = 0 or (x + 3) = 0

x = 0/12 or x = -3

So, x = 0 or x = -3

The jeweler made 4 bracelets and 6 necklaces using 172 grams of gold.

An equation is a mathematical statement that shows that two expressions are equal. It contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Equations are used to solve problems by finding the values of the variables that make the equation true.

Let x be the number of bracelets made, and y be the number of necklaces made.

Then, we can create the following system of equations:

Equation 1: 7x + 24y = 172 (the total amount of gold used is 172 grams)

Equation 2: y = x + 2 (the number of necklaces made is 2 more than the number of bracelets made)

So, the variables are x (the number of bracelets made) and y (the number of necklaces made).

We can solve this system of equations to find the values of x and y. We can use substitution or elimination to solve for one variable and then plug it into the other equation.

Substitution method:

From Equation 2, we have y = x + 2. Substituting this into Equation 1, we get:

7x + 24(x+2) = 172

7x + 24x + 48 = 172

31x = 124

x = 4

So, the jeweler made 4 bracelets.

Plugging this into Equation 2, we get:

y = x + 2 = 4 + 2 = 6

So, the jeweler made 6 necklaces.

Therefore, the jeweler made 4 bracelets and 6 necklaces using 172 grams of gold.

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"Problem:

Jenny earns a base pay of $1000 per month plus a 5% commission on all sales. Last month she sold $8000 worth of products. What was her total earnings for the month?

Solution:

Jenny's commission for the month is 5% of $8000, which can be calculated as:

Commission = 5% of $8000 = (5/100) * $8000 = $400

Her total earnings for the month can be found by adding her base pay and commission:

Total Earnings = Base Pay + Commission

Total Earnings = $1000 + $400

Total Earnings = $1400

Therefore, Jenny's total earnings for the month were $1400." (ChatGPT, 2023)

Answer:

To find the total balance after three years, we can use the formula for continuous compounding:

A = Pe^(rt)

Where A is the total balance, P is the principal (initial amount), e is Euler's number (approximately 2.71828), r is the annual interest rate as a decimal, and t is the time in years.

In this case, P = $12,000, r = 0.0125 (1.25% expressed as a decimal), and t = 3. Plugging these values into the formula, we get:

A = $12,000 x e^(0.0125 x 3)

A = $12,000 x e^(0.0375)

A = $12,000 x 1.038163

A = $12,458.54

Therefore, the total balance after three years is $12,458.54.

Answer:

The probability that exactly 5 of the 7 selected smartphone users use their phones in meetings or classes is approximately 0.0127 or 1.27%.

Answer:

6 times

Step-by-step explanation:

24/4 = 6

Answer:

Line DB and AC intersect at point E.

∠AEB = ∠DEC because they are vertical angles.

∠EAB and ∠ECD are alternate interior angles because AB and CD are parallel lines and line AC crosses through both parallel lines.

m∠EAB = m∠ECD because they are alternate interior angles.

ΔABE ≅ ΔCDE because there are two sets of congruent angles.

Answer:

∂z/∂x = 3x^2 + 6y

∂z/∂y = 6x - 3y^2

Now, we need to solve the system of equations:

3x^2 + 6y = 0

6x - 3y^2 = 0

From the first equation, we get:

y = -x^2/2

Substituting this into the second equation, we get:

6x - 3(-x^2/2)^2 = 0

Simplifying, we get:

6x - 3x^4/4 = 0

Multiplying by 4 and rearranging, we get:

3x^4 - 24x = 0

Factoring out 3x, we get:

3x(x^3 - 8) = 0

Therefore, the critical values of x are x = 0 and x = 2.

For x = 0, we have y = 0 (from y = -x^2/2). So, one critical point is (0, 0).

For x = 2, we have y = -2. So, the other critical point is (2, -2).

To find the critical values of the function, we need to evaluate the function at each critical point:

z(0, 0) = 0^3 + 6(0)(0) - 0^3 - 1 = -1

z(2, -2) = 2^3 + 6(2)(-2) - (-2)^3 - 1 = -13

Therefore, the critical values of the function are -1 and -13.

Answer:

The answer is D

Step-by-step explanation:

Function A

f(x)  =  1 / ( x - 3)

The  vertical asymptote is  the value of x that makes x - 3 = 0 ⇒   x = 3

The vertical asymptote of B   is  x = 4

So....

Function A has a vertical asymptote at x = 3.

Function B has a vertical asymptote at x = 4.

Answer: 2x-6

Step-by-step explanation:

2) They tοgether will take 10.9 hours οr 120/11

3) jοsh will take 21/11 hrs οr 1.9 hrs

4) Tοgether they will take 36/13 οr 2.8 hrs

The place values οf the digits in a decimal number are displayed οn the decimal place value chart. We knοw that a digit in a number represents a numerical value οr place value.

Decimal place value charts are used tο determine the prοper placement οf each digit in a decimal number.

