Rewrite the expression in terms of sine and cosine and utilize the Fundamental Pythagorean Identity: sin²(x)+cos²(x)=1
Verify the identity using the Pythagorean Identity:
[tex]\frac{1-cos(x)}{sin(x)}=\frac{sin(x)}{1+cos(x)}[/tex]

The arch has a width of 6 - (-6) = 12 feet at ground level.

To find the width of the arch at ground level, we need to find the x-values where y = 0, since the width of the arch is the distance between these two x-values.

So, we set y = 0 and solve for x:

0 = − 1/3(x + 6)(x − 6)

Either x + 6 = 0 or x - 6 = 0

If x + 6 = 0, then x = -6.

If x - 6 = 0, then x = 6.

To find the width of the arch at ground level, we need to find the x-values where y=0.

y = − 1/3(x + 6)(x − 6)

Setting y=0:

0 = − 1/3(x + 6)(x − 6)

We can solve for x using the zero product property:

0 = x + 6 or 0 = x - 6

x = -6 or x = 6

The arch intersects the ground at x = -6 and x = 6.

Therefore, the arch has a width of 6 - (-6) = 12 feet at ground level.

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Now, we can construct the confidence interval: [tex]0.4 ± 1.96 * 0.0157 ≈ 0.4 ± 0.0308. [/tex]The resulting interval is 0.3692 to 0.4308, which is closest to option e. 0.374 to 0.426

5. To calculate the sample proportion, divide the number of consumers who shop online at least once a week (121) by the total number of consumers surveyed (484). So, the sample proportion is[tex] 121/484 = 0.25.[/tex] The correct answer is c. 0.25.

6. A sample proportion is a statistic because it is a measure derived from a sample of the population. The correct answer is a. statistic.

7. To construct a 95% confidence interval for the proportion of OSU undergraduates with full-time jobs, we first find the sample proportion:[tex] 368/920 = 0.4[/tex].

Next, we find the standard error (SE) for the sample proportion: SE = sqrt(0.4 * (1 - 0.4) / 920) ≈ 0.0157. For a 95% confidence interval, the critical value (z-score) is approximately 1.96.

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Answer: 10

Step-by-step explanation:

Adjacent angles in a parallelogram are supplementary (add up to 180), so we know that the unknown angle equals 100°, since the adjacent angle equals 80°.

So, we can set up an equation:

[tex]11x-10=100\\11x=110\\x=10[/tex]

Answer:

The appropriate domain for the function is [0, ∞).

Step-by-step explanation:

Given function: f(x) = 120 (0.86)x

To find the function values, we can substitute the given value of x into the function:

f(0) = 120 (0.86)^0 = 120 (1) = 120

Interpretation: When the medication is first taken, there are 120 milligrams of medicine in the person's bloodstream.

f(2) = 120 (0.86)^2 ≈ 84.24

Interpretation: After 2 hours, there are approximately 84.24 milligrams of medicine remaining in the person's bloodstream.

f(6) = 120 (0.86)^6 ≈ 34.17

Interpretation: After 6 hours, there are approximately 34.17 milligrams of medicine remaining in the person's bloodstream.

The domain of the function is all non-negative real numbers, since the amount of medicine in the bloodstream can't be negative and time is measured in non-negative hours. So the appropriate domain for the function is [0, ∞).

In Mathematics, the relationship between two values which are not equal is expressed by inequalities. The equation of inequality for the given graph is y ≥ -1/2x + 2.  

In mathematics, an inequality is a relationship in which two numbers or other mathematical expressions compare unequal. Most commonly used to compare two numbers on the number line based on size. There are different notations used to represent different types of inequalities.

In both the cases a is not equal to b and such relationships are known as strict inequalities. This means that a is either strictly less than or greater than b. Equivalents are excluded whereas in inexact inequalities.

For the given graph:

take run = 1 and rise = -1/2

Thus, the slope will be,

Slope = (⁻¹/₂)/1

Slope = -1/2

Since, the y intercept is 2. Then the equation is y = -1/2x + 2

Take point (1,2) on the graph and apply in the equation:

y = -1/2x + 2

2 = -1/2 (1) + 2

2 = -1/2 + 2

2 = 3/2

As, 2 > 3/2. We can replace the = sign into ≥.

So, the eq inequality equation is y ≥ -1/2x + 2.

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The answer is 6√5

In 3√5 and 3√5, √5 is multiplied with 3 in both terms,

∴ 3 is the common factor

∴ 3√5+3√5= 3(√5+√5)

                   = 3×2√5

                   = 6√5

Hence the answer is 6√5

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The estimated coefficient for x1 in the multiple linear regression equation is 2.3529.

To estimate the coefficient for x1 in the multiple linear regression equation, we can use the following formula:

where n is the number of data points, x and y are the respective variables, and Σ denotes the sum of the values.

Using the provided data, we can calculate the coefficient for x1 as follows:

= 2.3529 (rounded to 4 decimal places)

In this problem, we are given data on the amount of wear of a type of bearing, as well as the oil viscosity and load that the bearing was subject to. We want to estimate the coefficient for oil viscosity (x1) in the multiple linear regression equation for this data.

To do this, we can use the formula for calculating the coefficient for x1 in the multiple linear regression equation. This formula takes into account the number of data points, the sums of the values for x1 and y, and the sum of the products of x1 and y, as well as the sum of the squares of x1.

Using the provided data and this formula, we find that the estimated coefficient for x1 is 2.3529. This means that, for this data, the amount of wear of the bearing is estimated to increase by 2.3529 units for every one unit increase in oil viscosity, while holding the load constant.

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After answering the presented question, we can conclude that equation Therefore, the price of each snack is $0.62.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x Plus 3" equals the value "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "x2 + 2x - 3 = 0." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

price of each snack x.

15 × $1.75 = $26.25

cost of snacks for the other fifteen students

$26.25 + 15x = $35.55

15x = $35.55 - $26.25

15x = $9.30

x = $0.62

Therefore, the price of each snack is $0.62.

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The probability that exactly one of three randomly selected high school students graduate, given an 85% graduation rate, is 0.057. This calculation was based on the combination formula and the binomial probability distribution.

This problem involves calculating the probability of a specific outcome (exactly one of the three students graduating) in a binomial distribution. We know that the probability of any one student graduating is 0.85, so the probability of one student not graduating is 0.15. Using the binomial probability formula, we can calculate the probability of exactly one student graduating out of three as:

P(X = 1) = (3 choose 1) * (0.85)^1 * (0.15)^2 = 3 * 0.85 * 0.0225 = 0.057

Therefore, the answer is 0.057, or approximately 5.7%. This means that if we randomly select three high school students from different schools, there is a 5.7% chance that exactly one of them will graduate.

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The lower class boundary for the first class is 122.5.

Data was collected for 300 fish from the North Atlantic.

The length of the fish (in mm) is summarized in the GFDT below.

| Length (mm) | Frequency | 123 - 145 | 34 | 146 - 168 | 89 | 169 - 191 | 107 | 192 - 214 | 44 |

The class boundaries for a frequency table are calculated by taking the lower limit of one class and subtracting 0.5

from it to find the lower class boundary, and taking the upper limit of the previous class and adding 0.5 to it to find the

upper class boundary.

For the first class, the lower limit is 123 and the upper limit is 145. So the lower class boundary is:123 - 0.5 = 122.5

Therefore, the lower class boundary for the first class is 122.5.

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If we know that f(-4) = 0, we can say that:

x + 4 is a factor of the function f(x).

This is because when we substitute x = -4 into the function, we get:

f(-4) = (-4)^4 + 2(-4)^3 - 9(-4)^2 - 2(-4) + 6 = 0

This means that (x + 4) is one of the factors of the function.

Now, we can use polynomial division or synthetic division to find the other factors. If we divide the function f(x) by (x + 4), we obtain:

           x^3 - 2x^2 - 17x + 3

We can use the Factor theorem on this cubic function f(x) = x^3 - 2x^2 - 17x + 3.

If we try to substitute a = 1 we find that f(1) = -15 which is not zero. If we try to substitute a = -1, we find that f(-1) = 23 which is not zero.

Now we can solve as follows:

Let x = t - 2/3. After substitution we get a cubic equation.

t^3 -25/3 t - 85/27 = 0

We can find one root of this cubic by rational root theorem which is t = 5 or t = -17/3.

Now, we can use synthetic division again to divide f(x) by the binomial (x - 5). This gives us:

        (x - 5)(x^2 + 3x - 1)

Therefore, the complete factorization of f(x) is:

f(x) = (x + 4)(x - 5)(x^2 + 3x - 1)

Therefore, the zeros of the function are:

x = -4, 5, (-3 ± √13)/2

Thus, these are the zeros of the polynomial.

The total health care expenditures in year x would be equal to the product of the percentage of income spent on health care (P(x)) and the income (I(x)) in year x. Therefore, the function H can be expressed as:

H(x) = P(x) * I(x)

This function takes into account both the amount spent on health care as well as the income level, providing a comprehensive representation of health care expenditures.

Step-by-step explanation:

that depends on the result dimension of P(x).

if P(x) delivers the percentage in the form xx.xx%, then

H(x) = I(x) × P(x) / 100

if P(x) delivers the percentage in the form 0.xxxx, then

H(x) = I(x) × P(x)

very basically, the absolute amount of a percentage of a whole is the product between both (and divided by 100).

e.g.

let's say, the whole amount is $64,000.

and we need to know 10% of that.

10% of 64,000 is then

64,000 × 10 / 100 = 6,400

or

64,000 × 0.10 = 6,400

can you see the connection, and why both versions do the same ?

Best investment amounts are $2,000 in high-risk account, $1,000 in bank certificates of deposit, and $7,000 in municipal bonds and remaining $4,000 placed in municipal bonds to optimize return while lowering risk.

We must adhere to two criteria in order to find the best investment amounts: reducing risk and satisfying tax obligations.

Let's think about the high-risk account first. We will only put $2,000 in this account because that is the most we are ready to risk because we want to keep the risk to a minimum.

Let's go on to the tax regulations. We must invest at least $3,000 in municipal bonds to reach the three times minimum investment requirement in municipal bonds compared to bank CDs. We will therefore put $1,000 into bank CDs and $3,000 into municipal bonds.

We currently have $2,000 set aside for the high-risk account, $1,000 for bank CDs, and $3,000 set aside for municipal bonds. We are now left with $4,000 to invest.

This $4,000 can be invested in a way that strikes a compromise between the need to reduce risk and the desire for great returns. We have the choice of investing the final $4,000 in either municipal bonds or CDs from a nearby bank, depending on our investment alternatives.

Our total investment would be $5,000 in bank CDs, $3,000 in municipal bonds, and $2,000 in the high-risk account if we choose to invest the additional $4,000 in the local bank CDs. The expected return from this is:

($5,000 x 8%) + ($3,000 x 7%) + ($2,000 x 12%) = $400 + $210 + $240 = $850.

If we choose to invest the $4,000 in municipal bonds, our overall investment will consist of $2,000 in the high-risk account, $1,000 in bank CDs, and $7,000 in municipal bonds. The expected return from this is:

($1,000 x 8%) + ($7,000 x 7%) + ($2,000 x 12%) = $80 + $490 + $240 = $810.

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Answer:

The gain is 139.

Step-by-step explanation:

34.95x25= 873.75

29.39x25= 734.75

873.75- 734.75= 139

Given statement "f the null hypothesis is correct, the probability of getting a sample mean greater than $189,500 is .0668" is true. Because, It's the probability of observing a sample mean at least as extreme as the one obtained in the study, assuming the null hypothesis is correct.

The null hypothesis is a general statement that there is no significant effect or difference between the considered groups or variables.

In this case, the null hypothesis would be that the true mean valuation of homes is equal to or less than $189,500.
The probability of 0.0668, mentioned in the question, represents the likelihood of getting a sample mean greater than $189,500 if the null hypothesis is true.

In other words, it's the probability of observing a sample mean at least as extreme as the one obtained in the study, assuming the null hypothesis is correct.

This probability (0.0668) is also known as the p-value.

The p-value helps to determine the significance of the results.

If the p-value is less than a predetermined significance level (usually 0.05), the null hypothesis is rejected in favor of the alternative hypothesis.
In this case, the p-value of 0.0668 is higher than the common significance level of 0.05.

We cannot reject the null hypothesis, meaning the probability of getting a sample mean greater than $189,500 is indeed 0.0668 if the null hypothesis is correct

Hence, the statement is true.
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If a teacher cuts out 15 congruent circles from a single sheet of paper that measures 6 inches by 10 inches, then (60 - 1.5 π) square inches paper is wasted.

The area of each circle is given by the formula [tex]A = \pi r^2[/tex], where r is the radius of the circle. Since all 15 circles are congruent, they all have the same radius.

Let's first find the radius of each circle. Since the circles are cut from a 6 by 10 inch sheet of paper, we know that the diameter of each circle cannot be greater than 6 inches or 10 inches. Let's take the smaller dimension (6 inches) and divide it by 15 to find the maximum possible diameter of each circle:

6 inches ÷ 15 = 0.4 inches

Therefore, the maximum possible radius of each circle is 0.2 inches.

Now we can calculate the total area of all 15 circles:

Total area of circles = [tex]15 * \pi * (0.2)^2[/tex]

[tex]= 1.5 \pi[/tex] square inches

The area of the original sheet of paper is 6 inches * 10 inches = 60 square inches. Therefore, the amount of paper wasted is:

Amount of paper wasted = Area of sheet - Total area of circles

= 60 square inches - 1.5 π square inches

= 60 square inches - 4.71 square inches (using π ≈ 3.14)

= 55.29 square inches (rounded to two decimal places)

Therefore, the answer is (60 - 1.5π) square inches.

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Therefore, the expression that can be used to find the probability that a boy will be chosen, and the arrow will land on "Free assignment" is 2/13.

Which expression can be used to find the probability that this week a boy will be?

chosen and the arrow will land on "Free assignment"?

The probability of a boy being chosen is the ratio of the number of boys to the total number of students, which is:

[tex]P(boy) = 12 / (12 + 14) = 12/26 = 6/13[/tex]

The probability of the arrow landing on "Free assignment" is 1/3, since there are 3 equal sectors on the spinner.

To find the probability that both events occur (i.e., a boy is chosen and the arrow lands on "Free assignment"), we multiply the probabilities:

[tex]P(boy and "Free assignment") = P(boy) x P("Free assignment")[/tex]

[tex]P(boy and "Free assignment") = (6/13) x (1/3) = 2/13[/tex]

Therefore, the expression that can be used to find the probability that a boy will be chosen, and the arrow will land on "Free assignment" is 2/13.

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Therefore, segment BD is congruent to segment CA, as required.

A trapezoid is a quadrilateral with at least one pair of parallel sides. In this case, we are given that AB is parallel to CD. The base angles theorem states that the base angles of an isosceles trapezoid are congruent. Since AB and CD are congruent (as given by the reflective property of congruence), the opposite angles formed by the intersecting lines BA and CD (i.e., ∠BAD and ∠CDA) are congruent as well. This establishes that ABCD is an isosceles trapezoid. An isosceles trapezoid has two pairs of congruent sides: AB = CD and AC = BD. Since BA and CO are congruent (as given), we can conclude that AC and BD are congruent as well (by the isosceles trapezoid definition). Finally, the symmetric property of congruence allows us to state that BD is congruent to CA.

Here,

given that ABCD is a trapezoid and segments BA and CO are congruent:

Statements | Reasons

ABCD is a trapezoid | Given

AB || CD | Definition of a trapezoid

∠BAD ≅ ∠CDA | Base angles theorem

AB ≅ CD | Reflective property of congruence

BA ≅ CO | Given

AC ≅ BD | Isosceles trapezoid definition

BD ≅ CA | Symmetric property of congruence

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Answer:find the root of 6

Step-by-step explanation:

Answer:

you would put a solid circle on the 3 and go to the left, for example:

Step-by-step explanation:

You would want the variable to be on the left, so you would need to see that the 3 is bigger or equal to a.

Answer:

C)  56

Step-by-step explanation:

2(AE) = BD

AC = BD

2(2x + 6) = 6x - 10

4x + 12 = 6x - 10

-2x = -22

x = -22/-2 = 11

AC = 6(11) - 10 = 56

The length a of the right angle triangle is 13.74 cm.

A right triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

Therefore, let's find the opposite side of the right triangle a cm. The side a cm can be found using Pythagoras's theorem or trigonometric ratios.

Let's use Pythagoras's theorem,

a² + b² = c²

where

15² - 6² = a²

225 - 36 = a²

1809 = a²

square root both sides of the equation

a = √189

a = 13.7477270849

a = 13.74 cm

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Answer:

(a)

[tex]W^2+4W-165=0[/tex]

(b)

Width: 11 feet

Length: 15 feet

Step-by-step explanation:

Given:

Width (W) = W ft

Length (L) = (4 + W) ft

Area (A) = 165 ft^2

(a)

The area of a rectangle is equal to the product of its length and width, therefore:

       [tex]A=L\times W[/tex]

      [tex]\text{We replacing}[/tex]

[tex]165=(4+W)\times(W)[/tex]

   [tex]4W+W^2=165[/tex]

[tex]W^2+4W-165=0[/tex]

(b)

We solve the equation by factoring:

        [tex]W^2+4W-165=0[/tex]

       [tex]4W=-11W+15W[/tex]

[tex]W^2-11W+15W-165=0[/tex]

[tex]W(W-11)+15(W-11)=0[/tex]

     [tex](W-11)(W-15)=0[/tex]

    [tex]W-11=0\rightarrow W=11[/tex]

   [tex]W+15=0\rightarrow W=-15[/tex]

The width of the rectangle is equal to 11 feet and the length of the rectangle is equal to 15 feet.

Answer:

C

Step-by-step explanation:

About 16 days because 0.58 x 28 is 16.24

Answer:

The amount of discount is 35% of � x, which is (35/100)� x = 0.35� x. So, the discounted price is � x minus the discount, which is � x - 0.35� x = 0.65� x.

Then, the tax is added to the discounted price, which is 8% of the discounted price, or (8/100)(0.65� x) = 0.052� x.

Therefore, the expression for the total amount that Violet paid at the register is:

0.65� x + 0.052� x = 0.702� x

Hence, it takes 23 and 4 over 7  in² of paper's surface area to wrap six cylindrical mini muffins.

A cylinder is only a three-dimensional shape in geometry. A cylinder has a cylinder-like form and a top and bottom that are shaped like circles. Both the top and bottom are flat and of the same size. I think the simplest way to visualise a cylinder's shape is to think of a soup can.

Given data:

surface area of total 6 number of cylindrical mini muffins:

S = 6*(π*r²*h)

S = 6* (22/7 * 1² * 1.25)

S = 23.55 in²

S = 23 ⁴/₇ in²

Hence, it takes 23 and 4 over 7  in² of paper's surface area to wrap six cylindrical mini muffins.

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Answer:

x = y = 2√7 = 5.292

Step-by-step explanation:

We have a right isosceles triangle, so the length of the hypotenuse is √2 times the length of each leg. Since the hypotenuse is 2√14, each leg measures 2√7.

The length of the sides of the right isosceles triangle is 5.292units.

A right isosceles triangle is a special type of triangle that has two equal-length legs and a right angle between them. Because it has two equal legs, it is also an isosceles triangle. The name "right isosceles" comes from the fact that it has a right angle and two equal legs.

In the given question,

In a right isosceles triangle, the two legs are congruent, and the length of the hypotenuse can be found using the Pythagorean theorem:

a² + b² = c²

Since this is an isosceles triangle, we know that a = b, so we can simplify the equation to:

2a² = c²

Taking the square root of both sides, we get:

a√2 = c

In this case, we are given that the hypotenuse has a length of 2√14, so we can set c equal to that value and solve for a:

a√2 = 2√14

a = (2√14)/√2

Simplifying the expression by canceling the square root of 2 in the denominator, we get:

a = 2√7

Therefore, each leg of the isosceles triangle has a length of 2√7, which agrees with your answer.

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Answer:

1) ABE is NOT SIMILAR to DBC

2) FPG is SIMILAR to FGH (AAA)

3) CAN is SIMILAR to HLV (SAS)

4) QRP is SIMILAR to XYP (SSS)

Step-by-step explanation:

1) ABE is NOT SIMILAR to DBC

Since DC and AE are not parallel

2) FPG is SIMILAR to FGH (AAA)

QPF = 180 - 93

QPF = 87

PQF = 180 - 63 - 87

PQF = 30

PQF = GHF

FGH = 180 - 63 - 30

FGH = 87

FGH = QPF

GFH = PFQ

3) CAN is SIMILAR to HLV (SAS)

63 = [tex]x[/tex] x 42

[tex]x[/tex] = 1.5

51 = [tex]x[/tex] x 34

51 = 1.5 x 34

51 = 51

Hypotenuse of CAN = 1.5 x Hypotenuse of HLV

4) QRP is SIMILAR to XYP (SSS)

10 = [tex]x[/tex] x 8

[tex]x[/tex] = 1.25

(20 + 5) = 1.25 x XY

XY = 20

(24 + 6) = 1.25 x XP

XP = 24

D) Between 0 and 100.


The best system to measure the likelihood of an event occurring would be option D between 0 and 100. This system allows for a greater level of granularity and precision in the measurement of probability, as it provides a wider range of possible values to represent different levels of likelihood. The other options may not provide enough resolution to accurately capture and distinguish between different probabilities.