Supplementary angles

23. By the diagram, the triangle is a [tex]45^\circ - 45^\circ-90^\circ[/tex] triangle. Therefore, the triangle is isosceles, so [tex]x=6.[/tex] And by the Pythagorean theorem, [tex]y^2=x^2+6^2=6^2+6^2=72,[/tex] and taking the square root yields [tex]y = 6\sqrt{2}.[/tex]

24. Notice that by the diagram, the triangle is a [tex]30^\circ-60^\circ-90^\circ[/tex] triangle. It is well known that in these special types of triangles, if [tex]a[/tex] is the side length opposite the [tex]30^\circ[/tex] angle, then the side length opposite the [tex]90^\circ[/tex] angle is [tex]2a,[/tex]  and finally the side length opposite the [tex]60^\circ[/tex]  is [tex]a\sqrt{3}.[/tex] This is shown in the image I've attached below, and can also be easily proven by the fact that two of these triangles combine to form an equilateral triangle.

So using this fact, we know that [tex]x = \frac{8}{2} = 4,[/tex] and [tex]y = x\sqrt{3} = 4\sqrt{3}.[/tex]

25. Tell whether the given side lengths can represent the sides of a right triangle: 10, 12, 20

For this question, we simply use the converse of the Pythagorean theorem. One part of its statement says that if a triangle has sides of length [tex]a, b, c[/tex] such that [tex]a^2 + b^2 = c^2[/tex], and [tex]c[/tex] is the longest side, then the angle opposite the side of length [tex]c[/tex] is a right angle.

Thus, if the triple [tex](a, b, c)[/tex] does not satisfy the equation, the triangle it forms is not a right triangle. In our case, the corresponding values for [tex](a, b, c)[/tex] are [tex](10, 12, 20),[/tex] because [tex]20[/tex] is the largest number in this triple. But [tex]10^2+12^2=244\neq 400 = 20^2,[/tex] so the triangle with side lengths [tex]10, 12, 20[/tex] is not a right triangle.

26. Another statement of the converse of the Pythagorean theorem is that if a triangle has side lengths [tex]a, b, c[/tex] such that [tex]c^2 > a^2+b^2[/tex] , with [tex]c[/tex] being the longest side, then the triangle is an obtuse triangle. So the corresponding values for [tex]a, b, c[/tex] are [tex]10, 11, 14,[/tex] and because [tex]14^2 = 196 < 221 = 10^2+11^2,[/tex] the triangle is an obtuse triangle.

Answer:

264π

Step-by-step explanation:

It will take 4. 09 weeks for the tires to be rotated at the current rate.

When rotating a car's tires every 10,000 miles, each tire must complete this distance before being moved.

25, 200 rotations / day x 7 days / week = 176, 400 rotations per week

To solve this predicament, dividing 10,000 miles by the number of rotations/mile (which equates to the tire's circumference) is necessary. The formula for calculating a tire's circumference can aid in obtaining this number along with the weeks required to perform the rotation process:

C = 2πr

Rotation per mile is:

1 mile / 87.96 inches / rotation = 72.26 rotations / mile

Number of weeks:

10, 000 miles x 72. 26 rotations / mile = 722, 600 rotations

722, 600 rotations / 176, 400 rotations/week = 4.09 weeks

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Answer:

The y-intercept represents Adam's base salary

Step-by-step explanation:

Since it's y = 0.28x + 38,000, when x is 0, the salary is 38,000 without commission.

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There are 1140 different possible ways to select 3 restaurants out of 20 for the promotional program.

Combination and permutation are two alternative strategies in mathematics to divide up a collection of components into subsets. The subset's components can be listed in any order when combined. The components of the subset are listed in a permutation in a certain order.The combination formula, which is: can be used to resolve this issue.

n C r = r! * (n-r)!

where r is the number of restaurants to be chosen, and n is the total number of restaurants (20 in this case). (3 in this case).

By replacing these values, we obtain:

20 C 3 = 20! / (3! * (20-3)!) = 20! / (3! * 17!) = (20 * 19 * 18) / (3 * 2 * 1) = 1140

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Given

ABCΔ  is a triangle.

D is a point on [BC]

|BD|=4

|AD|=|CD|

m(CBAˆ)=α=30∘

m(ACBˆ)=β=20∘

To Find

What is |AC|=x?

Solution

Geometric approaches gave me nothing. Letting |AD|=|CD|=a, |AB|=b

and applying law of cosines to all three triangles yields;

respectively, from ACDΔ, ABDΔ and ABCΔ. Directly isolating x from these equations gives an expression involves trigonometric function(s), and i cannot find a way to annihilate trigonometric functions in these equations to reach the answer which is x=4

The complete question is -

In the given diagram find the value of x

a) 1

b) 2

c) 3

d) 4

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Answer:

Step-by-step explanation:

Okay, here are the steps to solve this problem:

* Jen earns a weekly salary of $250

* She gets a 10% bonus on any sales

* In the past 4 weeks, she sold $312 worth of phones and accessories

* 10% of $312 is $31.20

* So in 4 weeks, her bonus would be $31.20 * 4 = $124.80

* Her weekly salary is $250

* So in 4 weeks, her salary would be $250 * 4 = $1000

* Her total income over 4 weeks is $1000 + $124.80 = $1124.80

* The choices are:

** $1,031.20 ** $562.00 ** $312.00 ** $281.20 ** $31.20

So the correct choice is $1,031.20.

The Sum of interior angles of the given polygon is: 540°

The value of the angle x is: 85°

The interior angle-sum formula for an irregular polygon is the same as the sum of interior angle in a regular polygon.

Sum of interior angles in a n-sided polygon is:

Sum = (n - 2)180

where:

n is the number of sides of the irregular polygon.

This is a 5-sided irregular polygon

Thus:

Sum = (5 - 2)180

Sum = 540°

Thus:

x + 115 + 115 + 130 + 95 = 540

x + 455 = 540

x = 540 - 455

x = 85°

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we know that,

[tex] \sf \dashrightarrow \: cos(a - b) = cos a \times cos b + sin a \times sin b[/tex]

Therefore,

[tex] \sf \leadsto \: cos \: 15 \degree[/tex]

[tex] \sf \leadsto \: cos \: (45 \degree - 30 \degree)[/tex]

[tex] \sf \leadsto \: cos \: 45 \degree . \: cos \: 30 \degree + \: sin \: 45 \degree . \: sin \: 30 \degree[/tex]

[tex] \sf \leadsto \: \frac{1}{ \sqrt{2} } \: . \: \frac{ \sqrt{3} }{2} + \: \frac{1}{ \sqrt{2} } \: . \: \: \frac{1}{2} \\ [/tex]

[tex] \sf \leadsto \: \frac{1. \: \sqrt{3} }{ \sqrt{2} \: . \: 2} \: + \: \frac{1 \: .\: 1}{ \sqrt{2} \: .\: 2} \\ [/tex]

[tex] \sf \leadsto \: \frac{ \: \sqrt{3} }{ 2\sqrt{2} \: } \: + \: \frac{1}{ 2\sqrt{2} \: } \\ [/tex]

[tex] \sf \leadsto \: \frac{ \: \sqrt{3} + 1 }{ 2\sqrt{2} \: } \: \\ [/tex]

Therefore Value of cos 15° is [tex] \sf \: \frac{ \: \sqrt{3} + 1 }{ 2\sqrt{2} \: } \: [/tex]

To the nearest tenth of a year, the person must leave the money in the bank for 15.5 years.

We can use the formula for compound interest:

A = P[tex](1 + r/n)^{nt}[/tex]

where A is the final amount, P is the principal (initial investment), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.

We know P = 8000, r = 0.0475, and we want A = 20400. We also know that the interest is compounded daily, so n = 365.

Let's solve for t:

20400 = 8000[tex](1 + 0.0475/365)^{(365t)}[/tex]

2.55 = [tex](1.00013)^{(365t)}[/tex]

Taking the natural logarithm of both sides, we get:

after solving

Using a calculator, we find:

t ≈ 15.5

So, to the nearest tenth of a year, the person must leave the money in the bank for 15.5 years.

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Please refer to the attachment

The median is the average that is most representative of the data.

The median of a data-set is the middle value of a data-set, the value of which 50% of the measures are less than and 50% of the measures are greater than. Hence, the median also represents the 50th percentile of a data-set.

In this problem, we have that smaller values have a higher number of observations, meaning that the data-set is not symmetric.

As the data-set is not symmetric, it means that the median is the average that is most representative of the data.

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In a case whereby Richard is saving money for a down payment on a car He models his savings plan with the equation y = 200x + 500 based on his current monthly savings rate  y-intercept of linear model represent B. The starting amount in his savings account.

It should be noted that in a  linear equation the y intercept  can be seen as one that is starting point  whereby x is zero, which implkies that starting balance he had is 1000 in the bank.

Therefore, from the  equation y = 200x + 500, the  y-intercept of linear model serves as the starting amount in his savings account.

