13.15.Verify that the six trigonometric function are well-defined. That is, show that it does not matter which right triangle with interior angle theta you choose- these six rayios will not change

Answer:

1)   13    **Explained below **

2)   4    **Explained below **

3)   6

4)   8

5)   15

6)   2.64575    **Explained below **

Step-by-step explanation:

All 6 problems can be solved using the Pythagorean theorem using the following two rules

If x is the hypotenuse and a, b are the shorter sides then
[tex]x = \sqrt{a^2 + b^2}[/tex]

If x is one of the shorter sides and a (or b) is the other shorter side and c is the hypotenuse then

[tex]x = \sqrt{c^2 - a^2\\[/tex]  if a is the other shorter side

[tex]x = \sqrt{c^2 - b^2\\[/tex]  if b is the other shorter side

I will just demonstrate for  #1 , #2 and #6. You can figure out the reasoning behind the rest.

#1. x is the hypotenuse, 5 and 12 are legs(shorter sides)

[tex]x = \sqrt{5^2 + 12^2}\\\\x = \sqrt{25 + 144}\\\\x = \sqrt{169}\\\\x = 13\\\\[/tex]

# 2. x is one of the shorter sides (leg)

5 is the hypotenuse and 3 is the other shorter side

[tex]x = \sqrt{5^{2} - 3^{2}}\\\\x = \sqrt{5^{2} - 3^{2}}\\\\x = \sqrt{25 - 9}\\\\x = \sqrt{16}\\\\x = 4[/tex]

#6  hypotenuse is 4, short side = 3

[tex]x = \sqrt{4^{2} - 3^{2}}\\\\x = \sqrt{4^{2} - 3^{2}}\\\\x = \sqrt{16 - 9}\\\\x = \sqrt{7}\\\\x = 2.64575\\\\[/tex]

(a) 20% of $224,000 is $44,800.

(b) 3 points on a $224,000 mortgage is $6,720.

(c) $44,800 + $6,720 is $51,520. So the total down payment and closing costs are $51,520.

(d) The mortgage amount is $224,000 - $51,520 = $172,480.

(e)

Monthly payment = $172,480 * (7%/12%)(30 years) = $1,234.15

Total interest paid over 30 years = $172,480 * 7% * 30 years = $243,720

The expression 8x^3/2 × -7x^2/3 can be simplified to give

To reduce the given expression, we can employ the principles of exponents and carry out the multiplication.

The given expression is:- 8x^(3/2) * (-7x^(2/3))

Utilizing this property, combining two terms with exponents entails adding the exponents when they feature the same bases. Thus, we establish:

8 * (-7) = -56 (coefficient multiplication)

x^(3/2 + 2/3) results in x^(13/6) (exponent addition)

Bringing it all together, we have:

8x^(3/2) * (-7x^(2/3)) = -56x^(13/6)

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Answer:

  y = 2x

Step-by-step explanation:

You want the simplified form of y = |x -5| +|x +5| if x > 5.

The graph of the whole function will have turning points where the absolute value expressions are 0:

  x -5 = 0   ⇒   x = 5

  x +5 = 0   ⇒   x = -5

For values of x > 5, we are concerned with that portion of the graph that is to the right of both of these turning points. Hence, both absolute value expressions are positive and unchanged by the absolute value bars.

  y = (x -5) +(x +5) . . . . . . if x > 5

  y = 2x . . . . . . . . . . . . . . collect terms

The simplified function is y = 2x.

__

Additional comment

The attached graph shows y=2x and the given function for x > 5. They are identical. (The y=2x graph is shown dotted, so you can see the red graph of the given function.)

The value of (2x^4+x^3-x^2+3)+(4x^4-2x^3-x^2+10x+2) is 6x⁴-x³-2x²+10x+5

Expression is a combination of numbers, variables, functions (such as addition, subtraction, multiplication or division etc.)

Adding two expressions together requires to collect like terms. like terms are terms with similar variable. For example, in 2x²+6x+x² , x² and 2x² are like terms.

