13. [-12 Points] DETAILS HARMATHAP12 10.4.017. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER A rectangular field with one side along a river is to be fenced. Suppose that no fence is needed along the river, the fence on the side opposite the river costs $40 per foot, and the fence on the other sides costs $10 per foot. If the field must contain 28,800 square feet, what dimensions will minimize costs?

The exact value of sin(α + β) is -16/65.

To find the exact value of sin(α + β), we can use the trigonometric identities and the given information about the values of sinα and sinβ.

We are given:

sinα = 3/5 (α in quadrant I)

sinβ = 5/13 (β in quadrant II)

Let's use the sum formula for sine:

sin(α + β) = sinα * cosβ + cosα * sinβ

To evaluate sinα, we need to find cosα using the Pythagorean identity:

sin²α + cos²α = 1

Given sinα = 3/5, we can solve for cosα:

(3/5)² + cos²α = 1

9/25 + cos²α = 1

cos²α = 1 - 9/25

cos²α = 16/25

cosα = ± √(16/25)

cosα = ± 4/5

Since α is in quadrant I, cosα is positive:

cosα = 4/5

Similarly, we need to find cosβ using the Pythagorean identity:

sin²β + cos²β = 1

Given sinβ = 5/13, we can solve for cosβ:

(5/13)² + cos²β = 1

25/169 + cos²β = 1

cos²β = 1 - 25/169

cos²β = 144/169

cosβ = ± √(144/169)

cosβ = ± 12/13

Since β is in quadrant II, cosβ is negative:

cosβ = -12/13

Now, substitute the values of sinα, cosα, sinβ, and cosβ into the sum formula for sine:

sin(α + β) = (3/5) * (-12/13) + (4/5) * (5/13)

sin(α + β) = -36/65 + 20/65

sin(α + β) = -16/65

Therefore, the exact value of sin(α + β) is -16/65.

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Step-by-step explanation:

Break it up into two trapezoids as shown

area = trap1 + trap2

      = 2 *   (7+3) / 2   + 3 *  ( 7 + 3) / 2 = 10 + 15 = 25 units^2

Answer:

x= 26

Step-by-step explanation:

Im not sure but i had this same quistion on a quiz and i got it correct

Therefore, the diameter of the circle with an area of pi is 2. We can round this to the nearest tenth to get 2.0, since there are no decimal places beyond the first digit after the decimal point. Thus, the diameter of the circle is 2.0.

To find the diameter of a circle with an area of pi, we need to use the formula for the area of a circle which is A = πr^2, where A is the area of the circle, π is pi, and r is the radius of the circle.

Since we are given the area of the circle as pi, we can substitute that into the formula as follows:A = πr^2pi = πr^2Dividing both sides of the equation by pi gives us

:r^2 = 1r = sqrt(1)

Since the radius is equal to the square root of 1, which is 1, we can now use the formula for the diameter of a circle to find the diameter. The formula for the diameter of a circle is

:d = 2rd = 2(1) = 2

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Answer:

78 Degrees

Step-by-step explanation:

Complimentary Angles add up to 90 degrees so the equation below solves for x and then finds angle A.

3x+42+x=90

4x=48

x=12

3(12)+42=78

Answer:

Step-by-step explanation:

B

Answer:  \(\frac{8x^3}{27y^6}\)

This is the fraction 8x^3 all over 27y^6

On a keyboard, we can write it as (8x^3)/(27y^6)

===========================================================

Explanation:

The exponent tells you how many copies of the base to multiply with itself.

We'll have three copies of \(\left(\frac{2x}{3y^2}\right)\) multiplied with itself due to the cube exponent on the outside.

So,

\(\left(\frac{2x}{3y^2}\right)^3 = \left(\frac{2x}{3y^2}\right)*\left(\frac{2x}{3y^2}\right)*\left(\frac{2x}{3y^2}\right)\\\\\left(\frac{2x}{3y^2}\right)^3 = \frac{2x*2x*2x}{(3y^2)*(3y^2)*(3y^2)}\\\\\left(\frac{2x}{3y^2}\right)^3 = \frac{(2*2*2)*(x*x*x)}{(3*3*3)*(y^2*y^2*y^2)}\\\\\left(\frac{2x}{3y^2}\right)^3 = \frac{8x^3}{9y^6}\\\\\)

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

Or another approach you could take is to cube each component of the fraction. The rule I'm referring to is \(\left(\frac{a}{b}\right)^c = \frac{a^c}{b^c}\)

Applying that rule will lead to:

\(\left(\frac{2x}{3y^2}\right)^3 = \frac{(2x)^3}{(3y^2)^3}\\\\\left(\frac{2x}{3y^2}\right)^3 = \frac{2^3*x^3}{3^3*(y^2)^3}\\\\\left(\frac{2x}{3y^2}\right)^3 = \frac{8x^3}{27y^{2*3}}\\\\\left(\frac{2x}{3y^2}\right)^3 = \frac{8x^3}{27y^6}\\\\\)

Either way you should get 8x^3 all over 27y^6 as one fraction.

Answer:

33, 9

Step-by-step explanation:

I33-21I=12

I9-21I=

I-12I=12

Every SUM of an absolute value will always be positive!!

Answer:

3.43

Step-by-step explanation:

This because 22.89 multiplied by %15 is 3.4335 but of course just put the first 3 numbers and the place holder.  

Answer: last one i’m pretty sure

Step-by-step explanation:

There are 2,139,792,767,488,000 ways to assign scores to the problems if the sum of the scores is 100 and each question is worth at least 5 points.

To solve this problem, we will use the concept of combinations with repetition.

First, let's subtract 5 points from each question, since each question is worth at least 5 points. Now, we need to distribute the remaining 50 points (100 - 10*5) among the 10 questions.

Imagine that we have 50 identical balls (points) to distribute among 10 distinct boxes (questions). To do this, we can use a technique called "stars and bars". We need to place 9 dividers (bars) between the 50 balls (stars) to divide them among the 10 questions.

We will have a total of 59 positions (50 stars + 9 bars). We need to choose 9 of these positions for the bars, and the stars will fill the remaining positions.

The number of ways to do this is given by the combination formula:

C(n, k) = n! / (k! * (n - k)!)

In our case, n = 59 (total positions) and k = 9 (dividers). So, we need to calculate C(59, 9):

C(59, 9) = 59! / (9! * (59 - 9)!)
C(59, 9) = 59! / (9! * 50!)
C(59, 9) = 2,139,792,767,488,000

Therefore, there are 2,139,792,767,488,000 ways to assign scores to the problems if the sum of the scores is 100 and each question is worth at least 5 points.

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The reason for why the confidence intervals help researchers attain their purpose of using a sample to understand a population is given below .

In the question ,

we have been asked how does the confidence interval help researchers to attain the purpose of using a sample to understand a population ,

we know that , the confidence interval is calculated from an estimate of how far away our sample mean is from actual population mean .

the confidence interval are useful because ,

(i) by calculating the confidence intervals around any data we collect, we have additional information about the likely values we are trying to estimate .

(ii) they make data analyses richer and help us to make more informed decisions about the research questions .

Therefore , the reason how confidence interval helps is mentioned above.

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The correct answer is  there are 72 ways to fill the two vacancies with one man and one woman.

a. To fill the two vacancies with any two of the nominees, we have 7 nominees to choose from. We need to select 2 nominees out of the 7. We can use the combination formula for this. The number of ways to choose 2 nominees out of 7 is given by:

C(7, 2) = 7! / (2! * (7-2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21

Therefore, there are 21 ways to fill the two vacancies with any two of the nominees.

b. To fill the two vacancies with any two of the women, we have 4 women to choose from. We need to select 2 women out of the 4. Again, we can use the combination formula:

C(4, 2) = 4! / (2! * (4-2)!) = 4! / (2! * 2!) = (4 * 3) / (2 * 1) = 6

Therefore, there are 6 ways to fill the two vacancies with any two of the women. c. To fill one vacancy with a man and one vacancy with a woman, we have 3 men and 4 women to choose from. We need to select 1 man out of 3 and 1 woman out of 4. Again, we can use the combination formula:

C(3, 1) * C(4, 1) = (3! / (1! * (3-1)!)) * (4! / (1! * (4-1)!)) = (3 * 2) * (4 * 3) = 6 * 12 = 72

Therefore, there are 72 ways to fill the two vacancies with one man and one woman.

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Answer: 0.3, 30%, or 3/10

Step-by-step explanation:

Andrea ate 1/4 of the pizza.

1/4 = 0.25

Her brother ate 35% of the pizza.

35% = 0.35

Her mom at 0.10 of the pizza.

1 - 0.25 - 0.35 - 0.10 = 0.3 of the pizza

0.3 = 30% = 3/10

The spinner that could be used to simulate pulling a bead out of the bag without looking would have three sections: 6 white, 1 red, and 1 pink.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.

To simulate pulling a bead out of the bag without looking, we need a spinner with three sections, each section representing one of the three colors: white, red, and pink. The size of each section should be proportional to the number of beads of that color in the bag.

The total number of beads in the bag is 18 + 3 + 3 = 24.

Therefore, the proportion of white beads is 18/24 = 3/4, the proportion of red beads is 3/24 = 1/8, and the proportion of pink beads is 3/24 = 1/8.

To create a spinner with these proportions, we could divide a circle into 8 equal sections, color 6 of them white, 1 of them red, and 1 of them pink.

Hence, the spinner that could be used to simulate pulling a bead out of the bag without looking would have three sections: 6 white, 1 red, and 1 pink.

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Answer:

6776

Step-by-step explanation:

8℅ = 0.08

0.08 x 42350 = 3388

3388 x 2 = 6776

Answer:

5

Step-by-step explanation:

Let a = number of pieces of chocolate bought by Amin.

Let b = number of pieces of chocolate bought by Bob.

b = 2a

(a - 3)(b - 3) = 14

ab - 3a - 3b + 9 = 14

a(2a) - 3a - 3(2a) = 5

2a^2 - 3a - 6a = 5

2a^2 - 9a - 5 = 0

(2a + 1)(a - 5) = 0

2a + 1 = 0  or  a - 5 = 0

a = -1/2  or  a = 5

Amin cannot have bought -1/2 pieces of chocolate, so we discard the soluion a = -1/2.

a = 5

Answer: 5

Determine the angle of depression.

Answer: 0.0120 degree

Concept: Proportion, to find the km/h return journer

3 hours to 84 km/hr = 3.5 hours to x km/hr

(since its inverse proportion, multiply the 1st to 2nd and 3rd to 4th)

Inverse proportion: as the other quantities increase, the other one decreases

3 : 84 = 3.5 : x

3(84) = 3.5x

252 = 3.5x                (divide 3.5 both sides)

x = 72

Step-by-step explanation:

This is equivalent to finding e such that \((x - 1)e = -yi\). Similarly, \(e(x - 1) = yi\). Hence,\(e = (-y + xi)/(1 - x²)\) is an identity element for G.

To determine if \(G = {a+bi | a² + b² = 1}\) is a group under multiplication, we need to verify the following conditions for any a, b, c, d ∈ R:

Closure: For all a, b ∈ G, ab ∈ G.

This is true because

if \(a = x + yi and b = u + vi\),

then\(ab = (xu - yv) + (xv + yu)i.\)

Since \(x² + y² = 1 and u² + v² = 1\),

then\((xu - yv)² + (xv + yu)² = 1.\)

Hence, ab ∈ G.

