what is the lowest depth the diver reaches using the function d(t)=3t^2-12t+0

a)0.2407 is the point estimate of the proportion.

b) The 95% confidence interval for the proportion that beat estimates is (0.4858, 0.6746).

c) A sample size of 754 is needed to achieve a desired margin of error of 0.05.

(a) What is the point estimate of the proportion?

The point estimate of the proportion that fell short of estimates is 39/162 = 0.2407. Rounded to four decimal places, this is pshort = 0.2407.

(b) How to determine the margin of error and provide a 95% confidence interval for the proportion that beat estimates?

To determine the margin of error and provide a 95% confidence interval for the proportion that beat estimates, we need to first find the point estimate and standard error of the proportion that beat estimates.

The point estimate of the proportion that beat estimates is 94/162 = 0.5802.

The standard error can be calculated as:

SE = √(p×(1-p)/n)

where p is the point estimate of the proportion that beat estimates and n is the sample size. Substituting the values we get:

SE = √(0.5802×(1-0.5802)/162) = 0.0482

To calculate the margin of error, we use the formula:

ME = zSE

where z is the z-score corresponding to the desired confidence level. For a 95% confidence level, z* = 1.96 (from the standard normal distribution table).

Substituting the values we get:

ME = 1.96×0.0482 = 0.0944

Therefore, the margin of error is 0.0944. To find the 95% confidence interval, we add and subtract the margin of error from the point estimate of the proportion that beat estimates:

CI = 0.5802 ± 0.0944

CI = (0.4858, 0.6746)

Therefore, the 95% confidence interval for the proportion that beat estimates is (0.4858, 0.6746).

(c) How large a sample is needed if the desired margin of error is 0.05?

To find the sample size needed for a desired margin of error of 0.05, we use the formula:

n* = (z*/ME)² × p×(1-p)

where z* is the z-score corresponding to the desired confidence level (we use 1.96 for a 95% confidence level), ME is the desired margin of error (0.05), and p is an estimate of the proportion (we use the point estimate of 0.5802 for the proportion that beat estimates).

Substituting the values we get:

n* = (1.96/0.05)² × 0.5802×(1-0.5802) = 753.16

Rounding up to the next integer, we get n* = 754. Therefore, a sample size of 754 is needed to achieve a desired margin of error of 0.05.

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The total width for the figure is given as follows:

13 ft 2 in.

The total width for the figure is obtained applying the proportions in the context of the problem.

The measures are given as follows:

The sum of the measures is given as follows:

Each feet is composed by 12 inches, hence:

26 inches = 2 feet and 2 inches.

Hence the sum is given as follows:

11 + 2 = 13 feet and 2 inches.

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Answer:

The standard form of the equation of a circle with center at (h, k) and radius r is:

(x - h)^2 + (y - k)^2 = r^2

We are given that the center of the circle is (4, -1), so h = 4 and k = -1. We also know that the circle passes through the point (-4, 1), which means that the distance from the center of the circle to (-4, 1) is the radius of the circle.

The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

So the radius of the circle is:

r = sqrt((-4 - 4)^2 + (1 - (-1))^2) = sqrt(100) = 10

Now we can substitute the values of h, k, and r into the standard form equation of a circle:

(x - 4)^2 + (y + 1)^2 = 10^2

Expanding the equation gives:

x^2 - 8x + 16 + y^2 + 2y + 1 = 100

Simplifying and putting the equation in standard form, we get:

x^2 + y^2 - 8x + 2y - 83 = 0

Therefore, the equation in standard form of the circle with center at (4, −1) and that passes through the point (−4, 1) is:

x^2 + y^2 - 8x + 2y - 83 = 0

The middle 95% of Jace's finishing times in the 400 meter race is between 82 and 92 seconds.

Approximately 68% of the data falls within 1 standard deviation of the mean (i.e., between μ - σ and μ + σ).

Approximately 95% of the data falls within 2 standard deviations of the mean (i.e., between μ - 2σ and μ + 2σ).

Approximately 99.7% of the data falls within 3 standard deviations of the mean (i.e., between μ - 3σ and μ + 3σ).

We want to find the interval of times that represents the middle 95% of Jace's finishing times, which means we want to find the interval that falls between μ - 2σ and μ + 2σ.

Substituting the given values, we get:

μ - 2σ = 87 - 2(2.5) = 82

μ + 2σ = 87 + 2(2.5) = 92

Therefore, the middle 95% of Jace's finishing times in the 400 meter race is between 82 and 92 seconds.

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Answer:

x=70°

Step-by-step explanation:

angles on straight line add to 180 so...

180-120=60°

angles in triangle add to 180° so...

180°-50°-60°=70°

so x=70°

If a single letter from the word SLEEPLESS is chosen, the probability of choosing a P or an S as a fraction in lowest terms is 5/8.

To find the probability of choosing a P or an S from the word SLEEPLESS, we need to first determine the total number of letters in the word and then count how many of those letters are P's or S's.

There are 8 letters in the word SLEEPLESS, so there are 8 possible outcomes when selecting a single letter at random. Of these 8 letters, 2 are P's and 3 are S's.

Therefore, the probability of choosing a P or an S is the sum of the probabilities of choosing a P and choosing an S:

P(P or S) = P(P) + P(S)

P(P) = 2/8 = 1/4, since there are 2 P's out of 8 total letters.

P(S) = 3/8, since there are 3 S's out of 8 total letters.

P(P or S) = 1/4 + 3/8

P(P or S) = 5/8, which is the final probability expressed as a fraction in lowest terms.

Alternatively, we can convert the fraction to a decimal and round to the nearest millionth:

P(P or S) = 0.625

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Answer:  To show that the equation "3sin () tan () = 5cos () - 2" is equivalent to the equation "(4 cos() - 3)(2 cos () + 1) = 0", we need to simplify the first equation and check if it has the same solutions as the second equation.

Starting with the first equation:

3sin () tan () = 5cos () - 2

Using the identity tan () = sin () / cos (), we can write:

3sin () (sin () / cos ()) = 5cos () - 2

Multiplying both sides by cos (), we get:

3sin^2 () = (5cos () - 2)cos ()

Using the identity sin^2 () + cos^2 () = 1 and rearranging, we get:

3(1 - cos^2 ()) = 5cos^2 () - 2cos ()

Expanding and rearranging, we get:

5cos^2 () - 2cos () - 3 + 3cos^2 () = 0

Simplifying, we get:

8cos^2 () - 2cos () - 3 = 0

Now, we can use the quadratic formula to solve for cos ():

cos () = [2 ± sqrt(2^2 - 4(8)(-3))]/(2(8))

cos () = [2 ± sqrt(100)]/16

cos () = (1/4) or (-3/8)

Substituting these values back into the original equation, we can verify that they satisfy the equation.

Now, let's consider the second equation:

(4 cos() - 3)(2 cos () + 1) = 0

This equation is satisfied when either 4cos() - 3 = 0 or 2cos() + 1 = 0.

Solving for cos() in the first equation, we get:

4cos() - 3 = 0

cos() = 3/4

Substituting this value back into the original equation, we can verify that it satisfies the equation.

Solving for cos() in the second equation, we get:

2cos() + 1 = 0

cos() = -1/2

Substituting this value back into the original equation, we can also verify that it satisfies the equation.

Therefore, we have shown that the equation "3sin () tan () = 5cos () - 2" is equivalent to the equation "(4 cos() - 3)(2 cos () + 1) = 0".

Points are (0,-8) and (-6,-3) Slope m = rise/run = Δy/Δx. [(-3 - (-8)]/[-6 - 0] = 5/-6 = -5/6

y = mx + b Using the (0,-8) point: -8 = 0x + b, so b = -8

The line is y = -5x/6 - 8 or (-5/6)x - 8. You can check via the Desmos Graphing Calculator site

I used the slope-intercept form, but you could also use the point-slope form (y2 - y1) = m(x2 - x1)

By putting the value it proved that csc²x+cot²x/csc⁴x-cot⁴x=1

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. Trigonometry is found all throughout geometry, as every straight-sided shape may be broken into as a collection of triangles.There are six type of Trigonometry

According to question,

we have prove that

(csc²x + cot²x) / (csc⁴x - cot⁴x) = 1

Now by manipulating the left-hand side of the equation using trigonometric identities.

Then, we can simplify the denominator using the identity:

a² - b² = (a + b)(a - b)

In this case we get , a = csc²x and b = cot²x, so:

csc⁴x - cot⁴x = (csc²x + cot²x)(csc²x - cot²x)

By substituting this expression into the given equation, we get:

(csc²x + cot²x) / [(csc²x + cot²x)(csc²x - cot²x)] = 1

By solving the numerator, we get:

1 / (csc²x - cot²x) = 1

Now, we can use the identity:

csc²x - cot²x = 1 / sin²x - cos²x / sin²x

= (1 - cos²x) / sin²x

= sin²x / sin²x

= 1

Substituting this expression back into the equation, we get:

1 / 1 = 1

Hence, we have proved that:

(csc²x + cot²x) / (csc⁴x - cot⁴x) = 1.

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The area of a trapezoid =1/2× (sum of the parallel sides)×h

=1/2×(a + b)×h=1/2×(3+6)×4=9×2= 18 square units.

A trapezoid is a geometric shape with four sides, where two of the sides are parallel and the other two sides are not parallel. The parallel sides are called bases, and the non-parallel sides are called legs. The height or altitude of the trapezoid is the distance separating the two bases.

There are two main types of trapezoids: isosceles and non-isosceles. In an isosceles trapezoid, the legs are congruent, and the base angles (the angles formed between each base and a leg) are also congruent. In a non-isosceles trapezoid, the legs and base angles are not congruent.

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Answer:

The formula for the surface area of a cylinder is:

SA = 2πr^2 + 2πrh

where r is the radius and h is the height.

Since the diameter of the tube is 2 cm, the radius is 1 cm.

Substituting the given values, we get:

SA = 2π(1^2) + 2π(1)(8)

SA = 2π + 16π

SA = 18π

To find the surface area in square centimeters, we need to multiply by the conversion factor (1 cm)^2/(1 π) = 1 cm^2/3.14.

SA = 18π × (1 cm^2/3.14)

SA ≈ 18 × 0.3185

SA ≈ 5.73 cm^2

Rounding to the nearest whole number, we get:

SA ≈ 6 cm^2

Therefore, the answer is B. 57 cm^2.

Step-by-step explanation:

The value of the derivative of variable x with respect to parameter t is equal to dx / dt = - 1 / 2.

In this problem we need to find the derivative of variable x with respect to parameter t. This can be done by the following expression:

dy / dx = (dy / dt) / (dx / dt)

If we know that y = 4 · x² + 4, x = - 1 and dy / dt = 4, then the exact value of dy / dt is:

dy / dx = 8 · x

[8 · (- 1)] = 4 / (dx / dt)

- 8 = 4 / (dx / dt)

dx / dt = - 1 / 2

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The data item in this distribution corresponding to a z-score of -7 will be 290.

Here, we have,

The standard score in statistics is the number of standard deviations that a raw score's value is above or below the mean value of what is being observed or measured.

Raw scores that are higher than the mean have positive standard scores, whereas those that are lower than the mean have negative standard scores.

The Z-score measures how much a particular value deviates from the standard deviation.

The Z-score, also known as the standard score, is the number of standard deviations above or below the mean for a given data point.

The standard deviation reflects the level of variability within a particular data collection.

Here,

z=(X-μ)/σ

-7=(X-500)/30

-210=X-500

X=290

The data item in this distribution that corresponds to we given z-score

z equals to -7 will be 290.

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Answer:  Option C

Step-by-step explanation: To compare the rates at which the pipes are leaking, we can find the amount of water each pipe leaks per minute.

For Pipe A:

3/4 gallon ÷ 30 minutes = 1/40 gallon per minute

For Pipe B:

1 1/4 gallon ÷ 45 minutes = 5/36 gallon per minute

For Pipe C:

1/2 gallon ÷ 20 minutes = 1/40 gallon per minute

Therefore, Pipes A and C are leaking at the same rate, which means option C is true. Option A and D are false, as Pipe B has the fastest leak rate. Option B is false, as both Pipe A and Pipe C are leaking at the same rate.

Answer:

29.2

Step-by-step explanation:

10.30+18.90=29.2

Answer:

$29.20

Step-by-step explanation:

$10.30

+$18.90

=$29.20

The word that describes the probability that Janelys reaches in without looking and pulls out a strawberry chew is "the probability of selecting a strawberry chew randomly from the bag".

The word or phrase that describes the probability of pulling out a strawberry chew without looking is "the probability of selecting a strawberry chew at random" or "the probability of picking a strawberry chew by chance".

This probability is calculated by dividing the number of strawberry chews in the bag by the total number of chews in the bag, since each chew is equally likely to be selected.

In this case, there are 15 strawberry chews and 5 cherry chews, so the probability of selecting a strawberry chew at random is 15/(15+5) or 3/4, which is approximately 0.75 or 75%.

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The solutions to the equation (x - 5)² = 25 are x = 10 and x = 0.

An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides separated by an equal sign (=). Each side of the equation can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

It is a shape with four sides of equal length and four right angles. It is also a number multiplied by itself.

Solve the equation (x - 5)² = 25:

1. Take the square root of both sides of the equation to remove the exponent of 2 on the left side.

  √[(x - 5)²] = √25

2. Simplify the left side by removing the exponent of 2 and keeping the absolute value.

  |x - 5| = 5

3. Write two separate equations to account for both possible values of x when taking the absolute value.

  x - 5 = 5   or   x - 5 = -5

4. Solve for x in each equation.

x = 10   or   x = 0

  So the solutions to the equation (x - 5)² = 25 are x = 10 and x = 0.

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Answer:

The answer is A. $1,166.40

Step-by-step explanation:

Answer: 288.62

Step-by-step explanation:

10.4^2 + 15.3^2 = 342.25,

c^2 = 342.25

c (hypotenuse of the right triangle & length of rectangle) = 18.5

7(8) + 10.4(15.3) = 288.62

The situations that represent a proportional relationship are:

A). Natalia is selling fresh eggs at the local farmer's market. She sells 6 eggs for $3.12, a dozen eggs for $6.24, and eighteen eggs for $9.36.

C). Graph 1 provided in the pictures. This graph shows a straight line passing through the origin, which indicates a proportional relationship.

D). Graph 2 provided in the pictures. This graph also shows a straight line passing through the origin, which indicates a proportional relationship.

E). Azul bought several different packages of 8-inch by 10-inch art canvases for a craft project at her family reunion. The number of canvases in a package and the cost of the package is shown in the table.

Therefore, the situations A, C, D, and E represent proportional relationships.

A proportional relationship is a relationship between two quantities where one quantity is a constant multiple of the other quantity. In other words, if one quantity increases or decreases by a certain factor, then the other quantity will increase or decrease by the same factor. This relationship can be represented by a straight line passing through the origin on a graph.

For example, if the price of gasoline is proportional to the number of gallons purchased, then buying twice as many gallons would cost twice as much money. Similarly, if the distance traveled is proportional to the time taken, then traveling twice as long would cover twice the distance.

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In increasing order according to their capacity:

The capacity of a container, such as a cup, bottle, or tank, is the amount of fluid or material it can hold. It is usually stated in volume quantities such as liters, milliliters, gallons, or ounces.

Capacity is a necessary concept in engineering, manufacturing, and transportation, where it is used to design and run systems involving the storage, movement, and processing of fluids or materials.

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If will be positive if:

-24(a-t) > 0

-(a-t) > 0

-a + t > 0

t > a

Using derivatives, it is found that the particle is moving to the right for t > a , that is, values of t in the interval (a, ∞)

A particle is moving to the right if its velocity is positive.

The position of the particle is given by:

x(t) = 12(a -t)^2

The velocity is the derivative of the position, hence:

v(t) = -24(a-t)

If will be positive if:

-24(a-t) > 0

-(a-t) > 0

-a + t > 0

t > a

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Answer: 2 Radians

Step-by-step explanation:

    A full circle is 2 full radians, also known as 2π. Half of a circle is equal to 180 degrees, which is equal to one radian, π.

    I have attached a picture of the unit circle below, this picture comes form www.remind.com, but you can find many online.

On adding the measurement of "4-boards", the total length of the 'four-boards" is  154 inches.

In order to add the lengths of these four boards, we first need to convert all the measurements to the same units.

So, Let us convert all the measurements to inches:

(i) 3 feet 4 inches = (3×12) + 4 = 40 inches,    ...because 1 feet = 12 inch;

(ii) 27 inches = 27 inches;

(iii) 1(1/2) yards = (1.5 × 3 × 12) = 54 inches;

(iv) 2(3/4) feet = (2.75 × 12) = 33 inches;

Now, we can add the lengths of the four-boards:

⇒ 40 + 27 + 54 + 33 = 154 inches,

Therefore, the total length of all four boards is 154 inches.

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Answer:

40 students are in the class :)

Step-by-step explanation:

35%=14

1%=14/35=0.4

0.4*100=40=100%

The profit for one small cake is £1.31.

Let's first determine the overall count of parts demonstrated in the ratio 5:7:12, which is equal to 24. Subsequently, each part would hence correspond to a quantity of nine cakes as per the equation below:

216 cakes / 24 = 9 cakes

5 parts x 9 cakes/part = 45 cakes

Likewise, similarly calculating the number of medium cakes leads us to 63 cakes, while that of large cakes sums up to 108 cakes:

7 parts x 9 cakes/part = 63 cakes

12 parts x 9 cakes/part = 108 cakes

Considering what the profit for one small cake is denoted by "x", and similarly, the profit for one medium cake is "2x" and the profit for one large cake is "3x", we can express the total gain as follows:

45x + 63(2x) + 108(3x) = 648.45

After similifying and solving for "x", we obtain an answer of £1.31 after rounding it off to two decimal places:

495x = 648.45

x = 1.31 (rounded to two decimal places)

Thus, the income earned from selling one single small cake is estimated at £1.31.

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The height of the thrown bottle becomes zero at the t = 1/2 that is half a second and at t = 3 seconds

An equation is formed by expressing the relationship between two or more mathematical expressions. It can be formed by using mathematical symbols, operations, and variables. Typically, equations are used to solve problems or find unknown quantities by setting up an equation that represents the problem and then manipulating the equation using algebraic operations until the solution is obtained. For example, to solve the equation 2x + 3 = 7, you would use algebraic operations to isolate the variable x on one side of the equation, giving you the solution x = 2.

To find when the height of the thrown bottle is equal to 0, we need to solve the given equation:

h = -16t² + 56t - 24 = 0

Takin -8 common out we get,

-8(2t² - 7t + 3) = 0

Now we can use the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero:

-8 = 0 or 2t² - 7t + 3 = 0

But -8 ≠ 0, so we only possible one is,

2t² - 7t + 3 = 0

(2t - 1)(t - 3) = 0

Using the zero product property, we get,

2t - 1 = 0 or t - 3 = 0

Solving for t we get,

t = 1/2 or t = 3

Therefore, the height of the thrown bottle is equal to 0 at t = 1/2 seconds and t = 3 seconds.

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The height of the thrown bottle becomes zero at the t = 1/2 that is half a second and at t = 3 seconds

An equation is formed by expressing the relationship between two or more mathematical expressions. It can be formed by using mathematical symbols, operations, and variables. Typically, equations are used to solve problems or find unknown quantities by setting up an equation that represents the problem and then manipulating the equation using algebraic operations until the solution is obtained. For example, to solve the equation 2x + 3 = 7, you would use algebraic operations to isolate the variable x on one side of the equation, giving you the solution x = 2.

To find when the height of the thrown bottle is equal to 0, we need to solve the given equation:

h = -16t² + 56t - 24 = 0

Takin -8 common out we get,

-8(2t² - 7t + 3) = 0

Now we can use the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero:

-8 = 0 or 2t² - 7t + 3 = 0

But -8 ≠ 0, so we only possible one is,

2t² - 7t + 3 = 0

(2t - 1)(t - 3) = 0

Using the zero product property, we get,

2t - 1 = 0 or t - 3 = 0

Solving for t we get,

t = 1/2 or t = 3

Therefore, the height of the thrown bottle is equal to 0 at t = 1/2 seconds and t = 3 seconds.

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The mean amount of money the insurance company can expect to lose on each policy is $142.00 with a standard deviation of $1,243.67.

What is an insurance?

Let X be the random variable representing the amount of money the insurance company will lose on a policy. Then we can calculate the expected value (mean) of X and the standard deviation of X as follows:

Expected value:

E(X) = 30,000(1/1,000) + 10,000(1/250) + 0(1 - 1/1,000 - 1/250) = $142.00

The first term in the sum corresponds to the probability of a client dying (1/1,000) multiplied by the payout ($30,000), the second term corresponds to the probability of a client becoming disabled (1/250) multiplied by the payout ($10,000), and the third term corresponds to the probability of neither event occurring (1 - 1/1,000 - 1/250).

Standard deviation:

To calculate the standard deviation, we need to find the variance of X first:

Var(X) = [30,000 - E(X)]²(1/1,000) + [10,000 - E(X)]²(1/250) + [0 - E(X)]²(1 - 1/1,000 - 1/250)

= $1,547,797.56

The first term in the sum corresponds to the squared difference between the payout for a client dying and the expected payout, multiplied by the probability of a client dying, and so on for the second and third terms.

Then, we can take the square root of the variance to find the standard deviation:

SD(X) = √[Var(X)] = $1,243.67

Therefore, the mean amount of money the insurance company can expect to lose on each policy is $142.00 with a standard deviation of $1,243.67.

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Answer:

Step-by-step explanation:

A is correct