Therefore, 2x⁴+x³-x²+3 + 4x⁴-2x³-x²+10x+2 can be simplified as;

2x⁴+4x⁴+x³-2x³-x²-x²+10x+2+3

= 6x⁴-x³-2x²+10x+2

therefore, the value of 2x⁴+4x⁴+x³-2x³-x²-x²+10x+2 is 6x⁴-x³-2x²+10x+5.

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Answer:

Step-by-step explanation:

To find the average mass of the six people, we need to divide the total mass by the number of people.

The total mass of the six people is 1/2 tonne, which is equivalent to 500 kg (since 1 tonne = 1000 kg).

So, the average mass of the six people is:

500 kg / 6 = 83.33 kg (rounded to two decimal places)

Therefore, the average mass of each person is approximately 83.33 kg.

Answer:

y = f(x) + 3

Hope this helps!

Step-by-step explanation:

Because the graph went up 3...

Answer:

y = f(x) + 3

Hope this helps!

Step-by-step explanation:

Because the graph went up 3...

Answer:

1.  Not true, so (5,2) is not a solution

2. Not true, so (-3,-6) is not a solution.

3. Not true, so (0, [tex]\frac{2}{3}[/tex]) is not a solution.

Step-by-step explanation:

Substitute 5 for x and 2 for y

1 - y > x - 6

1 - 2 > 5 - 6

-1 > -1

This is not true. -1 is not greater than -1

y < x - 1

Substitute -3 for x and -6 for y

2 - y [tex]\leq[/tex] 2x

2 - (-6) [tex]\leq[/tex] 2(-3)

8 [tex]\leq[/tex] -6  

This is not true. 8 is not less than -6.

y [tex]\geq[/tex] x

Substitute 0 for x and [tex]\frac{2}{3}[/tex]

3 - [tex]\frac{1}{2}[/tex] x + 3y < 8

3 - [tex]\frac{1}{2}[/tex] (0) + 3([tex]\frac{2}{3}[/tex]) < 8

3 - 0 + [tex]\frac{6}{3}[/tex] < 8

3 + 2 < 8

6 < 8

This is true. 6 is less than  8

y [tex]\geq[/tex] 1

[tex]\frac{2}{3}[/tex] [tex]\geq[/tex] 1

This is not true. [tex]\frac{2}{3}[/tex] is not greater than or equal to one.  

The ordered pair has to be a solution for both equations for the ordered pair to be a solution for the system.

Helping in the name of Jesus.

Answer:

[tex] - 9 {g}^{3} + {g}^{2} + 10g - 5[/tex]

The area under the curve is 522 units².

The area under the curve is calculated by integrating the function and finding the definite integra as shown below.

f(x) = 8x + 10

The integra of the function;

f(x)' = 4x² + 10x

Find the definite integra at the boundaries, x = 19 and x = 22

f(19)' = 4(19)² + 10(19)

F(19)' = 1,634

f(22)' = 4(22)² + 10(22)

F(22)' = 2,156

Area = 2,156 - 1,634 = 522 units²

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The equation to find the total number of lessons is T = 6 + 12 = 18, and the ratio to find the total cost of the lessons is c:T = 72:18 or  4:1.

The equation to find the total number of lessons (T) can be written as:

T = number of salsa lessons + number of rumba lessons

Substituting the given values, we get:

T = 6 + 12 = 18

The ratio to find the total cost of the lessons (c) can be written as:

c = cost per lesson x total number of lessons

Substituting the given values, we get:

c = $4 x 18 = $72

Therefore, the equation to find the total number of lessons is T = 6 + 12 = 18, and the ratio to find the total cost of the lessons is c:T = 72:18 = 4:1.

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ANSWER:

They have $1,100 altogether.

EXPLANATION:

- Originally, Mike had $500, while Joel had $600. That adds up to $1,100, and it makes Joel have $100 more than Mike.

- Half of Joel’s $600 is $300, which he gave to Mike. That makes Joel now have $300 himself.