A decimal number cοnsists οf a whοle number and a fractiοnal cοmpοnent, separated by the decimal pοint, a dοt.

Fοr instance, the decimal number 4.37 has twο parts: an actual number pοrtiοn οf 4 and a fractiοnal pοrtiοn οf 37.

Sοlutiοns tο the abοve prοblem

2) Tοm takes = 12 hrs

Huck takes= 10hrs

We can aply unitary methοd

tοgether they will take= 1/12+1/10=2/t

11t=120;

t=120/11 οr 10.9 hrs

3) Tοgether Kim and jοsh can clean the hοuse tοgether in = 3 hrs

Kim takes = 7hrs

Jοsh will take= 1/7+1/x=2/3

3x+21=14x

21=11x

x=21/11 οr 1.9 hrs

4) Jack makes baseball grοunds in= 2 hrs

Mike takes= 3 hrs

Chris takes= 4 hrs

Tοgether they will take= 1/2+1/3+1/4=3/x

13x=36

x=36/13 οr 2.8 hrs

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Answer:

Step-by-step explanation:

The possible rational zeros of a polynomial function are all the possible values of x, where x is a factor of the constant term of the function divided by a factor of the leading coefficient of the function.

For the polynomial function f(x) = x^4 - 5x^3 + 8x^2 - 20x + 16, the constant term is 16 and the leading coefficient is 1. Therefore, the possible rational zeros are all the possible values of x, where x is a factor of 16 divided by a factor of 1. That is, the possible rational zeros are:

±1, ±2, ±4, ±8, ±16

To check if any of these possible zeros are actually zeros of the function, we can use synthetic division or long division to test each one.

Answer: To evaluate 13 - 0.75w + 8x when w = 12 and x = 1/2, we substitute these values into the expression:

13 - 0.75w + 8x = 13 - 0.75(12) + 8(1/2)

Simplifying the expression inside the parentheses first:

13 - 9 + 4 = 8

Substituting this value into the expression:

13 - 0.75w + 8x = 13 - 0.75(12) + 8(1/2) = 13 - 9 + 4 = 8 + 4 = 12

Therefore, the value of 13 - 0.75w + 8x when w = 12 and x = 1/2 is 12.

Answer:

8

Step-by-step explanation:

Given expression is
[tex]13-0.75w\:+\:8x[/tex]

and we are asked to evaluate this expression when
[tex]w = 12 ,\: x = \dfrac{1}{2}[/tex]

[tex]0.75w = 0.75 \times 12 = 9[/tex]

[tex]8x = \8 \times \dfrac{1}{2} = 4[/tex]

[tex]\text{So the expression at $w = 12 \; , x= \dfrac{1}{2}$ evaluates to}[/tex]

13 - 9 + 4 = 8

Answer: 8

Answer:21>15+2x

Step-by-step explanation:

Therefore , the solution of the given problem of angles comes out to be   A is larger than B in terms of size and scale variables can be used with this technique.

A tilt is a form in Euclidean geometry made up of two sides, but rather cylindrical faces, that divide at the barrier's centre and apex. At their junctions, two rays may combine to form an angle. Angle is another outcome of two entities interacting. Dihedral shapes best describe them. A two-dimensional curve can be created by arranging two line beams in various configurations at their ends.

Here,

Consider Polygon A to be a square with the dimensions 2 feet by 5 feet.

=> Size = 2 feet

=>Size = 51 t

The dimension of A can be multiplied by the same scale factor to obtain potential scaled duplicates.

Consider for instance, the scale factor k is:

A scaled-down version of rectangle A is 6 feet by 15 feet, for example.

You obtain this by:

=> Length: 2 feet, 3 inches. 5 feet 15 feet wide

Area A consists of:

=> Area = 2 feet × 5 feet 2

B's size is 90 feet²

=> 2 Arca = 15 feet²

Comparison

A is larger than B in terms of size.

Scale variables can be used with this technique.

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we can see that a larger proportion of male students preferred mathematics (0.48) compared to female students (0.22), while a larger proportion of female students preferred English (0.18) compared to male students (0.12).

HOW TO SOLVE THE QUESTION?

To create a two-way relative frequency table, we can use the following table:

       | English | Mathematics | Total

--------|---------|-------------|-------

Male    |   x     |     24      |   30

Female  |   y     |     11      |   20

--------|---------|-------------|-------

Total   |   35    |     35      |   50

Here, x and y represent the number of male and female students who preferred English, respectively.

To calculate the values for x and y, we can use the fact that there were 35 students who preferred mathematics, and 24 of them were male. Therefore, the number of females who preferred mathematics is 35 - 24 = 11. Since there are a total of 20 female students, the number of females who preferred English is 20 - 11 = 9. Similarly, the number of male students who preferred English is 30 - 24 = 6.

To calculate the relative frequencies, we can divide each cell by the total number of students (50). For example, the relative frequency of male students who preferred mathematics is 24/50 = 0.48.

The resulting two-way relative frequency table is as follows:

       | English | Mathematics | Total

--------|---------|-------------|-------

Male    |   0.12  |     0.48    |  0.60

Female  |   0.18  |     0.22    |  0.40

--------|---------|-------------|-------

Total   |   0.30  |     0.70    |  1.00

From this table, we can see that a larger proportion of male students preferred mathematics (0.48) compared to female students (0.22), while a larger proportion of female students preferred English (0.18) compared to male students (0.12). However, it would not be safe to assume that all females prefer English and all males prefer mathematics based on this data alone, as there may be individual variations and exceptions to this trend.

It's also important to note that the sample size is relatively small, with only 50 students surveyed, and may not be representative of the larger population. Further research with a larger and more diverse sample size would be needed to make more accurate conclusions about gender-based preferences in this population

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In exercise 1:

Situation: The account holder saves $30 each month

The first three terms of the sequence are: $120, $150, $180.

In exercise 2:

Situation: The account earns 25% interest each month.

The first three terms of the sequence are: $120, $150, $187.50.

A sequence in mathematics is described as an enumerated collection of objects in which repetitions are allowed and order matters.

Explaining  exercise 1:

In this exercise, the account balance for each month should form an arithmetic sequence with a common difference of $30

February 1 balance: $150

March 1 balance: $180

Hence, we have the first three terms of the sequence as:  $120, $150, $180.

Explaining exercise 2:

In this exercise, the account balance for each month should form a geometric sequence with a common ratio of 1.25.

The account earns 25% interest each month is the situation here.

February 1 balance: $150

March 1 balance: $187.50

The first three terms of the sequence are: $120, $150, $187.50.

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5. The quadratic function for the height of the rock, h = -16·t² + 79·t + 50, indicates;

a. The height of the rock will be 125 feet above the beach at 1.28 seconds and 3.66 seconds after it is thrown

b. The maximum height reached is about 147.5 feet

The time it takes the rock to reach the maximum height is about 2.47 seconds

A quadratic function is a function of the form; f(x) = a·x² + b·x + c, where, a, b, and c are numbers, and a ≠ 0.

The initial velocity of the rock = 79 ft/s

Height of the cliff above the beach from which the rock is thrown = 50 feet

The function for the height of the rock above the beach is; h = -16·t² + 79·t + 50

Please find attached the graph of the height of the rock as a function of time, created with MS Excel.

Required; When the height of the rock will be 125 feet above the beach

When the height of the rock is 125 feet, we get;

h = 125 = -16·t² + 79·t + 50

-16·t² + 79·t + 50 - 125 = 0

-16·t² + 79·t - 75 = 0

The value of t obtained using an online tool are;

t = (79 + √(1441))/32 ≈ 3.66 and t = (79 - √(1441))/32 ≈ 1.28

The times at which the height of the rock will be 125 feet above the beach are; 1.28 seconds and 3.66 seconds after the rock is thrown

b. The maximum height of the rock can be obtained from the formula for finding the maximum height of a quadratic equation of the form; f(x) = a·x² + b·x + c, which is;

At the maximum point, x = -b/(2·a)

The function for the height; h = -16·t² + 79·t + 50, indicates that we get;

a = -16, b = 79, and c = 50

Therefore;

The time it takes the rock to reach the maximum height, t(max), is therefore;

t(max) = -79/(2 × (-16)) ≈ 2.47

It takes about 2.47 seconds for the rock to reach the maximum height

The maximum height, h(max) = -16 × 2.46875² + 79×2.46875 + 50 ≈ 147.5

The maximum height reached by the rock is about 147.5 feet

Possible part of the question, obtained from a similar online question, includes;

a. The time it takes the rock to reach a height of 125 feet above the beach
b. The maximum height reached by the rock and the duration it takes the rock to reach the maximum height.

Please find attached the possible graph in the question, created with MS Excel, using the function for the height of the rock.

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The assertion that the percentage of males who own cats is lower than the percentage of women who do not own cats is unsupported by sufficient data.

Hence, the Alternate hypothesis would be:

H0: Pm = Pf

H1 = Pm < Pf

Alternate hypotheses are used in statistical hypothesis testing. The hypothesis of a test always predicts no effect or no association between variables, but the alternative hypothesis reflects the effect or relationship that your research predicts.

Given information:

The test claims proportion of men who owns cat is smaller.

The significance level = 0.005

As the hypothesis always gets ten statements of equality but in the case given alternate hypothesis will claim.

Now calculating the Z-value:

P= 10+14/80

= 0.3

So, the p-value according to the obtained Z value is 0.1365.

Since, 0.1365>0.005, we fail to reject the hypothesis.

Hence, there is not enough proof to support the claim that the proportion of man who owns the cat is less than that of the proportion of women who owns the cat.

Therefore, the Alternate hypothesis would be:

H0: Pm = Pf

H1 = Pm < Pf

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