A cylinder is sliced at an angle, leaving the shape shown at right. The shorter height
is 12 cm while the longer height is 18 cm. The radius of the base is 4 cm.
What is the volume of this sliced cylinder?

Answer:

Step-by-step explanation:

False.

The correct identity is:

cos^2(x/3) = 1/2[1+cos(2x/3)]

To see why, use the double angle formula for cosine:

cos(2x/3) = 2cos^2(x/3) - 1

Substitute this into the original equation:

cos(x/3)cos(x/3) = 1/2[1+2cos^2(x/3)-1]

Simplify:

cos^2(x/3) = 1/2[1+cos(2x/3)]

Answer:

Statement is true!

Step-by-step explanation:

Required to Prove:

[tex]\Large \textsf{$\cos \left(\frac{x}{3}\right)\cos \left(\frac{x}{3} \right)=\frac{1}{2} \left[1+\cos(\frac{2x}{3})\right]$}[/tex]

This is a special property, used in integral calculus, that can be derived and hence proved, from the double angle formula of cosine.

[tex]\large \textsf{Given that cos(A+B) = cosA\,cosB $-$ sinA\,sinB,}\\ \\\large \textsf{Hence cos(A+A) = cosA\,cosA $-$ sinA\,sinA}\\ \\\large \textsf{$\therefore$ cos2A = cos$^2$A $-$ sin$^2$A}\\ \large \textsf{$\rm \phantom{\therefore cos^2A}=$ 1 $-$ 2sin$^2$A}\\ \large \textsf{$\rm \phantom{\therefore cos^2A}=$ 2cos$^2$A $-$ 1 (using Pythagorean Identity $\Rightarrow cos^2A+sin^2A = 1$)}[/tex]

This property, can be quoted in exams and only has to be derived, not proved. Now using the Cos2A property, we can manipulate the formula:

[tex]\large \textsf{$\cos2\rm A = \cos^2A - \sin^2A$}\\ \\ \large \textsf{$\rm \phantom{\cos 2A}=2\cos^2A-1$}\\ \\ \large \textsf{$\rm \therefore \cos2A+1 = 2\cos^2A$}\\ \\ \large \textsf{$\rm \cos^2A=\frac{\cos2A+1}{2}$}\\ \\ \large \textsf{$\rm \phantom{\cos^2A}=\frac{1}{2}(\cos2A+1)$}\\ \\ \large \textsf{$\rm \phantom{\cos^2A}=\frac{1}{2}(1+\cos2A)$}[/tex]

And since:

[tex]\large \textsf{$\cos \left(\frac{x}{3}\right)\cos\left(\frac{x}{3}\right)=\cos^2\left(\frac{x}{3}\right)$}[/tex]

Therefore, inputting the value of A = [tex]\Large \textsf{$\frac{x}{3}$}[/tex] into the formula we derived above, hence:

[tex]\Large \boxed{\boxed{\textsf{$\cos \left(\frac{x}{3}\right)\cos \left(\frac{x}{3} \right)=\frac{1}{2} \left[1+\cos(\frac{2x}{3})\right]$}}} \Large \textsf{ , as required}[/tex]

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the answer is the fourth option

a < -4 1/2

The amount spent on advertising that will yield a monthly profit of $9,000 is $8.57.

Profit is the amount of money or financial gain that a business or an individual makes after deducting all the expenses and costs associated with producing or providing a product or service.

According to the model, the profit P (in dollars) is dependent on the sum x (in dollars) that the business invests in advertising:

P = 130x² - 550

We want to find the amount spent on advertising that will yield a monthly profit of $9,000. In other words, we want to solve for x when P = 9000:

130x² - 550 = 9000

Adding 550 to both sides, we get:

130x² = 9550

Dividing both sides by 130, we get:

x² = 73.46

x = ±8.57

Since we are dealing with a real-world scenario, the amount spent on advertising must be a positive value. Therefore, we take the positive root:

x = 8.57

Therefore, the amount spent on advertising that will yield a monthly profit of $9,000 is $8.57.

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The probability that exactly 19 men and 7 women are chosen is 0.053107%.

Probability:

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.

Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.

[tex]C_n_,_x[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_n_,_x=\frac{n!}{x!(n-x)!}[/tex]

Desired outcomes:

19 men, from a set of 21

7 women, from a set of 11

[tex]D= C_2_1_,_1_9[/tex]  × [tex]C_1_1_,_7[/tex] [tex]=\frac{21!}{19!2!}[/tex] ×  [tex]\frac{11!}{7!4!}[/tex][tex]=69,300[/tex]

Total outcomes:

26 people from a set of 21 + 11 = 32.

[tex]T=C_3_2_,_2_6=\frac{32!}{26!6!}[/tex][tex]=13,049,164,800[/tex]

The probability is :

P = [tex]\frac{D}{T}= \frac{69,300}{13,049,164,800} = 5.3107[/tex]

0.053107% probability that exactly 19 men and 7 women are chosen.

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Answer:

The image you provided appears to be a rectangular prism. To find the surface area of a rectangular prism, we need to add up the areas of all of its faces.

The rectangular prism has dimensions of 4 cm x 6 cm x 8 cm.

Each face of the rectangular prism is a rectangle, so the area of each face can be found by multiplying the length by the width.

The surface area of the rectangular prism is:

2(4 cm x 6 cm) + 2(4 cm x 8 cm) + 2(6 cm x 8 cm)

= 48 cm^2 + 64 cm^2 + 96 cm^2

= 208 cm^2

Therefore, the surface area of the rectangular prism is 208 square centimeters.

have a good day and stay safe

The volume of the of box is derived to be 12 cubic inches, which makes option B correct.

The volume of the box also known as a cuboid can be calculated using the formula:

V = l x w x h

where:

V is the volume of the cuboid

l is the length of the cuboid

w is the width of the cuboid

h is the height of the cuboid

We shall evaluate for the volume of the box as follows:

Volume of the box = 3 in × 2 in × 2 in

Volume of the box = 12 in²

Therefore, the volume of the of box is derived to be 12 cubic inches.

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The volume of the of box is derived to be 12 cubic inches, which makes option B correct.

The volume of the box also known as a cuboid can be calculated using the formula:

V = l x w x h

where:

V is the volume of the cuboid

l is the length of the cuboid

w is the width of the cuboid

h is the height of the cuboid

We shall evaluate for the volume of the box as follows:

Volume of the box = 3 in × 2 in × 2 in

Volume of the box = 12 in²

Therefore, the volume of the of box is derived to be 12 cubic inches.

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The Amplitude is 1/2 and period is π/2.

We have the function as

y= 1/2 cos π/4 x

As, The general equation of a Cosine function is

y=A cos (B(x−D))+C

where A is Amplitude , D is the shift.

So, the amplitude is 1/2

Period = 2π / 4= π/2

and, the phase shift is not possible to determine.

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1). There are 5,245,786 different number combinations that could win. The likelihood of taking home the lottery's grand prize is 1 in 5,245,786 or roughly 0.000019%.

2). There are 42,467,328,000 different winning number combinations that a gambler can choose from.

Combinations are the various ways, independent of their sequence, in which a group of things or objects can be chosen.

The formula n! / (r! * (n-r)! can be used to determine the number of potential combinations of r items from a collection of n items, which is symbolised by the symbol C(n,r).

1. Six numbers are chosen at random from a pool of 42 numbers in a 6/42 lottery. The formula for combinations can be used to determine how many winning number combinations a gambler has the option of choosing:

C(42, 6) = 42! / (6! * (42-6)!)

= 5,245,786

2. The number of winning number combinations that a bettor may choose to select can be determined using the formula for permutations with repetition if the 6/42 Lottery permits repeat of numbers:

[tex]42^6[/tex] = 42 * 42 * 42 * 42 * 42 * 42 = 42,467,328,000

There are therefore 42,467,328,000 different ways to pick winning numbers. The odds of taking home the lottery's grand prize are 1 in 42,467,328,000, or roughly 0.000000002%.

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Answer:

4. n + 24

5. b - 11

6. d/5

The equations solved for the variables T₁ and P₁ are:

We start with the equation:

(P₁*V₁)/T₁ = (P₂*V₂)/T₂

And we want to solve this for T₁, we can multiply both sides by T₁ and divide both sides by the expression in the right side.

(P₁*V₁) = T₁*[ (P₂*V₂)/T₂]

(P₁*V₁)*[T₂/(P₂*V₂)] = T₁

That is the equation solved for T₁.

34: Now we have the same equation but we want to solve it for P₁, to do so, just multiply both sides by T₁/V₁

We will get:

(T₁/V₁)*(P₁*V₁)/T₁= (T₁/V₁)*(P₂*V₂)/T₂

P₁ = (T₁/V₁)*(P₂*V₂)/T₂

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The real distance between City X and City Y is 17 miles.

We know that the scale of the drawing is:

1 inch = 17 miles.

Now, if you look at the diagram for cities X and Y, you can see that the distance between City X and City Y is exactly 1 inch.

And we know that 1 inch is equivalent to 17 miles, then we can conclude that the actual distance between the two cities is exactly 17 miles.

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Answer:

Last number line

Step-by-step explanation:

Solving |d| > 3,

d^2 > 9

d = +-3

Using the graph y=x^2,

d < -3, d > 3

Hence, it's the last number line i.e. the one with blank dots.

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The sets that is part of Set A are:

To be able to get the set A, a set need to have the same elements as {5, 7, 11, 13, 17, 19}.  So the set that has six number is one that can be the set A.

Hence:

The set {5, 7} exclusively comprises elements present in the initial set.

Any set contains the empty set within its subsets.

The set {17} consists of a single element that is present in the initial set.

The set {5, 7, 11, 13, 17, 19} is a subset of the original set as it encompasses all of its elements.

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See text below

Select all of the following sets that could be the set A if A ⊆⊆ {5, 7, 11, 13, 17, 19}.

{5, 7}

{}

{7, 8, 9}

{17}

{5, 7, 11, 13, 17, 19}

{4, 5, 6}

The answer choice that matches the calculated confidence interval is 62.9 to 83.1

In statistics, the mean is a measure of central tendency that represents the average value of a set of numerical data. It is also known as the arithmetic mean, and it is calculated by adding up all the values in the dataset and dividing the sum by the number of values.

The mean is a useful measure of central tendency because it is easy to calculate, and it provides a single value that represents the center of the dataset. It is affected by outliers, which are extreme values that are far from the other values in the dataset, so it may not accurately represent the typical value of the data if there are outliers present.

To construct a 95.44% confidence interval for the mean score, u, of all students taking the test, we can use the formula:

CI = x ± t(alpha/2, n-1) * (s / √(n))

where CI is the confidence interval, x is the sample mean (73), t(alpha/2, n-1) is the t-value for the given alpha level (0.0278) and degrees of freedom (n-1=4) from the t-distribution table, s is the sample standard deviation, and n is the sample size.

The sample standard deviation is not given, so we will assume that it is the same as the population standard deviation, which is 15. Thus, s = 15.

Using the t-distribution table with 4 degrees of freedom and an alpha level of 0.0278, we find that the t-value is approximately 3.747.

Plugging in the values into the formula, we get:

CI = 73 ± 3.747 * (15 / √(5))

Simplifying, we get:

CI = 73 ± 16.27

Therefore, the 95.44% confidence interval for the mean score, u, of all students taking the test is:

CI = (73 - 16.27, 73 + 16.27)

CI = (56.73, 89.27)

Rounding to one decimal place, we get:

CI = (56.7, 89.3)

Therefore, the answer choice that matches the calculated confidence interval is:

62.9 to 83.1

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The complete question is:

The mean score, x on an aptitude test for a random sample of 5 students was 73, assuming that 0 = 15, construct a 95.44% confidence interval for the mean score, u of all students taking the test. answer choices, 43 to 103, 59.6 to 86.4, 62.9 to 83.1, and 67.0 to 79.0.

The limit of the trigonometric function f(x) = (1 - cos x) / x is equal to 0.

In this problem we need to determine the limit of a trigonometric function for x → 0. This can be done by simplifying the expression by trigonometric formulas. First, write the trigonometric function:

f(x) = (1 - cos x) / x

Second, modify the expression by means of algebra properties and trigonometric formulas:

f(x) = (2 / x) · (1 - cos x) / 2

f(x) = sin² (x / 2) / (x / 2)

f(x) = sin (x / 2) · [sin (x / 2) / (x / 2)]

For u = x / 2:

f(u) = sin u · (sin u / u)

Third, use limits to evaluate the trigonometric function:

f(u) = 0 · 1

f(u) = 0

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The work required to pump the oil out through a hole to the top of the tank is approximately 6,476,160π/3 foot-pounds.

We can find the work required to pump the oil out of the tank by using the formula:

W = ∫[V1, V2]ρgh dV

Where

First, we need to find the density of the oil. We are told that the oil weighs 50 pounds per cubic foot, so:

ρ = 50 lb/ft^3

Next, we need to find the height of the oil column being pumped. The tank is half full, so the height of the oil column is:

h = r - (r/2) = r/2 = 8/2 = 4 feet

Now, we need to find the volume of oil being pumped. Since the tank is half full, the volume of oil is:

V = (1/2)(4/3)πr^3 = (1/2)(4/3)π(8)^3 = 1,024π/3 cubic feet

Finally, we can integrate the work formula to find the total work required:

W = ∫[V1, V2]ρgh dV

W = ∫[0, 1,024π/3] (50 lb/ft^3)(32.2 ft/s^2)(4 ft) dV

W = (6,476,160π/3) ft-lb

Therefore, the work required to pump the oil out through a hole to the top of the tank is approximately 6,476,160π/3 foot-pounds.