- Adding $300 to Mike’s $500 is $800, which means Mike now has $800.

- $800 (Mike’s new amount) minus $300 (Joel’s new amount) is $500, which works because Mike now has $500 more than Joel.

-$300 + $800 is, of course, still $1,100.

Answer:

Yes, because while the total number of professors increases, the number of pure math professors decreases.

Step-by-step explanation:

Using Hamilton's method:

pure math 4/28 = 0.142857

applied math 12/28 = 0.42857

statistics = 12/28 = 0.42857

10 × 0.142857 = 1.42857

10 × 0.42857 = 4.2857

10 × 0.42857 = 4.2857

numbers of professors:

pure math 1

applied math 4

statistics 4

total 9

Add 1 to pure math

Final original numbers of professors using Hamilton's method

pure math 2

applied math 4

statistics 4

total 10

Add 1 professor to department.

11 × 0.142857 = 1.5714

11 × 0.42857 = 4.71428

11 × 0.42857 = 4.71428

numbers of professors:

pure math 1

applied math 4

statistics 4

total 9

Add 1 to applied math and 1 to statistics

Final new numbers of professors using Hamilton's method

pure math 1

applied math 5

statistics 5

total 11

Despite the addition of 1 professor to the department, the field of pure mathematics went from 2 professors to 1 professor. This is an example of the Alabama paradox.

Answer: Yes, because while the total number of professors increases, the number of pure math professors decreases.

Step-by-step explanation:

We can use the two data points given to find the equation of the line, which gives the monthly cost (y) in terms of the calling time in minutes (x).

First, we can find the slope of the line:

slope = (change in y) / (change in x)

slope = (20.66 - 18.45) / (56 - 39)

slope = 0.219

Next, we can use one of the data points and the slope to find the y-intercept (b). Let's use the data point (39, 18.45):

y - y1 = m(x - x1)

y - 18.45 = 0.219(x - 39)

y - 18.45 = 0.219x - 8.541

y = 0.219x + 9.909

So the equation for the monthly cost is y = 0.219x + 9.909.

To find the monthly cost for 50 minutes of calls, we plug in x = 50:

y = 0.219(50) + 9.909

y ≈ $21.44

Therefore, the monthly cost for 50 minutes of calls is approximately $21.44.

if Dwayne made 10 hours of long-distance calls at a rate of $0.10 per minute, his bill for that month would be $60. However, if the rate is different, the bill will be different as well.

The cost of long-distance calls can vary depending on the service provider and the country being called. Without that information, it's difficult to give an accurate answer to this question.

Assuming a standard rate of $0.10 per minute for long-distance calls, we can calculate Dwayne's bill as follows:

10 hours = 600 minutes (since there are 60 minutes in an hour)

600 minutes x $0.10 per minute = $60

Therefore, if Dwayne made 10 hours of long-distance calls at a rate of $0.10 per minute, his bill for that month would be $60. However, if the rate is different, the bill will be different as well.

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The total number of each zone for a three-week period is between 1200 and 1300, 1:2 is the ratio of customer from Zone 1 to customers from Zone 3.

​

1) During 3 weeks bound

Customers from zone 3 = 900

Customers from zone 4 = 320

So, total customers = (900 + 320)

= 1220

Hence, option D is correct i.e between 1200 and 1300.

2) Customer from zone 1 = 450

Customers from zone 3 = 900

So, Customer from zone 1 / Customer from zone 3 = 450/900

= 1/2

Hence, option 2 i.e., 1:2 is correct.

3) 3/5 pound of clay is used to make = 1 bowl

So, 1 bound of clay is used to make = 5/3 bowl

By unitary method,

10 pounds of clay is used to make = 10 *(5/3)

= 16.66

= 17 bowls (approx)

Hence, option d is correct answer.

Therefore, The total number of each zone for a three-week period is between 1200 and 1300, 1:2 is the ratio of customer from Zone 1 to customers from Zone 3.

