you local gym wants to estimate the mean amount of time (in hours) that their members spend each week at their center. They would like a 95% confidence interval with a margin of error of 30 minutes. based on previous data, they know that the population standard deviation is 2.2 hours. What is the smallest size that will allow this confidence interval to be built?

The cylinder's surface area is S = 452.4 cm²

Given data ,

Let the surface area of the cylinder with one base be S

Now , Surface area of cylinder = 2πr ( r + h )

To find the surface area of a cylinder, we need to calculate the sum of the areas of the curved surface (lateral surface) and the two circular bases.

where r = 8 cm

h = 5 cm

The formula for the lateral surface area of a cylinder is given by:

Lateral Surface Area = 2πrh

where r is the radius of the cylinder and h is the height.

The formula for the area of a circle is given by:

Base Area = πr²

Now , S = πr² + 2πrh

On simplifying , we get

S = π ( 8 )² + 2 ( π ) ( 8 ) ( 5 )

S = 64π + 80π

S = 452.389 cm²

Hence , the surface area is S = 452.4 cm²

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Answer:

c. 34.5 m

Step-by-step explanation:

Over horizontal ground, the range R for velocity v and launch angle θ is:

R=v^2sin2θ/g

=35^2sin16∘/9.8

= 34.5 meters (so the answer is (c))

The altitude of the kite is 53 meters.

Now, First, we need to identify the right triangle formed by Bob, the kite, and the ground.

Here, The angle of elevation of the kite is, 58 degrees

Which is opposite to the altitude of the kite that we need to find.

And,  The length of the string is, 63 m

Which is the hypotenuse of the right triangle.

Hence, By Using trigonometry, we can use the sine function to solve for the altitude of the kite:

sin(58) = altitude / 63

Solving for the altitude, we get:

altitude = 63 x sin(58)

Altitude = 53 meters

Thus, the altitude of the kite is 53 meters.

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The choice between the two depends on the specific problem and desired model characteristics.  The lasso penalty may be preferred when the goal is to identify a small number of important predictors, while the ridge penalty may be preferred when the goal is to retain all predictors in the model but reduce their impact.

The lasso penalty and the ridge penalty are two different approaches to shrinkage models in which the goal is to reduce the complexity of a model by shrinking the coefficients towards zero. The main difference between the lasso penalty and the ridge penalty is in the way they perform this shrinking.
The lasso penalty uses an L1 norm, which encourages sparsity by driving some coefficients to exactly zero. This means that the lasso penalty is able to perform variable selection by completely eliminating some predictors from the model. In contrast, the ridge penalty uses an L2 norm, which does not drive coefficients to exactly zero, but rather shrinks them towards zero. This means that the ridge penalty is not able to perform variable selection in the same way as the lasso penalty.
The effect of this difference on the estimated coefficient is that the lasso penalty tends to produce more sparse models with fewer non-zero coefficients, while the ridge penalty tends to produce models with more non-zero coefficients. The lasso penalty may be preferred when the goal is to identify a small number of important predictors, while the ridge penalty may be preferred when the goal is to retain all predictors in the model but reduce their impact.

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The function that models the number of bacteria, f(t), at time t, in hours, is f(t) = 100(∛4)^t

Based on the scatter plot, we can see that the data points roughly form an exponential curve.

Therefore, we can use the function of the form f(t) = ab^t to model the data, where a is the initial value of the function (the number of bacteria at t=0) and b is the growth factor.

To find the values of a and b, we can use the two data points given in the table: (0, 100) and (3, 400). Substituting these values into the equation, we get:

100 = ab^0  ->  a = 100

400 = ab^3

Dividing the second equation by the first equation, we get:

4 = b^3  ->  b = ∛4

Therefore, the function that models the number of bacteria, f(t), at time t, in hours, is f(t) = 100(∛4)^t

So the answer is f(t) = 100(∛4)^t.

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The descriptions that represent a categorical data set are: Color of chewing gum brands sold at Store A, Best selling chewing gum flavors, Names of stores that carry Brand X of chewing gum.

We know that,

A data set is a collection of data, usually presented in tabular form, that can be analyzed to draw conclusions or make decisions. It can be numerical or categorical and can come from various sources such as surveys, experiments, or observations.

A data set typically contains individual data points or observations, each representing a unit or subject being studied. The size of a data set can range from small (e.g., a few observations) to large (e.g., millions of observations).

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In the above scenario, The point that B is transformed to are coordinates (10, 6). This means that it is translated to the right by 9 units and upwards by 5 units.

In order to derive the new location of point B after the transformation by matrix A, we need to perform matrix multiplication of A with the column vector representing point B.

[tex]\left[\begin{array}{cc}8&2\\7&1\\\end{array}\right][/tex]   = [tex]\left[\begin{array}{cc}1\\1\\\end{array}\right][/tex]

[tex]\left[\begin{array}{cc}(8 * 1) + &(2 *1)\\(7 *1)&(1 *1)\\\end{array}\right][/tex] =  [tex]\left[\begin{array}{cc}10\\6\\\end{array}\right][/tex]

As a result, the converted point B is situated at (10, 6). As a result of the transformation provided by matrix A, point B has been translated to the right by 9 units (from x = 1 to x = 10) and upwards by 5 units (from y = 1 to y = 6).

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Answer:

[tex]\sin\beta=\dfrac{8}{17}[/tex]

[tex]\cos\beta=\dfrac{15}{17}[/tex]

[tex]tan\beta=\dfrac{8}{15}[/tex]

[tex]\csc\beta=\dfrac{17}{8}[/tex]

[tex]\cot\beta=\dfrac{15}{8}[/tex]

Step-by-step explanation:

The secant ratio is the reciprocal of the cosine ratio.

[tex]\sec \beta= \dfrac{1}{\cos \beta}[/tex]

Therefore, if sec β = 17/15 then:

[tex]\dfrac{1}{\cos \beta}=\dfrac{17}{15}[/tex]

[tex]\cos \beta=\dfrac{15}{17}[/tex]

The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse of a right triangle:

[tex]\cos\beta= \sf \dfrac{adjacent}{hypotenuse}[/tex]

Therefore, the length of the side adjacent angle θ is 15 and the length of the hypotenuse is 17.

[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]

We can use Pythagoras Theorem to calculate the length of the side opposite angle β:

[tex]15^2+O^2=17^2[/tex]

[tex]O^2=17^2-15^2[/tex]

[tex]O=\sqrt{17^2-15^2}[/tex]

[tex]O=\sqrt{64}[/tex]

[tex]O=8[/tex]

Therefore, the length of the side opposite angle β is 8.

Now we have the lengths of the three sides of the right triangle, we can find the other trigonometric function of angle β.

[tex]\boxed{\begin{minipage}{8cm}\underline{Trigonometric functions}\\\\$\sf \sin\beta=\dfrac{O}{H}\quad\cos\beta=\dfrac{A}{H}\quad\tan\beta=\dfrac{O}{A}$\\\\\\$\sf\csc\beta=\dfrac{H}{O}\quad\sec\beta=\dfrac{H}{A}\quad\cot\beta=\dfrac{A}{O}$\\\\\\where:\\\phantom{ww}$\bullet$ $\beta$ is the angle.\\\phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle.\\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle.\\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse.\\\end{minipage}}[/tex]

Given values:

Substitute these values into the six trigonometric functions:

[tex]\sin\beta=\dfrac{O}{H}=\dfrac{8}{17}[/tex]

[tex]\cos\beta=\dfrac{A}{H}=\dfrac{15}{17}[/tex]

[tex]tan\beta=\dfrac{O}{A}=\dfrac{8}{15}[/tex]

[tex]\csc\beta=\dfrac{H}{O}=\dfrac{17}{8}[/tex]

[tex]\sec\beta=\dfrac{H}{A}=\dfrac{17}{15}[/tex]

[tex]\cot\beta=\dfrac{A}{O}=\dfrac{15}{8}[/tex]

The required image T of triangle ABC under the transformation (x,y) → (x-4, y+1) is formed by connecting the vertices A'(a-4, b+1), B'(c-4, d+1), and C'(e-4, f+1)

To find the image T of triangle ABC under the transformation (x,y) → (x-4, y+1), apply this transformation to each of the vertices of triangle ABC and then connect the new vertices to form the image T.

Let's assume that the coordinates of the vertices of triangle ABC are A(a,b), B(c,d), and C(e,f).

To find the image of vertex A under the transformation, we substitute x = a and y = b into the transformation equation:

(x,y) → (x-4, y+1)

(a,b) → (a-4, b+1)

Therefore, the image of vertex A is A'(a-4, b+1).

Similarly, we can find the images of vertices B and C:

B'(c-4, d+1)

C'(e-4, f+1)

Therefore, the image T of triangle ABC under the transformation (x,y) → (x-4, y+1) is formed by connecting the vertices A'(a-4, b+1), B'(c-4, d+1), and C'(e-4, f+1)

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The coordinates of the pre-image, and the applied rotation transformation indicates;

3. a. The center of rotation is (0, 0), and the rotation is 90 degrees counter clockwise

b. The center of rotation is; (-1, 2), and the angle of rotation 90 degrees clockwise

A rotation transformation is one in which the the pre-image is rotated about a point, referred to as the center of rotation.

The coordinates of the vertices of the blue triangle are; (1, -1), (2, -1), (2, -3)

The coordinates of the corresponding vertices of the green triangle are; (1, 1), (1, 2), (3, 2)

The rotation transformation from the blue triangle to the green triangle therefore is (x, y) ⇒ (-y, x), which corresponds to a 90 degrees counterclockwise rotation about the origin.

b. The coordinates of the corresponding vertices of the orange triangle are; (-2, 0), (-1, 0), (-1, -2)

The 90 degrees clockwise rotation of a point (x, y) has an image at the point (y, -x)

The center of rotation can be found using the perpendicular bisectors method, which requires the perpendicular bisectors of the segment joining the corresponding points of the image and the pre-image as follows;

The equation of the line joining (1, 1), to (-2, 0) is; 3·y = x + 2

The slope is 1/3

The coordinates of the location of the perpendicular bisector is (1 + (-2))/2. (1 + 0)/2) = (-0.5, 0.5)

The slope of the perpendicular bisector = -3

The equation of the perpendicular bisector is; y - 0.5 = (-3)·(x - (-0.5))

y = -3·x - 1.5 + 0.5 = -3·x - 1

y = -3·x - 1

The equation of the line from (3, 2) to (-1, -2) is; y = x - 1

The slope of the perpendicular bisector is; -1

The midpoint is; (1, 0)

The equation is; y - 0 = -1·(x - 1)

y = -x + 1

The coordinate of the center of rotation is therefore;

-3·x - 1 = -x + 1

x = -1

y = -3 × (-1) - 1 = 2

The coordinate of the center of rotation is; (-1, 2)

The relative coordinates of the green triangle vertices with regards to the center of rotation are; (1 - (-1), 1 - 2)), (1 - (-1), 2 - 2), (3 - (-1), 2 - 2)

(1 - (-1), 1 - 2)) = (2, -1)

(1 - (-1), 2 - 2) = (2, 0)

(3 - (-1), 2 - 2) = (4, 0)

The coordinates following a 90 degrees clockwise rotation are;

(2, -1) ⇒ (-1, -2)

(2, 0) ⇒ (0, -2)

(4, 0) ⇒ (0, -4)

The coordinate of the image, relative to the origin  are therefore;

(-1 + (-1), -2 + 2) = (-2, 0)

(0 + (-1), -2 + 2) = (-1, 0)

(0 + (-1), -4 + 2) = (-1, -2)

The coordinates of the image corresponds with the coordinates of the orange triangle

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The equation of line p is x = 4 because it is parallel to the y-axis and passes through the point (4,7).

When a line is parallel to the y-axis, its slope is undefined. However, we know that line p passes through the point (4,7). Since it is parallel to the y-axis, we can conclude that its x-coordinate is always 4.

Therefore, the equation of line p is x = 4. This means that every point on the line will have an x-coordinate of 4 and can have any y-coordinate.

We can confirm this by checking that the point (4,7) satisfies the equation x = 4. Since the point lies on the line p, we can conclude that this is the equation for line p.

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Expression: 12x / 2

A coefficient is the number in front of a variable. In this problem, the coefficient is 12 because it is in front of the variable x.

Hope this helps!

Part A: Possible words to describe this data set include: discrete, unimodal, symmetric. Part B: the best measures of center and spread would be median and quartiles. Part C: the best measures of center and spread would be mean and standard deviation

Part A:

Data Set A: 1 3 3 2 4 0 4 3 3 5 2

This data set has a range of 5 with values ranging from 0 to 5. Possible words to describe this data set include: discrete, unimodal, symmetric.

Data Set B: 3 5 5 4 6 1 6 6 6 6 5

This data set has a range of 5 with values ranging from 1 to 6. Possible words to describe this data set include: discrete, bimodal, right-skewed.

Part B:

For Data Set A, the best measures of center and spread would be median and quartiles. The median would give an estimate of the center of the data, and the quartiles would give an estimate of the spread.

Part C:

For Data Set B, the best measures of center and spread would be mean and standard deviation. Mean would give an estimate of the center of the data, and standard deviation would give an estimate of the spread.

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If the dilation is centered at the origin, the vertices of polygon A′B′C′D′ are: D. A′(−2.4, 3.6), B′(−1.2, 1.2), C′(2.4, −1.2), D′(2.4, 2.4).

In Geometry, dilation refers to a type of transformation which typically changes the size of a geometric object, but not its shape.

Next, we would have to dilate the coordinates of the preimage by using a scale factor of 3/5 centered at the origin as follows:

Ordered pair A (-4, 6) → Ordered pair A' (-4 × 3/5, 6 × 3/5) = Ordered pair A' (-2.4, 3.6).

Ordered pair B (-2, 2) → Ordered pair B' (-2 × 3/5, 2 × 3/5) = Ordered pair B' (-1.2, 1.2).

Ordered pair C (4, -2) → Ordered pair C' (4 × 3/5, -2 × 3/5) = Ordered pair C' (2.4, -1.2).

Ordered pair D (4, 4) → Ordered pair D' (4 × 3/5, 4 × 3/5) = Ordered pair D' (2.4, 2.4).

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The number of hours they can play video games before the weekend, then the inequality that represents the situation will be:

8 - a ≤ 4

Noted that inequality is a mathematical statement that compares two quantities or expressions that are not equal using the inequality signs such as:

Given that Trini is allowed to play video games no more than 4 hours over the weekend.

A number of hours to be played during the weekend is given as, at least 4 hours. This means the number of hours to play at the weekend can be 4 hours or more.

If a represents the number of hours they can play video games before the weekend, then the inequality that represents the situation will be:

8 - a ≤ 4

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Answer:

answer in explanation below

Step-by-step explanation:

79 to 83      5

84 to 88      3

89 to 93      1

94 to 98      6

          total = 15

1. The velocity of P after 3 seconds is -15i - 5j m·s⁻¹

1b. To one decimal place, the speed is  15.8 m·s⁻¹

2. The displacement of the particle after 4 seconds is -20i + 8j meters

2b. The distance is 21.5 meters.

In kinematics, the velocity of an object at a given time can be calculated by V = U + at

V = U + at

V = (-3i - 8j) + 3×(-4i + j)

V = (-3i - 8j) + (-12i + 3j)

V = (-3-12)i + (-8+3)j

V = -15i - 5j m/s

Speed = √((velocity in i)² + (velocity in j)²)

Speed = √((-15)² + (-5)²)

Speed = √(225 + 25)

Speed = √(250)

= 15.8 m/s.

In kinematics, the displacement of an object at any given time can be calculated by

s = ut + 1/2at²

s = (i + 6j)4 + 1/2(-3i - 2j)×4²

s = (4i + 24j) + (-24i - 16j)

s = (4-24)i + (24-16)j

s = -20i + 8j m

using the Pythagorean theorem

Distance = sqrt((displacement in i)² + (displacement in j)²)

Distance = √(-20)² + 8²)

Distance =√(400 + 64)

Distance = √464

=  21.5 meters

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Answer: $26.52

Step-by-step explanation:

1 Box of cereal is $2.37 so 6 boxes is equal to 6 * $2.37 = $14.22

1 Pound of apples is $4.10 so 3 pounds is equal to 3 * $4.10 = $12.30

Total = $14.22 + $12.30 = $26.52

Answer: x=6

Step-by-step explanation:

Subtract x on both sides

-x - 1 = -7

Add 1 to both sides

-x = -6

Divide both sides by -1

x = 6

Answer:

3.5 y + 1.25 x = 30

Step-by-step explanation:

she can buy x number of fly tapes and y number of ant traps. this amount has to be less or equal to the $30 that she has brought along.

3.5 y + 1.25 x ≤ 30

this is an equality.

since it asks for an equation, 3.5 y + 1.25x = 30

Answer:

Step-by-step explanation:

svsvse

Two times a number c when subtracted from 6 times the number is equal to 4

Algebraic expressions are defined as expressions that are composed of terms, coefficients, variables, constants and factors.

These algebraic expressions are also made up of mathematical or arithmetic operations.

These arithmetic operations are;

From the information given, we have that;

6c - 2c = 4

The variables are c

The coefficient are 6 and 2

The constant is 4

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Answer:

Step-by-step explanation:

[tex]y=29\times2^x\\put~x=x+1\\y1=29\times2^{x+1}=29\times2^x\times2=y\times 2\\y_{n+1} =2y_{n}[/tex]

The percent increase in the average height of the corn plants during this one-week period is approximately 50%.

To calculate the percent increase in the average height of the corn plants, we need to find the difference between the final height and the initial height, and then express that difference as a percentage of the initial height.

Initial height = 16 1/2 in.

Final height = 24 7/10 in.

To find the difference, we subtract the initial height from the final height:

Difference = Final height - Initial height

Difference = 24 7/10 - 16 1/2

To perform the subtraction, let's convert the heights to a common denominator, which is 10:

Difference = (24 * 10 + 7) / 10 - (16 * 10 + 5) / 2

Difference = (240 + 7) / 10 - (160 + 5) / 10

Difference = (247 / 10) - (165 / 10)

Difference = 82 / 10

Difference = 8.2 in.

Now, let's calculate the percent increase:

Percent increase = (Difference / Initial height) * 100

Percent increase = (8.2 / 16.5) * 100

Percent increase ≈ 49.70

Rounding to the nearest whole percent, the percent increase in the average height of the corn plants during this one-week period is approximately 50%.

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The expression (fg)(x) is equal to 16x^4 + 72x^3 - 39x^2 + 5x.

To find (fg)(x), we need to evaluate the product of f(x) and g(x).

First, let's express f(x) and g(x) explicitly:

f(x) = 4x^2 + 19x - 5

g(x) = 4x^2 - x

Now, we can calculate their product:

(fg)(x) = f(x) * g(x)

= (4x^2 + 19x - 5) * (4x^2 - x)

To simplify the expression, we can use the distributive property and combine like terms:

(fg)(x) = 4x^2 * 4x^2 + 4x^2 * (-x) + 19x * 4x^2 + 19x * (-x) - 5 * 4x^2 - 5 * (-x)

Expanding and collecting like terms:

(fg)(x) = 16x^4 - 4x^3 + 76x^3 - 19x^2 - 20x^2 + 5x

Combining like terms:

(fg)(x) = 16x^4 + 72x^3 - 39x^2 + 5x.

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To find the dimensions of the rectangular base, we can use the formula for the volume of a rectangular box:

Volume = Length x Width x Height

Given that the volume is 2860 and the height is 40 cm, we can rearrange the formula to solve for the dimensions of the base:

2860 = Length x Width x 40

Now, let's find possible dimensions by factoring 2860:

2860 = 2 x 2 x 5 x 11 x 13

To find the dimensions of the base, we need to divide these factors into the length and width in different combinations. Here are a few possible options:

Option 1:

Length = 2 x 5 x 11 = 110 cm

Width = 2 x 13 = 26 cm

Option 2:

Length = 2 x 2 x 11 = 44 cm

Width = 5 x 13 = 65 cm

Option 3:

Length = 2 x 2 x 13 = 52 cm

Width = 5 x 11 = 55 cm

These are just a few examples, and there could be other combinations of factors that give a volume of 2860 with a height of 40 cm.

The volume of the silo is 17,128 cubic feet (ft³).

We have,

The volume of a cylinder is given by the formula:

V = πr²h

where V is the volume, r is the radius, and h is the height.

Substituting the given values, we get:

V = 3.14 × 8.5² × 49

V ≈ 17,128

Rounding to the nearest whole number, the volume of the silo is 17,128 cubic feet (ft³).

Thus,

The volume of the silo is 17,128 cubic feet (ft³).

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