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Which of the following radical expressions has an absolute value symbol in its simplified form?
√81x
O √64x³
O 16x
O V-125x³

Answer:$386,711.70

Step-by-step explanation:

To solve the problem, we can use the formula for the future value of an annuity:

FV = PMT x [(1 + r/n)^(n*t) - 1] / (r/n)

where:

FV = future value

PMT = payment per period

r = annual interest rate

n = number of compounding periods per year

t = number of years

First, we need to find the number of compounding periods and the interest rate per quarter:

n = 4 (quarterly compounding)

r = 0.12 / 4 = 0.03 (3% quarterly interest rate)

Next, we can plug in the values:

PMT = $500

n = 4

r = 0.03

t = 30

FV = $500 x [(1 + 0.03/4)^(4*30) - 1] / (0.03/4)

FV = $500 x [(1 + 0.0075)^120 - 1] / 0.0075

FV = $500 x [6.3207 - 1] / 0.0075

FV = $500 x 773.4234

FV = $386,711.70

Therefore, if you save $500 quarterly into an account that pays 12% compounded quarterly from your 21st to your 51st birthday, you will have approximately $386,711.70 in the account.

Step-by-step explanation:

You need the line in y = mx + b form:

y = -12x + 765        where x = minutes     y = height in feet

First, 90 + 90 = 180

Second, 320 - 180 = 140

Third, 140 / 2 = 70.

|                                90 * 70 = 6,300
|
|
| 90
|
|____________

         70

The statement "S′H′ is parallel to S′Q′" is true.

To find the image of the square after reflection across the x-axis, we need to flip each point of the square over the x-axis. The coordinates of the reflected points will have the same x-coordinate, but the y-coordinate will have the opposite sign.

The coordinates of the original square are:

S(−3,0)

H(2,0)

A(2,5)

Q(−3,5)

The coordinates of the reflected square are:

S′(−3,0)

H′(2,0)

A′(2,−5)

Q′(−3,−5)

Now, we can find the slopes of the sides of the reflected square to determine which sides are parallel.

S′H′¯¯¯¯¯¯¯¯¯¯¯: The slope of S′H′¯¯¯¯¯¯¯¯¯¯¯ is 0, since both points have the same y-coordinate. Therefore, S′H′¯¯¯¯¯¯¯¯¯¯¯ is parallel to the x-axis.

S′Q′¯¯¯¯¯¯¯¯¯¯: The slope of S′Q′¯¯¯¯¯¯¯¯¯¯ is 0, since both points have the same y-coordinate. Therefore, S′Q′¯¯¯¯¯¯¯¯¯¯ is also parallel to the x-axis.

H′A′¯¯¯¯¯¯¯¯¯¯¯: The slope of H′A′¯¯¯¯¯¯¯¯¯¯¯ is undefined, since the two points have the same x-coordinate. Therefore, H′A′¯¯¯¯¯¯¯¯¯¯¯ is parallel to the y-axis.

H′S′¯¯¯¯¯¯¯¯¯¯¯: The slope of H′S′¯¯¯¯¯¯¯¯¯¯¯ is also undefined, since the two points have the same x-coordinate. Therefore, H′S′¯¯¯¯¯¯¯¯¯¯¯ is parallel to the y-axis.

A′Q′¯¯¯¯¯¯¯¯¯¯: The slope of A′Q′¯¯¯¯¯¯¯¯¯¯ is 0, since both points have the same y-coordinate. Therefore, A′Q′¯¯¯¯¯¯¯¯¯¯ is parallel to the x-axis.

Q′S′¯¯¯¯¯¯¯¯¯¯: The slope of Q′S′¯¯¯¯¯¯¯¯¯¯ is 0, since both points have the same y-coordinate. Therefore, Q′S′¯¯¯¯¯¯¯¯¯¯ is also parallel to the x-axis.

Based on these calculations, we can see that S′H′¯¯¯¯¯¯¯¯¯¯¯ is parallel to S′Q′¯¯¯¯¯¯¯¯¯¯ (both are parallel to the x-axis). Therefore, the statement "S′H′¯¯¯¯¯¯¯¯¯¯¯ is parallel to S′Q′¯¯¯¯¯¯¯¯¯¯" is true.

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After answering the prοvided questiοn, we can cοnclude that As a result, the sοlutiοn tο the inequality is p greater than 32.