​

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Answer:

Step-by-step explanation:

By the law of cosines, [tex](59.7)^2=52^2+41^2-2\cdot 52 \cdot 41 \cos(\angle R).[/tex]

Thus, after solving for [tex]\cos(\angle R)[/tex], we get [tex]\cos(\angle R) = 0.1925211.[/tex] So taking the inverse cosine then yields [tex]\boxed{\angle R = 1.377 \text{ rad}}.[/tex]

Answer:

Step-by-step explanation:

By the law of cosines, [tex](59.7)^2=52^2+41^2-2\cdot 52 \cdot 41 \cos(\angle R).[/tex]

Thus, after solving for [tex]\cos(\angle R)[/tex], we get [tex]\cos(\angle R) = 0.1925211.[/tex] So taking the inverse cosine then yields [tex]\boxed{\angle R = 1.377 \text{ rad}}.[/tex]

Oliver withdraws an amount of $285.10 from his saving account and the interest is $26.60

Compound interest is a type of interest that is paid on the closing balance at the end of the previous year, which includes the interest paid in previous years.

His saving account earns 1.8% annually

The interest Oliver could have earned in five months:

Monthly interest = Annual interest ÷ 12

Monthly interest = 1.8% ÷ 12

Monthly interest = 0.018 ÷ 12 = 3/2000

A = P(1 + r)n

After five months = Principle × (1 + interest)ⁿ

After five months = 285.10 × (1 + 0.018)⁵

After five months = 311.70

Interest earned = 311.70 - 285.10 = $26.60

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Answer:

1) Michele

2) f(t)

3) Elisa's puppy

4) g(x)

Step-by-step explanation:

This is because Michele's situation has a rate of change of -10, f(t) has -6.2, Elisa's puppy's situation has a rate of change of +5, and g(x) has 8.1

Hence,

-10 < -6.2 < 5 < 8.1

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The town's emergency response planning committee has proposed to place four emergency response centers at the four corners of the town, with each center serving the people who live within 3 miles of the respective response center. The idea is to provide coverage to the entire town and ensure prompt emergency response for the residents. However, there are some problems with this idea.

Unequal coverage: Placing the emergency response centers at the four corners of the town may result in unequal coverage for the residents. Depending on the size, shape, and population distribution of the town, some areas may be farther away from the response centers, resulting in longer response times and reduced effectiveness in emergency situations.

Overlapping coverage: Placing four response centers in a small town may result in overlapping coverage areas, where the coverage areas of multiple response centers overlap with each other. This may lead to duplication of resources and inefficiencies in emergency response efforts.

Limited reach: Placing response centers only at the four corners of the town may result in limited reach for certain areas, especially those located in the middle or farther away from the corners. This may leave some residents outside the 3-mile coverage radius without access to timely emergency response services.

One potential solution to address these problems could be to use a more strategic approach to determine the locations of the emergency response centers. This could involve conducting a thorough analysis of the town's population density, geographical features, road network, and existing emergency resources. Based on this analysis, the response centers could be strategically placed at locations that provide the best coverage to the entire town, considering factors such as response time, resource allocation, and accessibility.

For example, instead of placing all the response centers at the corners of the town, they could be distributed more evenly across the town to ensure more equitable coverage. Additionally, the use of advanced GIS (Geographical Information System) technology and modeling techniques could help in identifying optimal locations for the response centers, taking into account various factors such as population density, road network, and travel time.

Furthermore, collaboration and coordination among the emergency response centers, along with proper communication and information sharing, can help in avoiding duplication of resources and improving the efficiency of emergency response efforts.

In conclusion, while the idea of placing four emergency response centers at the four corners of town may seem simple, there are potential problems such as unequal coverage, overlapping coverage, and limited reach. A more strategic and data-driven approach, considering factors such as population density, geographical features, and existing resources, can help in identifying optimal locations for the response centers and ensuring effective emergency response services for all residents of the town.

The volume of the cylinder in terms of b is expressed as: (20b³ + 12b²)π cubic units.

Where r represents the radius of a cylinder and h represents its height, the volume of the cylinder can be calculated using the formula:

V = πr²h.

Given the following:

Radius (r) = 2b

Height (h) = 5b + 3

Volume (V) = π * (2b)² * (5b + 3)

Volume (V) = π * 4b² * (5b + 3)

Volume (V) = π * 20b³ + 12b²

Volume (V) = (20b³ + 12b²)π cubic units

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Complete paragraph proof would be detailed proof.

Given that M is the midpoint of PK and PK ⊥ MB, we need to prove that △PKB is isosceles.

Proof,

Since M is the midpoint of PK, PM ≅ KM.

Also, since PK ⊥ MB, we have ∠PMB and ∠KMB are right angles.

Since right angles are congruent, we have ∠PMB ≅ ∠KMB.

Now, by the SAS congruence theorem, we have △PMB ≅ △KMB because they share side MB, and PM ≅ KM and ∠PMB ≅ ∠KMB.

Thus, we have BP ≅ BK because corresponding parts of congruent triangles are congruent.

Therefore, △PKB has two congruent sides and isosceles by definition.

Hence, we have proven that △PKB is isosceles.

Correct Question :

Complete the paragraph proof. Given: M is the midpoint of PK PK ⊥ MB Prove: △PKB is isosceles It is given that M is the midpoint of PK and PK ⊥ MB. Midpoints divide a segment into two congruent segments, so PM ≅ KM. Since PK ⊥ MB and perpendicular lines intersect at right angles, ∠PMB and ∠KMB are right angles. Right angles are congruent, so ∠PMB ≅ ∠KMB. The triangles share MB, and the reflexive property justifies that MB ≅ MB. Therefore, △PMB ≅ △KMB by the SAS congruence theorem. Thus, BP ≅ BK because . Finally, △PKB is isosceles because it has two congruent sides.

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For the function in the graph the range is option B: 40≤f≤100.

The collection of all potential output values (y-values) that a function could produce is known as the function's range. It is, in other words, the entire set of values that the function is capable of accepting as its input changes across its domain. The collection of all potential output values (y-values) that a function could produce is known as the function's range. It is, in other words, the entire set of values that the function is capable of accepting as its input changes across its domain.

The range of the function is the output values, or the y-coordinates of the function.

For the given graph, the output of the function is from 40 to 100 thus, the range is:

40≤f≤100

Hence, for the function in the graph the range is option B: 40≤f≤100.

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The value of h is ( -1/3, -2/3)

Quadratic equation is a type of equation in which the highest power of variable is 2. A quadratic equation is represented as ax²+bx + c = 0

Solving, 9h²+9h = -2

9h²+9h+2 = 0

9h² +6h +3h +2 = 0

(9h²+6h) ( 3h +2) = 0

3h( 3h + 2)+1 ( 3h+2) = 0

(3h+2)(3h+1) = 0

Therefore ;

3h+2 = 0

3h = -2

h = -2/3 or

3h+1 = 0

3h = -1

h = -1/3

therefore the value of h is ( -1/3, -2/3)

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To produce 60 liters of a 55% acid solution, we would need 40 liters of the 50% acid solution and 20 liters of the 65% acid solution.

Let number of liters of 50% acid solution needed be = "x", and

Let number of liters of 65% acid solution needed to produce 60 liters of a 55% acid solution. be = y,

To solve for x and y, we can use the following system of equations:

The "total-volume" of the solution is 60 liters,

⇒ x + y = 60,    ...equation(1)

We know that, the total amount of acid in the solution is equal to the concentration of the final solution times its volume;

⇒ 0.50x + 0.65y = 0.55(60)     ...equation(2),

On simplifying equation(2),

We get,

⇒ 0.50x + 0.65y = 33,

From equation(1), we have ⇒ x = 60 - y,

Now, we substitute "x" into the second equation and solve for y)

⇒ 0.50(60 - y) + 0.65y = 33,

⇒ 30 - 0.50y + 0.65y = 33,

⇒ 0.15y = 3,

⇒ y = 20

So, we need 20 liters of the 65% acid solution.

⇒ x + y = 60,

⇒ x + 20 = 60,

⇒ x = 40,

Therefore, we need 40 liters of the 50% acid solution and 20 liters of the 65% acid solution to produce 60 liters of a 55% acid solution.

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If Thabang and Ndivhu decided to share their money according to hours, then Thabang will receive R240.

Thabang worked for 3 hours and Ndivhu worked for 2 hours.

Together, they worked for a total of 5 hours. They were paid R400 for their work,

So, we calculate their hourly-rate by dividing the total amount by the total number of hours worked;

⇒ Hourly rate = (Total amount paid)/(Total number of hours worked),

⇒ Hourly rate = R400/5,

⇒ Hourly rate = R80 per hour,

Now, we find amount Thabang will receive if it is decided to share the money according to "hours-worked".

Thabang worked for 3 out of the total 5 hours, so he should receive 3/5 of the total amount paid,

⇒ Amount Thabang will receive = (Hours Thabang worked / Total hours worked) × Total amount paid,

⇒ Amount Thabang will receive = (3/5) × R400,

⇒ Amount Thabang will receive = R240,

Therefore, Thabang will receive R240 if they decided to share their money according to hours worked.

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The given question is incomplete, the complete question is

Thabang and his friend Ndivhu share a job of cleaning a neighbors yard and they were paid R400, Thabang worked 3 hours and Ndivhu worked for 2 hours. Determine the amount Thabang will receive if they decided to share their money according to hours​.

A recursive definition for the set S of ordered pairs of positive integers (a,b) where a is a factor of b can be given as follows:

1. The ordered pair (1, b) is in S for all positive integers b.

2. If a is a factor of b, then (a, b) is in S if and only if (a, b/a) is in S.

The first rule establishes the base case for the recursion, stating that (1, b) is in S for all b. The second rule provides the recursive step, stating that (a, b) is in S if and only if (a, b/a) is in S, where b/a is an integer division of b by a, and a is a factor of b. This recursive definition ensures that all pairs (a, b) in S have a as a factor of b.

The distance from the ball to the center of the green (CD) is approximately 13.2 yards.

Given:

AB = 110 yards (distance from marker to center of the green)

BC = 45 yards (distance from marker to golfer's ball)

angle BAC = 115° (angle formed when golfer turns towards the ball)

To find CD (distance from ball to center of the green):

Apply the cosine function to angle BAC:

cos(115°) = AB/BC  (∵cos θ = Adjacent side/ Hypotenuse)

Substitute the given values:

cos(115°) = 110/45

Calculate cos(115°):

cos(115°) ≈ -0.2924 (rounded to four decimal places)

Multiply both sides by BC to solve for AB:

AB = BC * cos(115°)

AB ≈ 45 * -0.2924

AB ≈ -13.16 yards (rounded to two decimal places)

Note: (The AB is negative, as the ball is moving in the opposite direction of the golfer's turn.)

CD = |AB| ≈ 13.16 yards (rounded to two decimal places)

CD ≈ 13.2 yards (rounded to one decimal point)

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The expression is equal to 2 is 5 x (one-third x 6) ÷ 5. The correct option is C.

A mathematical expression is the collection of mathematical symbols that results from the proper combination of numbers and variables using operations like addition, subtraction, multiplication, division, exponentiation, and other as-yet-unlearned operations and functions.

Can be simplified each of the expressions see if them equals 2:

4 x (one-half x 6) ÷ 3 = 4 x 3 ÷ 3 = 4

6 ÷ (one-fourth x 3 x one and one-fourth) = 6 ÷ (3/4 x 5/4) = 6 ÷ (15/16) = 96/15 ≠ 2

5 x (one-third x 6) ÷ 5 = 5 x 2 ÷ 5 = 2

10 − (one-fifth x 10) + 1 = 10 - 2 + 1 = 9 ≠ 2

Therefore, the correct option is c. 5 x (one-third x 6)

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