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The missing angle C is 103, length of a is 9 m, and length b is 16 m.

option B.

The missing angle C is calculated as follows;

A + B + C = 180 (sum of angles in a triangle)

26 + 51 + C = 180

C = 180 - 77

C = 103

The value of length a and length b is calculated as follows;

sin 26/a = sin 103/20

0.438/a = 0.0487

a = 0.438/0.0487

a = 9 m

b/sin51 = 20/sin103

b = 16 m

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true

Step-by-step explanation:

all square have the same features and properties like

all side are equal

The answers are explained in the solution.

Considering the triangle, ABC,

BC = √AC²-AB² [Pythagoras theorem]

BC = √126.4²-126²

BC = 10 ⇒ Height of the ramp at B,

Slope = tanBC/AB = 10/126

The slope is less than 1/12, hence, it will get ADA approval,

Let θ be angle of elevation,

θ = tan⁻¹(10/126)

= 4.5° < 5°

Hence the town met the given requirement.

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The probability that the number of times Luke will hit the inner ring of the target out of the 5 attempts is less than the mean of X is 0.951.

A discrete random variable with a countable number of potential values is said to have a discrete probability distribution. Each possible value of the random variable is given a probability by the probability distribution, and the sum of these probabilities is 1. The number of heads you get while flipping a coin or the number of cars that pass through a specific crossroads in a given hour are both examples of discrete random variables.

The mean that Luke will hit the inner ring is given as:

E(X) = np

Now, n = 5 and p = 0.90.

So, E(X) = 5 x 0.90 = 4.5

Now, the probability of less than 4.5 is given as:

P(X < 4.5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

P(X < 4.5) = 0.0005 + 0.0144 + 0.1361 + 0.4095 + 0.3915

P(X < 4.5) = 0.951

Hence, the probability that the number of times Luke will hit the inner ring of the target out of the 5 attempts is less than the mean of X is 0.951.

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The answer for the above expression is 16/9.

An expression is a combination of numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, division, exponentiation, and root extraction, that represents a mathematical quantity or a mathematical statement. An expression can be as simple as a single number or variable, or it can be a complex combination of several numbers, variables, and operations.

According to the given information:

The expression "[tex](\frac{3}{4} )^{2}[/tex]" represents the fraction "3/4" raised to the power of "-2". In mathematical notation, this is written as "[tex](\frac{3}{4} )^{-2}[/tex]".

To calculate this value, we can use the rule that a negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent. Therefore, "[tex](\frac{3}{4} )^{-2}[/tex]" is equal to the reciprocal of "3/4" raised to the power of "2", or "[tex]\frac{1}{(\frac{3}{4} )^{2}}[/tex]".

Evaluating this expression, we get:

[tex](\frac{3}{4} )^{-2}[/tex]= [tex]\frac{1}{(\frac{3}{4} )^{2}}[/tex] = [tex]\frac{1}{(\frac{9}{6})^{2} }[/tex]= 16/9

So, "[tex](\frac{3}{4} )^{-2}[/tex]" is equal to 16/9.

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Answer:

Two lines that are in general position divide a plane into four regions or parts.

When two lines intersect at a point, they divide the plane into four distinct regions, called quadrants. However, if the two lines are parallel, they do not intersect, and the plane is divided into only two regions, called half-planes.

In general position, two lines in a plane have different slopes and different y-intercepts, which means that they are neither parallel nor coincident. Therefore, the two lines must intersect at a point, dividing the plane into four regions.

Hope this helps!

a z-test for the proportional difference is the proper hypothesis test.

A hypothesis test can be used to find whether the deviation in proportions impacts the medication's effectivity. We may compare the secondary hypothesis—that the proportions are different—to the null hypothesis.

which states that the dimension of patients who show cut down blood pressure is the same for the two doses of the drug (50 mg and 100 mg).

Popular test statistics like the z-test can be applied to this hypothesis test and other statistical analyses.

[tex]z = (p1 - p2) / SE[/tex]

where p1 and p2, for the 50 mg and 100 mg doses, respectively, are the sample proportions of patients who show fallen blood pressure, and SE is the standard error of the difference between the proportions.

the samples are assumed to be independent or dependent, impacts the SE formula. The samples in this instance are presumed to be independent because they came from various patients. Consequently, the equation for SE is:

[tex]SE = \sqrt(p1 \times (1 - p1)/n1 + p2 *\times(1 - p2)/n2)[/tex]

here, the sample sizes for two doses is n1 and n2.

We can compute the z-test statistic based on the sample sizes and proportions and compare the result to a critical value or p-value to decide whether to accept or reject the null hypothesis.

Therefore, a z-test for the proportional difference is the proper hypothesis test.

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Area of the two right triangles:

A = 1/2(b)(h)

A = 1/2(10)(24)

A = 120

Total area = 240

Area of the left-most rectangle:

A = (b)(h)

A = (24)(25)

A = 600

Area of the right-most rectangle:

A = (b)(h)

A = (25)(26)

A = 650

Area of the base rectangle:

A = (b)(h)

A = (10)(25)

A = 250

Surface Area:

240 + 600 + 650 + 250

1740

Answer: 1740 cm^2

Hope this helps!

[tex]\sf SA=\boxed{\sf 1740cm^{2} }.[/tex]

Check attached 1 to see what parts we're referring to in this step.

This part forms a right triangle. Therefore, the formula to use to find it's area is the following:

[tex]\sf A=\dfrac{bh}{2}[/tex]; where "b" is the length of the base of the triangle, and "h" is its height.

Since we have another section identical to this part at the back, we multiply this area by 2 and calculate:

[tex]\sf A=2\dfrac{bh}{2}=(10cm)(24cm)=240cm^{2}[/tex]

Check image 2 to see this part highlighted.

This shape forms a rectangle. Therefore, use the following formula to calculate:

[tex]\sf A=lw[/tex]; where "l" is length, and "w" is width.

[tex]\sf A=(25cm)(10cm)=250cm^{2}[/tex]

Check image 3.

This shape also forms a rectangle, therefore its area is calculated like this:

[tex]\sf A=(24cm)(25cm)=600cm^{2}[/tex]

Check image 4.

This shape also forms a rectangle, therefore its area is calculated like this:

[tex]\sf A=(26cm)(25cm)=650cm^{2}[/tex]

The total surface area of this prism is given by the addition of all of its individual areas that we just calculated.

[tex]\sf SA=240cm^{2} +250cm^{2} +600cm^{2} +650cm^{2} =\boxed{\sf 1740cm^{2} }.[/tex]

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Answer:

I hope this helps...

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Answer:

Step-by-step explanation:

1 out of 100 is 1 percent. This is because percentage is always out of 100 so you don’t have to change anything. That means 1 is always 1 percent of 100. It is also 1 part of 100.

The probability of selecting an R or an E without replacement, and then selecting an S is 5/36

Because the word SEPTEMBER has 9 letters, there are 9 different alternatives for the initial letter.

The probability of selecting a R or an E without replacing is 2+3=5.

The odds of picking a R or an E on the initial draw are 5/9.

After the first letter is drawn, the bag contains eight letters, including one S. If the first letter is not replaced, there are only four letters that fit the requirement.

Given that a R or an E was selected without replacement on the first draw, the probability of selecting a S on the second draw is 4/8.

When we multiply these probability together, we get:

P(R or E, not replacing) * P(S after R or E, not replacing) = (5/9) * (4/8) = 10/72 = 5/36

Hence, the probability of selecting an R or an E without replacement, and then selecting an S is 5/36

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The approximate average number of miles she put on a rental car each day is C. 42.

The average is the quotient of the total value divided by the number of data items.

The average is also described as the mean data value.

The mean is one of the basic centers of measurement.

The total number of miles driven by Jackie's car for the five days = 209.1 miles

The number of days of driving undertaken by Jackie = 5 days

The average miles per day = 41.81 (209.1 ÷ 5)

41.81 miles per day is approximately = 42 miles per day

Thus, we can confidently conclude that Jackie's car drove 42 miles dai on the average.

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The value of x, based on the Alternate Interior Angles Theorem, is calculated as: x = 5.

The Alternate Interior Angles Theorem states that if two parallel lines are intersected by a transversal, then the pairs of alternate interior angles formed are congruent. In other words, if two lines are parallel and a third line intersects them, then the angles that are inside (or "interior" to) the two parallel lines and on opposite sides of the transversal are congruent.

Therefore, we have:

3x - 8 = x + 2 [based on the Alternate Interior Angles Theorem]

Combine like terms:

3x - x = 8 + 2

2x = 10

2x/2 = 10/2

x = 5

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