Associativity: For all \(a, b, c ∈ G, (ab)c = a(bc).\)

We need to show that there exists an element e such that for any element a ∈ G, ae = ea = a.

Let a = x + yi. Then \(ae = (x + yi)e = xe + yie and ea = e(x + yi) = xe + yie\). We need to find e such that\(xe + yie = x + yi.\)

Inverse:

For each a ∈ G, there exists an element b ∈ G such that \(ab = ba = e.\)

To verify this, let a = x + yi, and find an element \(b = c + di\) such that \((x + yi)(c + di) = 1, or xc - yd + (xd + yc)i = 1 + 0i.\)

Equating real and imaginary parts gives two equations:

\(xc - yd = 1 and xd + yc = 0.\)

Solving this system of equations yields \(b = (x - yi)/(x² + y²).\)

The above discussion proves that G is a group under multiplication.

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Answer:

x=0

Step-by-step explanation:

solve for  x  by simplifying both sides of the equation, then isolating the variable. which got me x=0

Answer:

x = 1/90

or

x = 90^-1

Step-by-step explanation:

-2/5x - 8/15x + 1/3x = -54

-9/15x = -54

-3/5x = -54

3x = 270x

270x = 3

x = 1/90, x ≠ 0

Answer:

The distributive property allows you multiply a sum in parenthesis by multiplying each addend separately, then add the products.

Step-by-step explanation:

How to use the distributive law example.

2(x+4) = 16

To use the distributive law in this example multiply 3 by all terms in the parenthesis. Multiply 2 and x, then 2 and 4 to open the parenthesis.

2x+8=16

That is how you use the distributive law.

To continue solving, subtract 8 from both sides.

2x+8-8=16-8

2x=8

Divide 2 from both sides.

2x/2=8/2

x=4

Hope this helps!

If not, I am sorry.

Using conversion from fraction to decimal and rational numbers, we have that:

7. The fraction that has a corresponding decimal greater than 1 is 4/3.

8. One rational number between 7 and 8 is given by option a, which is 7.83838383...

A fraction is converted to decimal dividing the numerator by the denominator.

In item 7, we consider fraction 4/3, for which:

Hence the corresponding decimal is given as follows:

4/3 = 1.333, which is the result of the division of 4 by 3, and is greater than 1.

The set of rational numbers is composed by all numbers that can be represented by fractions, such as integer, terminating decimals and repeating decimals.

In item 8, option a, we have the number 7.83, with the bar at 83. This means that 83 is the repeating decimal part, that is, the rational number is 7.83838383...

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Answer:

\(17 \le t \le 28\)

Step-by-step explanation:

Given

\(t = 22\frac{1}{2}\) --- average time

\(\triangle t = 5\frac{1}{2}\) --- the variation

Required

The inequality to represent the scenario

To do this, we simply add and subtract the variation from the average time.

i.e.

\(t \± \triangle t\)

So, the inequality is:

\(22\frac{1}{2} - 5\frac{1}{2} \le t \le 22\frac{1}{2} + 5\frac{1}{2}\)

Solve:

\(17 \le t \le 28\)

Answer:

the answer is \(\frac{5}{6}\)÷\(\frac{1}{12}=10\)

Step-by-step explanation:

Answer:

Made $840 in commissions for the past month.

Step-by-step explanation:

sales = 70 * 150 = 10500

commission = .08 * 10500 = 840

Answer:

$840

Step-by-step explanation:

8% = 0.08

.08 * 150 = 12 per unit sold

12 * 70 = 840.00

The length of PS is 34 inches. Points P,Q,S are points on a line, so that PQ is 6 in, The ratio PQ:QR is 3:4 and The ratio QR:RS is 2:5.

An equation is an expression containing numbers and variables linked together by mathematical operations such as addition, subtraction, division, multiplication and exponents.

The length of PQ is 6 in. The ratio PQ:QR is 3:4, hence:

PQ = PR * (3/7)

6 = PR * (3/7)

PR = 14 inches

QR = PR * (4/7)

QR = 14 * 4/7

QR = 8 inches

The ratio QR:RS is 2:5, hence:

QR = QS * (2/7)

8 = QS * 2/7

QS = 28 inches

Points P,Q,S appear in that order on a line. Therefore:

PS = PQ + QS

PS = 6 + 28

PS = 34 inches

The length of PS is 34 inches.

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Answer:

Discount : 3.60$

Discounted shirt price: 16.40$

Step-by-step explanation:

If there is a 18% then we can get the discounted price you get by multiplying 20$ by 18%

So we can change 18% to decimal if we need to by just moving a decimal point 2 decimal points to the left. so .18.

Now 20*.18 = 3.60$ | That is the discount you get on sale. To get the actual discount price you just need to subtract 20-3.6=  16.40$

Answer:

Step-by-step explanation:

To calculate the 18% discount, convert 18% into the equivalent decimal fraction 0.18 and then multiply the regular price ($20) by 0.18; the discount is thus $3.60.

Subtract the discount, $3.60, from the regular price ($20) to obtain the discount price:  $16.40.

********************************

Alternatively, take the shortcut of multiplying $20 by (1.00 - 0.18), or

$20 by 0.82.  The result (i. e., the discount price) is the same:  $16.40

The probability that a single bulb will burn out within the next 10 minutes is approximately 0.6321. The expected amount of time until the first bulb goes out is 10 minutes. The probability density function (pdf) of the random variable T, representing the time when you leave the room after the last light bulb goes out, is given by \(g(t) = 20 * (1/10) * e^{(-(1/10)t)} * (1 - e^{(-(1/10)t))^(19)}\).

To find the probability that a single bulb will burn out within the next 10 minutes, we can use the exponential distribution. The exponential distribution with a mean of 10 minutes has a rate parameter λ = 1/10.

The probability density function (pdf) for an exponential distribution is given by \(f(x) = λ * e^{(-λx)}\)

In this case, we want to find the probability that a bulb burns out within the next 10 minutes, which corresponds to the cumulative distribution function (CDF) at x = 10. The CDF is given by \(F(x) = 1 - e^{(-λx)\)

So, substituting the values, we have:

\(F(10) = 1 - e^{(-(1/10)*10)\)

\(= 1 - e^{(-1)\)

= 1 - 0.3678794412

≈ 0.6321

Therefore, the probability that a single bulb will burn out within the next 10 minutes is approximately 0.6321.

The amount of time until the first bulb goes out follows an exponential distribution with a rate parameter of λ = 1/10 (since it has a mean of 10 minutes).

The probability density function (pdf) for the time until the first bulb goes out is given by\(f(t) = λ * e^{(-λt).\)

To find the expected amount of time until the first bulb goes out, we need to calculate the mean (or expected value) of this distribution.

The expected value of an exponential distribution with rate parameter λ is equal to 1/λ. In this case, the expected value is 1/(1/10) = 10 minutes.

Therefore, the expected amount of time until the first bulb goes out is 10 minutes.

To find the probability density function (pdf) of the random variable T, which represents the time when you leave the room (after the last light bulb goes out), we need to consider the distribution of the maximum of the exponential random variables.

Since there are 20 light bulbs in the room, and each follows an exponential distribution with a rate parameter λ = 1/10, the time until the last bulb goes out can be modeled as the maximum of 20 exponential random variables.

The pdf of the maximum of independent exponential random variables with the same rate parameter λ is given by \(g(t) = n * λ * e^{(-λt)} * (1 - e^{(-λt))^(n-1)}\), where n is the number of random variables.

In this case, n = 20, and λ = 1/10. Thus, the pdf of T is \(g(t) = 20 * (1/10) * e^{(-(1/10)t)} * (1 - e^{(-(1/10)t))^(19)}\)

This expression represents the pdf of the random variable T, which denotes the time when you leave the room after the last light bulb goes out.

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