In mathematics, an inequality is a nοn-equal relatiοnship between twο expressiοns οr values. As a result, imbalance leads tο inequality. In mathematics, an inequality cοnnects twο values that are nοt equal. Inequality is nοt the same as equality. When twο values are nοt equal, the nοt equal sign is cοmmοnly used ().

Different inequalities, nο matter hοw small οr large, are used tο cοntrast values. Many simple inequalities can be sοlved by mοdifying the twο sides until οnly the variables remain. Hοwever, a number οf factοrs cοntribute tο inequality: Negative values are divided οr added οn bοth sides. Exchange left and right.

We must isοlate the variable p οn οne side οf the inequality sign in οrder tο sοlve it.

-16 negative οne-half p

When bοth sides οf the inequality are multiplied by -2 (and the inequality sign is reversed because we are multiplying by a negative number), we get:

p > 32

As a result, the sοlutiοn tο the inequality is p greater than 32.

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The ratio means that for every 2 motor boats, there are 3 sail boats. 6 sail boats must also enter so that the ratio remains the same. A possible missing information will be the total number of ships in the armada.

A ratio is a mathematical expression that represents the relationship between two quantities or numbers. It is a comparison of two numbers, often written in the form of a fraction or with a colon. Ratios are used to express how much of one thing there is in relation to another.

1. a. The ratio of motor boats to sail boats can be written in three ways:

As a fraction: 2/3

With a colon: 2:3

With the word "to": 2 to 3

This ratio means that for every 2 motor boats, there are 3 sail boats. Alternatively, it can be interpreted as for every 3 sail boats, there are 2 motor boats. The ratio does not specify the total number of boats, only the relative proportion of motor boats to sail boats.

b. To keep the ratio between motor boats and sail boats the same, we need to maintain the same ratio of motor boats to sail boats.

Currently, the ratio of motor boats to sail boats is 2:3.

Let x be the number of sail boats needed to maintain the ratio.

After x sail boats enter, the total number of boats will be 2 + x motor boats and 3 + x sail boats.

The ratio of motor boats to sail boats will still be 2:3, so we can write,

[tex]\frac{(2 + x)}{(3 + x)}[/tex] = [tex]\frac{2}{3}[/tex]

Cross-multiplying, we get,

2(3 + x) = 3(2 + x)

6 + 2x = 6 + 3x

x = 6

Therefore, 6 sail boats must also enter so that the ratio remains the same.

2. To find the total number of galleons and galleys in the Spanish armada, we need at least one additional piece of information. The ratio of 5:1 tells us that for every 5 galleons, there is 1 galley. However, we don't know the total number of galleons and galleys in the armada.

One possible missing information that could help us find the total number of galleons and galleys is the total number of ships in the armada. If we knew the total number of ships, we could find the number of galleons and galleys in the armada.

For example, let's say the total number of ships in the armada is 100. And let number of galleons = a and number of galleys = b. Then,

a + b = 100

[tex]\frac{a}{b}[/tex] = 5/1

b = a/5

Substituting this expression into the first equation, we get:

a+ (a/5) = 100

(6a/5) = 100

a = 500/6 ≈ 83.33

Then b = (1/5)(83.33) = 16.67

However, since we cannot have a fractional part of a ship, we can round up or down to get whole numbers. Therefore, we can conclude that the Spanish armada had approximately 83 galleons and 17 galleys, based on the given ratio of 5:1.

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Total area of garden after adding rectangular piece will be 55 ft² i.e. B.

What exactly is a rectangle?

A rectangle is a two-dimensional geometric shape that is characterized by having four sides and four right angles. Opposite sides of a rectangle are parallel and equal in length, while opposite angles are equal. The area of a rectangle can be calculated by multiplying its length by its width, while its perimeter is the sum of the lengths of all its sides. Rectangles are commonly used in many fields, such as architecture, engineering, and mathematics.

Now,

The rectangle has a length of |-21 - (-23)| = 2 and a width of |7 - 12| = 5. Therefore, the area of the new rectangular piece is 2 x 5 = 10 square feet.

Adding this to the area of the existing garden gives a total area of 45 + 10 = 55 square feet.

Therefore, the answer is (B) 55 ft².

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Answer:

Step-by-step explanation:

-8^2 and (-2)^3 are exponents so it should be solved first

-8^2 = -64 and (-2)^3 = -8

(-64)/(-8)

64/8

8

8

To solve this expression, we need to follow the order of operations, which is PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction) from left to right:

First, we need to simplify the exponents: -8^2 means -1 times 8 squared, which is -1 times 64 or -64. (-2)^3 means -2 multiplied by itself three times, which is -2 x -2 x -2 or -8.

Next, we need to perform the division and multiplication, again from left to right: -64 ÷ -8 equals 8.

Therefore, the final answer to the expression -8^2÷(-2)^3 is 8.

The number of ounces in solution A= 16, and the number of ounces in the solution B = 104

let x be the number of ounces of solution A

let y be the number of ounces of solution B

x+y=120

Substitute the value of y,

y=120-x

Solution A is 70% salt and B is 95% salt,

then

0.7x+.95y=.75(120)

0.7x+.95y=90

multiply both sides of the equation by 100 to remove the decimal points,

70x+95y=900

70x+95(120-x)=900

70x+1140-95x=900

-15x=900-1140

15x=240

x=16

and y=120-16

y=104

The number of ounces in solution A= 16

and the number of ounces in solution B = 104

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Answer:

142 square meters

Step-by-step explanation:

To find the Surface Area of a Cylinder you have to use the formula [tex]A= 2\pi rh+2\pi r^2[/tex]

Plug in the radius, height, and pi to get

[tex]A=(2*3.14*2.6*6.1)+(2*3.14*2.6^2)\\A= (99.6008)+(42.4528)\\A= 142.0536\\[/tex]

You can round the answer to 142

Answer:

The number of times the tire will have to turn in travelling the length of the street is 30.9 times.

To determine the number of times the tire will have to turn in travelling the length of the street, we will first calculate the circumference of the tire.  

Since the tire is circular, the circumference of the tire can be calculated from the formula for calculating the circumference of a circle.

The circumference of a circle is given by

C = πd

Where C is the circumference and d is the diameter

From the question d = 1.7m and π = 3.14

∴ C = 3.14 × 1.7

C = 5.338 m

Therefore, the circumference of the tire is 5.338 m

Now, for the number of times the tire will have to turn in travelling the length of the street, we will divide the length of the street by the circumference of the tire.

Number of times the tire will have to turn = Length of the street ÷ Circumference of the tire

Number of times the tire will have to turn = 165 m ÷ 5.338 m

Number of times the tire will have to turn = 30.91045 times

Number of times the tire will have to turn ≅ 30.9 times

Hence,  the number of times the tire will have to turn in travelling the length of the street is 30.9 times

Amount Due at Maturity a. $102,000, b. $30,100, c. $62,620, d. $42,930, e. $40,700.

Interest is the cost of borrowing money, typically expressed as a percentage of the borrowed amount, called the principal. It is the amount charged by a lender to a borrower for the use of money over a certain period of time. Interest can also refer to the amount earned on money that is invested, such as in a savings account or a bond. The rate of interest depends on factors such as the level of risk associated with the loan, the length of the loan period, and the market rate of interest at the time of borrowing. The amount of interest paid or earned can be calculated using various formulas, including simple interest and compound interest.

To determine the due date and amount of interest due at maturity for each note, we can use the following formula:

Interest = Principal x Rate x Time

where Principal is the face amount of the note, Rate is the annual interest rate, and Time is the time in years (based on a 360-day year).

a. January 5, $100,000, 6%, 120 days

Due Date: May 5

Time: 120/360 = 1/3 year

Interest = $100,000 x 0.06 x 1/3 = $2,000

Amount Due at Maturity = Principal + Interest = $100,000 + $2,000 = $102,000

b. February 15, $30,000, 4%, 30 days

Due Date: March 17 (assuming non-leap year)

Time: 30/360 = 1/12 year

Interest = $30,000 x 0.04 x 1/12 = $100

Amount Due at Maturity = Principal + Interest = $30,000 + $100 = $30,100

c. May 19, $62,000, 8%, 45 days

Due Date: July 3

Time: 45/360 = 1/8 year

Interest = $62,000 x 0.08 x 1/8 = $620

Amount Due at Maturity = Principal + Interest = $62,000 + $620 = $62,620

d. August 20, $42,400, 5%, 90 days

Due Date: November 18

Time: 90/360 = 1/4 year

Interest = $42,400 x 0.05 x 1/4 = $530

Amount Due at Maturity = Principal + Interest = $42,400 + $530 = $42,930

e. October 19, $40,000, 7%, 90 days

Due Date: January 17

Time: 90/360 = 1/4 year

Interest = $40,000 x 0.07 x 1/4 = $700

Amount Due at Maturity = Principal + Interest = $40,000 + $700 = $40,700

Therefore, the due date and amount of interest due at maturity for each note are as follows:

a. Due Date: May 5, Interest: $2,000, Amount Due: $102,000

b. Due Date: March 17, Interest: $100, Amount Due: $30,100

c. Due Date: July 3, Interest: $620, Amount Due: $62,620

d. Due Date: November 18, Interest: $530, Amount Due: $42,930

e. Due Date: January 17, Interest: $700, Amount Due: $40,700

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Answer: The answer is C.

Step-by-step explanation:

The equation that could represent a rational function is answer C.

I hope this helped! A brainilist would be amazing! <3

After simplifying the given expression we can write answer using positive exponent [tex]7^3 * 6^30[/tex].

What is expression?

In mathematics, an expression is a combination of numbers, variables, and symbols that are grouped together to represent a mathematical relationship or quantity. It can be a single term, or a combination of terms separated by mathematical operations such as addition, subtraction, multiplication, and division.

When we raise a power to another power, we need to multiply the exponents. So we can simplify the expression as follows:

[tex]7^3(6^6)^5 = 7^3 * 6^(6*5) = 7^3 * 6^30[/tex]

so we write the answer using positive exponents as:

[tex]7^3 * 6^30[/tex]

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Step-by-step explanation:

The probability of Meikel wearing khakis and sandals is

3 ÷ 12 which will give us

1/4

Answer:

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

Angle with the measure of 130° forms a vertical angle pair with the sum of x and 107° angle.

Since vertical angles are congruent we have:

The two angles, x and 2x together form a right angle, since they form a linear pair with another angle, marked as right angle. Therefore, x and 2x are complementary angles:

The maximum number of homes that can be placed on the street is 11.

first we need to divide the total length of the street by the width of each home.

The length of the street is 345 feet and the width of each home is 30 feet. Therefore, the number of homes that can be placed on the street is:

345 feet / 30 feet per home = 11.5 homes

Since we cannot have a fraction of a home, we need to round down to the nearest whole number.

Therefore, the maximum number of homes that can be placed on the street is 11.

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Using Bayes theorem, the probability that the person passed through the radar trap located at L2 given that she received a speeding ticket is 0.059 or 5.9%.

We can solve this problem using Bayes' theorem. Let A be the event that the person received a speeding ticket, and Bi be the event that the radar trap was set at location Li, for i = 1, 2, 3, or 4. Then we want to find the conditional probability of B22 given A, which is:

[tex]P(B_2 | A) = P(A | B_2) * P(B_2) / P(A)[/tex]

We can compute each of the probabilities on the right-hand side using the given information.

First, we have:

[tex]P(B_1) = 0.4, P(B_2) = 0.2, P(B_3) = 0.3, P(B_4) = 0.4[/tex]

Next, we have:

[tex]P(A | B_1) = 0.3, P(A | B_2) = 0.1, P(A | B_3) = 0.4, P(A | B_4) = 0.2[/tex]

Finally, we have:

[tex]P(A) = P(A | B_1) * P(B_1) + P(A | B_2) * P(B_2) + P(A | B_3) * P(B_3) + P(A | B_4) * P(B_4)\\= (0.3 * 0.4) + (0.1 * 0.2) + (0.4 * 0.3) + (0.2 * 0.4)\\= 0.34[/tex]

Plugging these values into Bayes' theorem, we get:

[tex]P(B_2 | A) = P(A | B_2) * P(B_2) / P(A)\\= (0.1 * 0.2) / 0.34\\= 0.059\\[/tex]

Therefore, the probability that the person passed through the radar trap located at L2 given that she received a speeding ticket is 0..059 or 5.9%.

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Therefore, we can't calculate the difference in blood pressure between the control and treatment groups. Difference in means: 244.9 - 226.5 = 18.4 mg/dL.

by the question.

The question appears to be incomplete. It asks for the "difference," but it doesn't specify what difference it's referring to. Based on the information provided, I can provide some possible interpretations and calculations:

Difference in cholesterol levels before and after treatment:

Since all patients started with a cholesterol level of 245 mg/dL, we can't calculate the difference in cholesterol levels before and after treatment. We don't have any data on the patients' cholesterol levels after treatment.

Difference in average cholesterol levels between the control and treatment groups:

Control group mean: (241+243+242+245+250+248+246+245+242+247)/10 = 244.9 mg/dL

Treatment group mean: (230+225+220+218+224+240+232+221+235+224)/10 = 226.5 mg/dL

Difference in means: 244.9 - 226.5 = 18.4 mg/dL

Difference in average blood pressure between the control and treatment groups:

The prompt states that the experiment tested whether the cholesterol medication lowers the overall blood pressure for male patients, but it doesn't provide any data on blood pressure.

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The steps that should be taken to evaluate the order of operations given are:

To solve with PEMDAS, you first need to solve what is in the brackets in the order of operations given which is 8÷2 (1+3) - 2^3.

= 8÷2 (1+3) - 2^3

= 8÷2 x 4 - 2^3

Then solve for the exponents :

= 8÷2 x 4 - 2^3

= 8÷2 x 4 - 8

Then because the division comes first, you solve for that:

= 8÷2 x 4 - 8

= 4 x 4 - 8

You then solve for the multiplication as this takes precedence over subtraction:

= 4 x 4 - 8

= 16 - 8

Then solve the subtraction:

= 16 - 8

= 8

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The difference between the means of the two data sets is 4.

Statistical measures are used to describe, analyze, and interpret data. There are several measures that are commonly used in statistics, such as mean, median, mode, standard deviation, etc.

The mean, also known as the average, is the sum of all the values in a dataset divided by the total number of values. It represents the central tendency of the data.

The difference between the means of the two data sets is:

difference between the means = (Class 2 mean) - (Class 1 mean)

difference between the means = 24 - 20 = 4

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Answer:

7:22

Step-by-step explanation:

Given: 2 hours and 22 minutes

5:00 + 2 hours

5 + 2 = 7

7:00 + 22 minutes

7:22

The probability that exactly 250 of those surveyed feel the state of morals is poor is approximately 0.9918.

(To find the probability that exactly 250 of those surveyed feel the state of morals is poor, we can use the normal approximation to the binomial with mean np = 500 * 0.45 = 225 and standard deviation √(npq) = √(500 * 0.45 * 0.55) ≈ 10.42, where q = 1 - p.

Then, we can standardize the value of 250 using the formula z = (x - np) / √(npq), where x is the number of people who feel the state of morals is poor.

z = (250 - 225) / 10.42 ≈ 2.4

Using a standard normal table or calculator, we find that the probability of z being less than or equal to 2.4 is approximately 0.9918.

Therefore, the probability that exactly 250 of those surveyed feel the state of morals is poor is approximately 0.9918.

(b) To find the probability that no more than 220 of those surveyed feel the state of morals is poor, we can use the normal approximation to the binomial with mean np = 500 * 0.45 = 225 and standard deviation √(npq) = √(500 * 0.45 * 0.55) ≈ 10.42, where q = 1 - p.

Then, we can standardize the value of 220 using the formula z = (x - np) / √(npq), where x is the number of people who feel the state of morals is poor.

z = (220 - 225) / 10.42 ≈ -0.48

Using a standard normal table or calculator, we find that the probability of z being less than or equal to -0.48 is approximately 0.3146.

Therefore, the probability that no more than 220 of those surveyed feel the state of morals is poor is approximately 0.3146.

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If Fran and Jill made a new years resolution to ride bikes.  it will take 3 weeks for Fran and Jill to have ridden the same number of miles..

Let's assume that it takes "x" number of weeks for Fran and Jill to ride the same number of miles.

In the first week, Fran rides 40 miles, and Jill rides 70 miles.

In the "x"th week, Fran will have ridden a total of 40 + 55x miles.

In the "x"th week, Jill will have ridden a total of 70 + 45x miles.

For Fran and Jill to ride the same number of miles in "x" weeks, we need to solve the following equation:

40 + 55x = 70 + 45x

Simplifying this equation, we get:

10x = 30

x = 3

Therefore, it will take 3 weeks for Fran and Jill to have ridden the same number of miles.

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The length segment of the circle indicated is 6 units.

A circle is a round-shaped figure that has no corners or edges. Therefore, let's find the length segment of the circle.

Therefore, the diameter of a circle is the line that passes through the centre and meets the circumference at opposite ends.

Therefore, the radius of a circle is the distance from the centre of the circle to any point on it's circumference.

The radius is half of a diameter.

Therefore, the length x is the radius of the circle and it is 6 units.

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Yes, 5:10 and 15:30 are equivalent ratios because they simplify to the same ratio of 1:2.

Ratios represent the relationship between two or more quantities or values. When two ratios have the same simplified form, they are considered equivalent because they represent the same relationship between the quantities being compared.

In the given example, the ratios 5:10 and 15:30 can be simplified to the same ratio of 1:2. This means that both ratios represent the same relationship between the quantities being compared. Specifically, both ratios represent a comparison between two quantities where the second quantity is twice as large as the first quantity.

To simplify a ratio, we divide both the numerator and denominator by their greatest common factor (GCF). The GCF is the largest number that divides evenly into both the numerator and denominator. In this case, the GCF of 5 and 10 is 5, and the GCF of 15 and 30 is 15. Dividing both ratios by their respective GCFs results in a simplified ratio of 1:2 for both ratios.

Therefore, 5:10 and 15:30 are equivalent ratios because they represent the same relationship between quantities, and have the same simplified form.

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21. The trainer raised the end of the treadmill from the floor by 0.62 ft or 7.4 in.

22. The ramp needs to be 1.19 ft tall at the highest point.

23. The springboard is 6.21 in long.

Trigonometric ratio is the ratio of two sides of a right triangle to each other, or the ratio of the sine, cosine, or tangent of an angle in a right triangle.

21: Incline = 7°

Length of walking surface = 5 ft

We need to calculate the change in height of the treadmill.

Let x be the height of the end of the treadmill from the floor.

We can use the trigonometric ratio, tangent (tan), to solve the problem.

tan(7°) = Opposite/Adjacent

=> x/5 = tan(7°)

=> x = 5tan(7°)

=> x = 5×0.124

=> x = 0.62 ft

22: Angle = 20°

Length of board = 3.5 ft

We need to calculate the height of the ramp at the highest point.

Let y be the height of the ramp at the highest point.

We can use the trigonometric ratio, sine (sin), to solve the problem.

sin(20°) = Opposite/Hypotenuse

=> y/3.5 = sin(20°)

=> y = 3.5sin(20°)

=> y = 3.5×0.342

=> y = 1.19 ft

23: Angle = 14.5°

Length of springboard coils = 6 in

We need to calculate the length of the springboard.

Let z be the length of the springboard.

We can use the trigonometric ratio, cosine (cos), to solve the problem.

cos(14.5°) = Adjacent/Hypotenuse

=> 6/z = cos(14.5°)

=> z = 6/cos(14.5°)

=> z = 6/0.966

=> z = 6.21 in

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The root of this given equation is 2, -3.

What is a quadratic equation?

Any equation that can be written in standard form as where x is an unknown value, a, b, and c are known quantities, and a 0 is a quadratic equation.

Here, we have

Given:  x³ - x² - 8x + 12 = 0

We have to solve this equation and find the roots.

x³ - x² - 8x + 12 = 0

= (x-2)(x²+ x - 6) = 0

= (x-2)(x² + 3x - 2x - 6)

= (x-2)(x(x+3)-2(x+3))

= (x-2)(x+3)(x-2) = 0

x = 2, -3

Hence, the root of this given equation is 2, -3.

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After solving  [tex]x^3 - x^2 - 8x + 12 = 0[/tex] we get x = 1,  x = √8, and x = -√8.

The equation[tex]x^3 - x^2 - 8x + 12 = 0.[/tex]

Identify the given equation.

The given equation is [tex]x^3 - x^2 - 8x + 12 = 0.[/tex]

Look for possible factorization.

In this case, we can factor the equation by grouping:

[tex]x^3 - x^2 - 8x + 12 = (x^3 - x^2) + (-8x + 12)[/tex]

Factor out the common terms in each group.

[tex]x^3 - x^2 - 8x + 12 = x^2(x - 1) - 8(x - 1)[/tex]

Factor out the common binomial.

[tex]x^3 - x^2 - 8x + 12 = (x^2 - 8)(x - 1)[/tex]

Solve the factors for x.

Set each factor to zero and solve for x:

[tex]x^2 - 8 = 0[/tex] and x - 1 = 0

[tex]x^2 = 8[/tex], x = 1

x = ±√8

The solutions to the equation [tex]x^3 - x^2 - 8x + 12 = 0[/tex] are x = 1, x = √8, and x = -√8.

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