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Answer:

D. Quadratic

Step-by-step explanation:

The vertex of the parent function y = x2 lies on the origin. It also has a domain of all real numbers and a range of [0, ∞). Observe that this function increases when x is positive and decreases while x is negative

a. K = E + (x + y). This implies that Kofi is (x + y) older than Esenam.

b. (i) Adoma = (5 + y) years

   (ii) Kofi = (5 + (x+y)) years

Word problem is a topic that requires the transformation of a mathematical statement into an equation, such that the variable is to be determined.

In the given question, we have;

Let the age of Kofi, Adoma, and Esenam be represented by K, A and E respectively. So that;

K = A + x ...... i

A = E + y........ ii

substitute i into ii

K = (E + y) + x

  = E + (x + y)

K = E + (x + y) ........ iii

Thus,

a. From equation iii, we can deduce that Kofi is (x + y) years older than Esenam.

b. The age of Adoma = 5 + y years

ii. The age of Kofi = 5 + (x + y) years

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Answer:

Parameter= 98ft

Area=180ft^2

Step-by-step explanation:

Parameter= Length times 2 plus width times 2 or

L2+W2=P 4(2)+45(2)=8+90=98

Area= LxW=A^2

4x45=180^2

Answer:

A. 4.5 hours

Step-by-step explanation:

We can use a proportion to solve this problem. Let x be the total time in hours for the train ride from Anchorage to Seward. We know that the train traveled 117 miles in x hours, and it traveled 13 miles in 0.5 hours (30 minutes). Therefore:

117 miles / x hours = 13 miles / 0.5 hours

We can cross-multiply to get:

117 miles * 0.5 hours = 13 miles * x hours

Simplifying:

58.5 = 13x

x = 58.5 / 13

x = 4.5 hours

Therefore, the non-stop train ride from Anchorage to Seward took 4.5 hours. Answer: A.

Hope this helps!

Answer:

A. 4.5 hours

Step-by-step explanation:

We can use a proportion to solve this problem. Let x be the total time in hours for the train ride from Anchorage to Seward. We know that the train traveled 117 miles in x hours, and it traveled 13 miles in 0.5 hours (30 minutes). Therefore:

117 miles / x hours = 13 miles / 0.5 hours

We can cross-multiply to get:

117 miles * 0.5 hours = 13 miles * x hours

Simplifying:

58.5 = 13x

x = 58.5 / 13

x = 4.5 hours

Therefore, the non-stop train ride from Anchorage to Seward took 4.5 hours. Answer: A.

Hope this helps!

Answer:

------------------------------

We are given three sides and need to find one of angles.

Use the law of cosines:

  [tex]cos(A) = \cfrac{ b^2 + c^2 - a^2}{2bc}[/tex]

Here we are given:

Substitute above values into the given equation:

   [tex]cos(R) = \cfrac{ 52^2 + 41^2 - 59.7^2}{2*52*41} =0.1925[/tex]

Find m∠R:

The values of the angles will be 6.67°, 156.78° Be 29.93° and the sides will be 10.01, 34 and 43.

SSA stands for side side angle postulate. In this postulate of congruence, we say that if two sides and an angle not included between them are respectively equal to two sides and an angle of the other triangle then the two triangles are equal.

By sine rule,

Y = sin156.78 × 44 / 34

= 29

sin Y = 0.499

Y = 29.93°

By the angle sum property. The last angle will be:

180 - 156.78 - 29.93

= 6.67°

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Applying the inscribed angle theorem, we have the missing measures as: 15. m<DEF = 127°         16. m(PL) = 62°

15. Angle DEF is an inscribed angle while arc FGD is the intercepted arc. Thus, according to the inscribed angle theorem, we have:

m<DEF = 1/2(m(FGD)

Plug in the values:

6x + 37 = 1/2(19x - 31)

2(6x + 37) = 19x - 31

12x + 74 = 19x - 31

12x - 19x = -74 - 31

-7x = -105

x = 15

m<DEF = 6x + 37 = 6(15) + 37 = 127°

16. m<LMP = m<LNP

Plug in the values:

5x - 19 = 2x + 11

5x - 2x = 19 + 11

3x = 30

x = 10

m(PL) = 2(LMP) [based on the inscribed angle theorem]

Plug in the values:

m(PL) = 2(5x - 19)

Plug in the value of x:

m(PL) = 2(5(10) - 19)

m(PL) = 62°

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The values  of x and y that make the triangles congruent by HL are:

x = 3 and y = 1

From the question, we are to determine the values of x and y that make the triangles congruent by HL.

Two triangles are said to be congruent if all three corresponding sides are equal and all the three corresponding angles are equal in measure.

Thus,

For the triangles to be right congruent by HL

x = y + 2

and

x + 1 = 4y

Solve the equations simultaneously

Substitute x = y + 2 in the second equation

x + 1 = 4y

y + 2 + 1 = 4y

y + 3 = 4y

3 = 4y - y

3 = 3y

Divide both sides by 3

3/3 = 3y/3

1 = y

Therefore,

y = 1

Substitute the value of y into the first equation

x = y + 2

x = 1 + 2

x = 3

Hence,

The values are:

x = 3 and y = 1

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the balance after 6 years is approximately $926.45.

What is Continuous compounding ?

Continuous compounding is the process of calculating interest and reinvesting it into an account's balance over an infinite number of periods.

We can use the formula for continuous compounding to find the balance after 6 years:

[tex]F = Pe^{(rt)}[/tex]

where F is the future value, P is the present value, e is the mathematical constant e (approximately 2.71828), r is the annual interest rate, and t is the time in years.

In this problem, P = $550, r = 8.5% = 0.085 (note that we need to use the decimal form of the interest rate), and t = 6 years.

[tex]F = 550e^{(0.085*6)}[/tex]

F ≈ $926.45

Therefore, the balance after 6 years is approximately $926.45.

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Therefore, the mean absolute deviation of the times taken to complete the obstacle course is 0.3125 minutes.

Mean absolute deviation (MAD) is a statistical measure that calculates the average distance between each data point in a dataset and the mean or median of that dataset. It measures the dispersion or variability of the data points in relation to the central tendency of the dataset.

To calculate the MAD, you first find the mean or median of the dataset, and then you calculate the absolute difference between each data point and the mean or median. The absolute differences are then averaged to give the mean absolute deviation.

MAD is expressed in the same units as the original data and is commonly used in finance, economics, and other fields to measure the variability of financial returns, economic indicators, or other datasets. MAD is a robust measure of dispersion that is less sensitive to extreme values or outliers compared to other measures of dispersion such as standard deviation.

Let's say the times taken by the four people are:

Diane: 2 minutes

Friend 1: 1.5 minutes

Friend 2: 2.5 minutes

Friend 3: 1.75 minutes

The mean of these times is:

Mean = (2 + 1.5 + 2.5 + 1.75) / 4 = 1.9375 minutes

Next, we need to find the absolute deviations of each time from the mean. To do this, we subtract the mean from each time and take the absolute value of the result:

|2 - 1.9375| = 0.0625

|1.5 - 1.9375| = 0.4375

|2.5 - 1.9375| = 0.5625

|1.75 - 1.9375| = 0.1875

Then, we find the mean of these absolute deviations:

MAD = (0.0625 + 0.4375 + 0.5625 + 0.1875) / 4 = 0.3125 minutes

Therefore, the mean absolute deviation of the times taken to complete the obstacle course is 0.3125 minutes.

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Answer:

b but I'm not exactly sureb

Z=+0.5, which has an absolute value of 0.5 and is the closest Z-score to the mean so option A is correct.

In mathematics, the absolute value (also known as modulus) of a real number is its distance from zero on the number line. It is denoted by enclosing the number between two vertical bars, such as |x|.

For example, the absolute value of 5 is 5, because 5 units to the right of zero on the number line is at a distance of 5 units from zero. Similarly, the absolute value of -5 is also 5, because 5 units to the left of zero on the number line is at a distance of 5 units from zero.

To determine which Z-score represents the location closest to the mean, we need to know the sign of each Z-score, as well as the absolute value of each Z-score.

The sign of the Z-score indicates whether the location is above (+) or below (-) the mean. The absolute value of the Z-score indicates how far away the location is from the mean, measured in standard deviations.

Looking at the options provided, we see that:

A. Z=+0.5 is positive, indicating a location above the mean, and has an absolute value of 0.5.

B. Z=+1.0 is positive, indicating a location above the mean, and has an absolute value of 1.0.

C. Z=-1.5 is negative, indicating a location below the mean, and has an absolute value of 1.5.

D. Z=-0.3 is negative, indicating a location below the mean, and has an absolute value of 0.3.

Since we want to find the Z-score closest to the mean, we should look for the option with the smallest absolute value. Therefore, the answer is A. Z=+0.5, which has an absolute value of 0.5 and is the closest Z-score to the mean.

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Answer:

The correct equation is C. d = 163t.

This equation shows a direct proportionality between the amount of time worked (t) and the amount of money earned (d). For each hour worked, Rajindri earns $163, so the total amount earned (d) is equal to the hourly rate of $163 multiplied by the number of hours worked (t).

Step-by-step explanation: