An ice-cream scoop has a shape of spherical cap. An engineer's goal is to maximize the volume of the scoop given that the surface area of the scoop is 36 cm2. Find the radius of the sphere and the height of the spherical cap of maximum volume. (Hint:
the volume of a spherical cap is [tex]Vcap=\frac{1}{3} \pi h^2(3R-h)[/tex], where R is the radius of the sphere and h is the height of the cap). Find the maximum volume of ice-cream scoop.
Write your answer in terms of π.

The volume of the cardboard box with given dimensions changes at the fastest average rate over the interval of  option c.  [1,5].

Length of the cardboard box = x + 1

Width of the cardboard box = 5  - x

Height of the cardboard box = x - 1

Volume of the box V (x) =  length × width × height

Substitute the values we have,

V(x) = (x+1)(5-x)(x-1)

Expanding this expression gives,

V(x) = -x³ + 5x² + x - 5

To know where Volume is changing at the fastest average rate,

Find the maximum value of the absolute value of the derivative of V(x) over each of the given intervals.

Taking the derivative of V(x), we get,

V'(x) = -3x²+ 10x + 1

Taking the absolute value of this expression, we get,

|V'(x)| = 3x² - 10x - 1

Now, we can calculate the maximum value of |V'(x)| over each interval,

At [1,2]

|V'(1)| = 8

|V'(2)| = 9

[1,3.5]

|V'(1)| = 8

|V'(3.5)| = 0.75

[1,5]

|V'(1)| = 1

|V'(5)| = 24

[0,3.5]

|V'(0)| = 1

|V'(3.5)| = 0.75

The maximum value of |V'(x)| occurs on interval [1,5].

Therefore, the volume of the box is changing at the fastest average rate over the interval option c . [1,5].

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The above question is incomplete, the complete question is:

A cardboard box manufacturing company is building boxes with length represented by x + 1, width by 5 − x, and height by x − 1. the volume of the box is modeled by the function below. over which interval is the volume of the box changing at the fastest average rate?

a. [1,2]

b. [1,3.5]

c. [1,5]

d. [0,3.5]

The measure of the diameter x for the circle is 18.6 units.

A circle is a geometric shape consisting of all points in a plane that are equidistant from a fixed point called the centre. A circle is defined by its radius, which is the distance from the centre to any point on the circle, or by its diameter, which is the distance across the circle passing through the centre and connecting two points on the opposite sides of the circle.

The diameter of a circle is a line segment that passes through the centre of the circle and connects two points on the opposite sides of the circle. The diameter is twice the length of the radius of the circle, and it is often used to express the size or scale of the circle.

The diameter of the circle will be calculated as,

x = 2 ( 3.2 + 6.1)

x = 2 ( 9.3)

x = 18.6 units

Hence, the diameter is 18.6 units.

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The  volume of rectangular prism is 90 unit³.

We can consider the 1 block = 1 unit.

Length of prism = 5 unit

width of prism = 6 unit

Height of prism = 3 unit

So, Volume of rectangular prism

= l w h

= 5 x 6 x 3

= 90 unit³

Thus, the volume of rectangular prism is 90 unit³.

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Answer:

the answer is (b)

because;

a^2-10a+24

product:24

sum:-10

factor:-6,-4

a(a-6)-4(a-6)

(a-6)(a-4)

therefore:two solutions but only one is given

If one pound 1.2 euros then 14 pounds is 21 euros.

A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity

Given that 1 pound is 1.2 euros

We have to find 14 pounds in euros

To do this we just need to multiply 14 with 1.2

14×1.5

21 euros

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The practical use of the confidence interval will be C. The confidence interval can be used to estimate that, with 90% confidence, the interval from true mean DNA base of all people

CI = 2.7 ± 1.645 * (1.0801/√10) ≈ (1.4, 4.0)

Therefore, the 90% confidence interval for the population mean µ is (1.4, 4.0).

The efficacy of confidence intervals lies in their aptitude to determine the probable range of values wherein the authentic populace mean probably resides, with a provided level of assurance. Consequently, we can assert with 90% sureness that the true population mean of DNA bases are likely to rest between 1.4 and 4.0.

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The answers to the box plot are given as follow;

1) the highest level of snow fall was 50inches. This is the maximum value on the plot
2) The data is most concentrated in the 3rd quarter. More cities has snow fall closer to 40 inches than to 13 inches.

3) the data is most spread all over the 4th quarter.

1) As per the data depicted in a box plot, it can be inferred that no urban center witnesses snowfall exceeding 50 inches.

2) The highest point on this chart supports this observation.

Furthermore, an overwhelming number of cities seem to report moderate levels of snowfall between 13-40 inches as indicated by most of these values lying within that particular section.

To buttress my argument further, I would note how close to its top edge is located the median line for this section. It is apparent that the data exhibits the highest degree of variability during the final quarter, signifying the top 25% of values.

The elongated vertical line extending from the upper end of the box up to the maximum value, which reaches 50 inches, is considerably lengthier than its counterpart at the bottom concluding with a minimum value of 5 inches.

This suggests that there is a greater disparity among snowfall measurements among cities receiving copious amounts.

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The input value that corresponds to an output value of 15 is x = -1/3.

To find the input value (x) that corresponds to an output value of 15, we need to derive an equation:

15 = 60 / (3x + 5)

Multiplying both sides by (3x + 5), we get:

15(3x + 5) = 60

Expanding the left side, we get:

45x + 75 = 60

Subtracting 75 from both sides, we get:

45x = -15

Dividing both sides by 45, we get:

x = -1/3

Therefore, the input value that corresponds to an output value of 15 is x = -1/3.

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The distribution is roughly symmetric, we can use the mean as a measure of center and the standard deviation as a measure of spread.

The distribution of miles ran each week by Elisa is relatively symmetric and centered around 30 miles per week, with a relatively tight spread of about 2.3 miles per week.

We have,

a.

Since the distribution is roughly symmetric, we can use the mean as a measure of center and the standard deviation as a measure of spread.

However, the distribution also has some slight right-skewness, so we may want to consider using the median instead of the mean, depending on the context and purpose of the analysis.

b.

The mean of the distribution is:

= (28 + 29 + 30 + 31 + 32 + 33 + 34)/7

= 30.1

So the center of the distribution is around 30 miles per week.

The standard deviation of the distribution is approximately 2.3 miles per week, which tells us that the spread of the distribution is relatively tight.

Alternatively, we could use the median as a measure of the center.

Since there are 7 data points, the median is the average of the 4th and 5th values (30 and 31):

= (30 + 31)/2

= 30.5

So the center of the distribution is also around 30 miles per week using the median.

Thus,

The distribution is roughly symmetric, we can use the mean as a measure of center and the standard deviation as a measure of spread.

The distribution of miles ran each week by Elisa is relatively symmetric and centered around 30 miles per week, with a relatively tight spread of about 2.3 miles per week.

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In 5 seconds of free fall at a speed of 165 feet per second, the parachute user will drop around 122.5 meters.

To convert feet per second (ft/s) to meters per second (m/s), we can use the following conversion factor:

1 m/s = 3.3 ft/s

Therefore, the parachutist's rate of 165 ft/s is equivalent to:

165 ft/s × (1 m/3.3 ft) = 50 m/s

Next, we can use the kinematic equation:

d = 1/2 * a * t²

Here d is the distance traveled (in meters), a is the acceleration due to gravity (9.8 m/s²), and t is the time (in seconds).

For 5 seconds of free fall, we have:

d = 1/2 * 9.8 m/s² * (5 s)² = 122.5 m

Therefore, the parachutist will fall approximately 122.5 meters during 5 seconds of free fall at a rate of 165 ft/s.

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Answer:

(-11,8)

Step-by-step explanation:

To find the coordinates of the fourth corner of the rectangular city block, we need to use the fact that opposite sides of a rectangle are parallel and equal in length.

Let's start by finding the length of one of the sides of the rectangle. We can use the distance formula for that:

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

For example, if we want to find the length of the side between (2,2) and (2,-4), we have:

d = sqrt((2 - 2)^2 + (-4 - 2)^2) = sqrt(6^2) = 6

We can also see from the given coordinates that the sides of the rectangle are parallel to the x or y axis, which means that their lengths are simply the difference between the corresponding x or y coordinates.

So now we know that the length of the sides of the rectangle are both 6 units. The missing corner must be positioned 6 units away from the given point (-5,-4) along the x-axis, since the fourth side of the rectangle is parallel to the x-axis. Therefore, its x-coordinate is -5 - 6 = -11.

Similarly, the fourth corner must be positioned 6 units away from the given point (2,2) along the y-axis, since the fourth side of the rectangle is parallel to the y-axis. Therefore, its y-coordinate is 2 + 6 = 8.

Therefore, the coordinates of the fourth corner are (-11, 8).

Answer: Letter C

Step-by-step explanation:

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Since c is saying that congruent arcs associated with chords are perpendicular its wrong.

By selecting 100 inhabitants who pay real estate tax, using the town's list of taxpayers and a random number generator, Option D offers an arbitrary population sample.

This approach guarantees that each taxpayer has an equal likelihood of being chosen for the study, resulting in a diversified sample representation.

Conversely, the other options have possible biases or restrictions; like surveying solely soccer player's guardians might narrow it down to one particular group or only conducting them in designated areas could lead to misrepresentation within the frame of the entire general opinion poll.

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The best conclusion to make based on the data is B The difference of the means is significant because the rerandomizations show that it is outside the range of what could happen by chance.

Given the details and dot plot, option B) is considered to be the most beneficial conclusion. This would imply that the difference between the means of fleas for two various treatment groups is not merely due to happenstance and is in fact statistically noteworthy.

Examining the randomizations further confirms that this deviation lies out of the expected parameter based on mere chance.

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Answer:

0.5 (any number between 0 and 1)

Step-by-step explanation:

34 * 0.5 = 17

34 > 17 > 0

The approximate population density of Monaco in persons per square mile in 1999 was 42,666.67.

To calculate the population density of Monaco in persons per square mile, we need to divide the population of Monaco by the area of the country in square miles.

First, we need to convert the area of Monaco from square miles to square feet because the population figure is in persons per square mile.

1 square mile = 5280 feet × 5280 feet = 27,878,400 square feet

So, the area of Monaco in square feet is approximately:

(3/4) × 27,878,400 = 20,908,800 square feet

Now, we can calculate the population density in persons per square mile:

Population density = Population / Area

Population density = 32,000 / (20,908,800 / 27,878,400)

Population density = 32,000 / 0.75

Population density = 42,666.67 persons per square mile (rounded to two decimal places)

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The angle of Z is 63.2 degree.

The word isosceles comes from the Ancient Greek term isosceles which literally means "equal-legged." An isosceles triangle has two of its sides of equal length, which means that the angles opposite these sides would also be of equal measure.

In Triangle ΔXYZ,

∠Y = 83°, XY = 9 and XZ = 10

To find the angle of Z

Using Sine rule :-  sine rule relates the sine of each angle to length of        opposite side.

[tex]\frac{Sin A}{a} =\frac{SinB}{b} =\frac{SinC}{c}[/tex]

Where, A, B , C are the angles of the triangles and a, b, c are the side of the triangles

Then, Apply it:

[tex]\frac{Sin83}{10}=\frac{SinZ}{9}[/tex]

Using cross multiplication:-

Sin 83 × 9 = Sin Z × 10

Sin Z = (Sin 83 × 9)/10

Sin Z = 0.893

∠Z = [tex]Sin^-^1(0.893)[/tex]

∠Z =  63.2°

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Rhys will have to pay £6 interest after the first year.

To calculate the interest, we can use the formula:

Interest = Principal x Rate

Where:

Principal = £200

Rate = 3% = 0.03 (in decimal form)

So, the interest Rhys will have to pay after the first year is:

Interest = £200 x 0.03 = £6

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To simplify the expression 2/(x-1) - x/(x^2+1) - 1/(x+1), we need to find a common denominator for all three terms.

The common denominator for the three terms is (x-1)(x^2+1)(x+1). We can rewrite each term using this denominator as follows:

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

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

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

Now, we can combine the three terms by subtracting the second and third terms from the first, and simplify as follows:

2*(x^2+1)*(x+1) / [(x-1)(x^2+1)(x+1)] - x*(x-1)*(x+1) / [(x-1)(x^2+1)(x+1)] - (x^2+1) / [(x-1)(x^2+1)(x+1)]

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

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

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

Therefore, the simplified expression is (x^3 + 2x - 1) / [(x-1)(x^2+1)(x+1)].

Answer:

Step-by-step explanation:5.7

Answer:

V=1/3*pi*radius(to the power of two)*height

Answer: 14 in squared

t < 8 is the inequality to solve the variable "t" as per the given data.

The cost C (in dollars) of renting the room for t hours can be expressed as:

C(t) = 4t + 18

To spend less than $50, we can set up an inequality:

C(t) < 50

Substituting the expression for C(t), we have:

4t + 18 < 50

Subtracting 18 from both sides, we get:

4t < 32

Dividing both sides by 4, we obtain:

t < 8

Therefore, the chemistry club could rent the meeting room for any number of hours less than 8 in order to spend less than $50.

The inequality solved for t is:

t < 8

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The quadratic value equation of triangle area is solved and x = 12 m

Given data ,

Let the triangle be represented as ΔABC

Now , the height of the triangle is h = x m

The base of the triangle is = ( x + 6 ) m

Area of triangle = ( 1/2 ) base x height

A = 108 m² = ( 1/2 ) ( x ) ( x + 6 )

216 = x² + 6x

Subtracting 216 on both sides , we get

x² + 6x - 216 = 0

On factorizing the equation , we get

( x + 18 ) ( x - 12 ) = 0

So , the solution is x = 12 m

Hence , the height of the triangle x = 12 m

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The slope intercept form of the given equation is,

y = 5x - 1/4

Given an equation of a line as,

5x - y = 1/4

We have to write the equation of the line in slope intercept form.

Equation of the line in slope intercept form is,

y = mx + c

Here m is the slope and c is the y intercept.

Rearranging the given equation,

5x = y + 1/4

5x - 1/4 = y

or,

y = 5x - 1/4

Hence the slope intercept form is y = 5x - 1/4.

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Answer:

To find out how much interest Jenny earns in one year, we can use the formula:

interest = principal x rate x time

where:

the principal is the amount of money in the savings account

rate is the interest rate per year, expressed as a decimal

time is the length of time the money is in the account, in years

In this case, Jenny's principal is $1200, the rate is 3% or 0.03 as a decimal, and the time is 1 year. Plugging these values into the formula, we get:

interest = 1200 x 0.03 x 1

interest = $36

Therefore, Jenny earns $36 in interest in one year.

Answer: h(-8) = 21

Step-by-step explanation:

Answer: 21

Step-by-step explanation:

You equation is h(y)  but you need to find h(-8) which means substitute y for -8 into equation

h(-8)= 4|(-8)+2| -3             do what's inside absolute value first

=4|-6| -3                           for absolute value, take positive of number

=4(6)-3                             multiply

=24-3                               subtract

=21

Answer:for the one in the middle, that one is 5.0, for the one in the top, that one is 7, and for the bottom one, that one is 50

Step-by-step explanation:

All the points of the vertices are in the shaded region

From the question, we have the following parameters that can be used in our computation:

s ≤ 10 - 0.5t

s ≥ 5 - t

s ≤ 20 - t

s ≥ 0

t ≥ 0

Next, we plot the graphs of the above inequalities

In the graph, we can see that all the points are in the shaded region

Hence, none of the points is not one of the vertices (s, t) of the shaded region

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Answer:

[tex](4 \times {10}^{4} ) \times (2.5 \times {10}^{6} )[/tex]

[tex] = 10 \times {10}^{10} = {10}^{11} [/tex]

Answer:

There is no square root function mentioned to compare with, but assuming you meant the parent square root function y = sqrt(x), the correct answer is:

A.) "stretched by a factor of 2, reflected over the x-axis, and translated 9 units right"

To see this, note that the equation y = 4x + 38 can be rewritten as y - 38 = 4(x - (-9.5)). This means that the graph of y = 4x + 38 is a transformation of the parent function y = sqrt(x) as follows:

So the correct answer is (a) stretched by a factor of 2, reflected over the x-axis, and translated 9 units right.

Answer:

answer is (a) stretched by a factor of 2, reflected over the x-axis, and translated 9 units right.

Step-by-step explanation: