The radius r of a sphere is increasing at a rate of 3 inches per minute. a) Find the rates of changes of the volume when r = 9 inches and r = 36 inches. b) Explain why the rate of change of the volume of the sphere is not constant even though dr/dt is constant.

To determine the approximate length of the underground pipeline needed from the lake to the water level in Apache Durham Park, we would know the distance between the lake and park, as well as the specific path that the pipeline would take from the lake to the park.

Length is a physical or conceptual measurement of the extent of something from one end to the other.

It refers to the distance between two points or the size of an object or entity in the direction of its longest dimension. In mathematics and geometry, length is a fundamental concept used to describe the size and shape of geometric figures and objects.

It is measured in units such as meters, feet, inches, or centimeters

Assuming that we have this information, we can use the distance between the two points as the approximate length of the pipeline. To calculate this distance, we can use the distance formula:

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

where (x1, y1) represents the coordinates of the lake and (x2, y2) represents the coordinates of Apache Durham Park.

Once we have the distance between the two points, we can round it to the nearest meter to get the approximate length of the pipeline needed.

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We would know the distance between the lake and park, as well as the precise route that the pipeline would travel from the lake to the park, if we knew the approximate length of the underground pipeline required from the lake to the water level in Apache Durham Park.

A gauge of length is one that shows how far something extends from one end to the other.

It describes the separation of two points or the size of an item or entity measured along its longest axis. Length is a basic notion in mathematics and geometry that is used to describe the size and shape of geometric figures and objects.

Its dimensions are expressed in terms of meters, feet, inches, or millimeters.

Assuming we have this knowledge, we can use the distance between the two locations to estimate the pipeline's length. We can use the following algorithm to determine this distance:

[tex]d=\sqrt{(x_{2}-x_{1}) ^{2}+(y_{2}-y_{1}) ^{2} }[/tex]

where [tex](x_{1},y_{1} )[/tex] stands for the lake's coordinates and [tex](x_{2} ,y_{2} )[/tex] for Apache Durham Park's coordinates.

Once we know how far apart the two locations are, we can round it to the closest meter to determine how long the pipeline should be roughly.

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The complete question is as follows:

Option D, which has 92° and 66° curves, is the correct choice. The total of these angles is 158°, which is needed for the final two angles.

The triangle with edges in the ratio $3:4:5 is therefore "yes" a right-angled triangle. Note: We can only apply Pythagoras' Theorem to right-angled triangles; it cannot be applied to any other triangles.

Let's add the given angle measures: 52° + 70° + 44° + 36° = 202°. This means that the sum of the measures of the remaining two angles in the two triangles is:

360° - 202° = 158°

As a result, since both options A and B have one angle that is higher than 90 degrees, we can rule them out.

Option C has angles of 88° and 68°, which add up to 156°.

Since the highest sum of two acute angles is 180°, the other two angles would have to have a sum of 204°, which is not feasible.

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The average rate of change of the function f(x) over the interval [-2, -9] is -6.

To determine the average rate of change of the function f(x) over the interval [-2, -9], we need to find the slope of the secant line that connects the points (-2, f(-2)) and (-9, f(-9)).

We first find the values of f(-2) and f(-9):

f(-2) = (-2)² + 5(-2) + 14 = 4 - 10 + 14 = 8

f(-9) = (-9)² + 5(-9) + 14 = 81 - 45 + 14 = 50

So, the two points are (-2, 8) and (-9, 50).

The slope of the secant line between these two points is:

slope = (f(-9) - f(-2)) / (-9 - (-2)) = (50 - 8) / (-9 + 2) = 42 / -7 = -6

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The ratio of ribbon to bouquets is 2:5, meaning for every 2 meters of ribbon, 5 bouquets can be made. For every 7 bouquets, 3 meters of ribbon are needed, meaning the ratio is 3:7.

A ratio is a comparison of two numbers or quantities that are related to each other in some way. It is the relationship between the sizes of two or more things, expressed as the quotient of one divided by the other.

Ratios can be used to compare different aspects of a set of data, such as the ratio of boys to girls in a classroom, or the ratio of red cars to blue cars on a street. They are also used in many real-world applications, such as finance, engineering, and science.

Ribbon to bouquets ratio is 2:5, which means that for every 2 units of ribbon, there are 5 bouquets.

With a ribbon to bouquet ratio of 1:3, the number of bouquets is three times the amount of ribbon.

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The cone and cylinder above have the same radius and height and the volume of the cone is 171 cubic inches.

Volume of a three-dimensional shape is the space occupied by the shape.

Given that,

Radius and height of the cylinder and the cone are same.

Let r and h be the radius and height of each of the cylinder and the cone.

Volume of the Cone = 1/3 π r² h

Volume of the cone = We have, volume of the cylinder is 57 cubic inches.

1/3 π r² h = 57

Multiplying both sides by 3,

π r² h = 57 × 3

π r² h = 171 cubic inches π r² h

Hence the volume of the cone is 171 cubic inches

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The magnitude of the electric field between the plates is 3.86 × [tex]10^7[/tex]N/C. The sign indicates the direction of the electric field, which depends on the sign of the charge on the plates.

The strength of the electric field between the plates can be calculated using the formula:

E = σ/ε0

where σ is the surface charge density, ε0 is the permittivity of free space, and E is the electric field strength.

The surface charge density is given by:

σ = Q/A

where Q is the charge on each plate and A is the area of each plate.

Substituting the given values, we get:

σ = ±0.0000470 C / 0.138 [tex]m^2[/tex] = ±0.000341 C/[tex]m^2[/tex]

The permittivity of free space is:

ε0 = 8.85 × [tex]10^{-12[/tex] F/m

Substituting the values, we get:

E = σ/ε0 = ±0.000341 C/[tex]m^2[/tex] / 8.85 × [tex]10^{-12[/tex] F/m = ±3.86 × [tex]10^7[/tex] N/C

The magnitude of the electric field between the plates is 3.86 × [tex]10^7[/tex]N/C. The sign indicates the direction of the electric field, which depends on the sign of the charge on the plates.

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From the figure, the value of x is given as 17 units

In mathematics, a ratio shows how many times one number contains another.

The ratio of the sides of the figure is as follows

3x-7/x+5 = 2/1

cross and multiply to have 3x - 7 = 2(x+5)

opening the brackets to have

3x -7 = 2x +10

collecting like terms to have

3x-2x = 10+7

x = 17 units

In conclusion, the value of x is 17 units

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The expression of the function using the specified formats are presented as follows;

Please find attached the required graph of the function f(x) = -2·(x - 4)² + 1 created with MS Excel

The table of values for the function can be presented as follows;

[tex]\begin{tabular}{|c|c|c|} \cline{1-2}x& f(x) \\ \cline{1-2}1 & -17\\ \cline{1-2} 2 & -7 \\ \cline{1-2} 3 & -1 \\ \cline{1-2} 4 & 1 \\ \cline{1-2} 5 & -1\\\cline{1-2} 6 & -7 \\ \cline{1-2} 7 & -17 \\ \cline{1-2}\end{tabular}[/tex]

The mapping diagram can be presented as follows;

x [tex]{}[/tex]  f(x)

1 → -17

2 → -7

3 → -1

4 → 1

5 → -1

6 → -7

7 → -17

The function expressed using set notation can be presented as follows;

f(x) = {-17, -7, -1, 1, -1, -7, -17} where x ∈ {1, 2, 3, 4, 5, 6, 7}

A function is a rule that assigns a unique value for the output for each input value.

The specified function is; f(x) = -2·(x - 4)² + 1

Please find attached the graph of the parabola, created with MS Excel, which is a parabola that opens downwards, with a vertex of (4, 1)

The table of values can be presented as follows;

 x  |  f(x)

-----|------

 1  |  -17

 2  |   -7

 3  |   -1

 4  |    1

 5  |   -1

 6  |   -7

 7  |  -17

Mapping diagram:

A mapping diagram is a diagram that illustrates how an element of a set is paired with elements in another set.

The mapping diagram for the function can be made to show how the x-values are mapped to the f(x) values as follows;

1  →  -17

2  →   -7

3  →   -1

4  →    1

5  →   -1

6  →   -7

7  →  -17

Set notation;

The function, f(x) = -2·(x - 4)² + 1 can be expressed using set notation for the interval 1 ≤ x ≤ 7, which is the set of integers between 1 and 7, inclusive, which is presented as follows;

Set of x-values; {1, 2, 3, 4, 5, 6, 7}

Set of f(x) values is therefore; {-17, -7, -1, 1, -1, -7, -17}

Therefore, we get f(x) = -2·(x - 4)² + 1 expressed using set notation can be presented as follows;

f(x) = {-17, -7, -1, 1, -1, -7, -17}, where, x ∈ {1, 2, 3, 4, 5, 6, 7}

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To find the approximate population of China, we need to multiply the population of Vatican City by 1.75 × 106:

Population of China = Population of Vatican City × 1.75 × 106

Population of China = 800 × 1.75 × 106

Population of China = 1.4 × 109

Therefore, the approximate population of China is 1.4 × 109, which is option D.

Therefore, the value of x in each triangle is approximately 5.4 and 11.1, respectively.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It involves the study of trigonometric functions such as sine, cosine, and tangent, which are used to relate the angles of a triangle to the lengths of its sides. Trigonometry has a wide range of applications in fields such as physics, engineering, navigation, surveying, and astronomy. For example, trigonometry is used in navigation to calculate the angles between a ship and a lighthouse, or between two ships, in order to determine their relative positions. In engineering, trigonometry is used to calculate the forces acting on structures such as bridges and buildings, and to design and build machines that rely on the principles of motion and energy.

Here,

For the first triangle, we have:

Using the sine ratio,

sin(36) = x / 9

Solving for x, we get:

x = 9 sin(36)

≈ 5.4

Rounding to the nearest tenth, we get:

x ≈ 5.4

For the second triangle, we have:

Using the tangent ratio,

tan(27) = x / 21

Solving for x, we get:

x = 21 tan(27)

≈ 11.1

Rounding to the nearest tenth, we get:

x ≈ 11.1

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The Solutions to 12m ² - 4mn-5n² are:

m = n/3 or m = -5n/3

To solve the quadratic equation 12m² - 4mn - 5n² = 0, we can use the quadratic formula:

m = (-b ± √(b^2 - 4ac)) / 2a

where a = 12, b = -4n, and c = -5n².

Substituting these values into the formula, we get:

m = (-(-4n) ± √((-4n)^2 - 4(12)(-5n²))) / 2(12)

Simplifying:

m = (4n ± √(16n² + 240n²)) / 24

m = (4n ± √(256n²)) / 24

m = (4n ± 16n) / 24

So the solutions are:

m = n/3 or m = -5n/3

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196 children's tickets and 328 adult tickets were sold.

What is a system of equations?

A finite set of equations for which common solutions are sought is referred to as a set of simultaneous equations, often known as a system of equations or an equation system.

Here, we have

Given: One night a theater sold 524 movie tickets. An adult's ticket costs $6.50 and a child's ticket cost $3.50. In all, $2881 was taken in.

Let the amount of the child's ticket be x

Let the amount of adults tickets be y

If the total number of tickets sold is 524 movie tickets, then;

x + y  = 524

x = 524 - y...(1)

If an adult ticket cost $6.50 and a child’s ticket cost $3.5 with a total of $2881 in all, then;

3.5x +  6.5 y = 2881

35x + 65y = 28810 ...(2)

Substitute equation 1 into 2:

35x + 65y = 2881

35(542-y) + 65y = 28810

18970 - 35y + 65y = 28810

30y = 9840

y = 328

Put the value of y in equation (1) and we get

x = 524 - 328

x = 196

Hence,  196 children's tickets and 328 adult tickets were sold.

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The distance of the fire from the road is d ≈ 6.15 miles + 2.94 miles ≈ 9.09 miles.

The two expressions for d give slightly different values. This is due to the fact that we rounded the values of x and y in our calculations.

In problem 9, we can use trigonometry to find the distance, d, of the fire from the road. Let x be the distance from Star Point Ranger Station to the fire, and let y be the distance from Twin Pines Ranger Station to the fire. Then, we have:

tan(34°) = d/x

tan(14°) = d/y

Multiplying both sides of each equation by the respective denominator and simplifying, we get:

d = x tan(34°)

d = y tan(14°)

Since the two expressions for d are both equal, we can set them equal to each other and solve for x and y:

x tan(34°) = y tan(14°)

x = (y tan(14°))/tan(34°)

Substituting the value we found for y in problem 7, which was y = 6.15 miles, we get:

x = (6.15 miles) * tan(14°) / tan(34°) ≈ 2.94 miles

Therefore, the distance of the fire from the road is d ≈ 6.15 miles + 2.94 miles ≈ 9.09 miles.

Now, let's check our answers using both expressions for d:

d = x tan(34°) ≈ 2.94 miles * tan(34°) ≈ 2.94 miles * 0.704 = 2.07 miles

d = y tan(14°) ≈ 6.15 miles * tan(14°) ≈ 6.15 miles * 0.249 = 1.53 miles

As we can see, the two expressions for d give slightly different values. This is due to the fact that we rounded the values of x and y in our calculations. If we use the exact values, we would get slightly different values for d, but they would still be very close.

In real life, it is important to be as accurate as possible when dealing with emergencies such as fires. However, in this case, the difference between the two values of d is relatively small, so it may not have a significant impact on the response to the fire. Nevertheless, it is important to use the most accurate values possible to ensure the safety of those involved.

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Answer:

2.1

Step-by-step explanation:all side lengths are the same (duh) use 4 and 3 (4 bc there is 4 sides). 4 as the diagonal support

4²=16

3²=9

16-9=7 then use square root for 7 the square root of 7 is 2.6457513111 or 2.1

(a) The distance that the tip of the hour hand covers in twelve hours is 28.29 cm.

(b)The distance that the tip of the minute hand covers in one hour is 34.57 cm.

(c) The tip of the minute hand covers a longer distance than the tip of the hour hand by 6.28 cm.

(a) To find the distance that the tip of the hour hand covers in twelve hours, we need to calculate the circumference of the circle that the tip of the hour hand traces in twelve hours. The circumference is given by:

C = 2πr

where r is the length of the hour hand

C = 2 x 22/7 x 4.5

C = 28.29 cm

(b) Also, the distance that the tip of the minute hand covers in one hour is:

C = 2 x 22/7 x 5.5

C = 34.57 cm

(c) We can see that the distance covered by the minute hand in one hour (34.57 cm) is greater than the distance covered by the hour hand in twelve hours (28.29 cm).

difference = 34.57 cm - 28.29 cm = 6.28 cm

Thus, the tip of the minute hand covers a longer distance than the tip of the hour hand by 6.28 cm.

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Answer:

Step-by-step explanation:
[tex]168yd^{2}[/tex]

Answer:

168 yd²

Step-by-step explanation:

4 × 7 yd × 12 yd / 2 = 168 yd²

The volume of the sphere is increasing at a rate of 209,439.51 mm³/s when the diameter is 100 mm

The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing (in mm3/s) when the diameter is 100 mm? (Round your answer to two decimal places).Formula to calculate the volume of a sphere = (4/3) × π × r³where r = radius of the sphere, π = pi = 3.14, d = diameter of the sphere. The diameter of the sphere, d = 100 mm.So, the radius of the sphere, r = d/2 = 100/2 = 50 mm.

Now, we need to find the rate of change of the volume of the sphere when the radius of the sphere is increasing at a rate of 4 mm/s.We know that the volume of the sphere is given by V = (4/3) × π × r³.We have to differentiate the above formula with respect to time (t).dV/dt = d/dt [(4/3) × π × r³]dV/dt = (4/3) × π × 3r² × dr/dt, substitute r = 50 mm and dr/dt = 4 mm/s in the above equation to find dV/dt.dV/dt = (4/3) × π × 3(50)² × 4dV/dt = 209,439.51 mm³/sTherefore, the volume of the sphere is increasing at a rate of 209,439.51 mm³/s .

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Step-by-step explanation:

La première baisse de prix est de 30%, ce qui signifie que le prix du vêtement est maintenant de 70% de son prix initial :

Prix après la première baisse = 100€ - (30% x 100€) = 70€

Ensuite, le prix est réduit à nouveau de 20%. Cela signifie que le prix final est de 80% du prix après la première baisse :

Prix après la deuxième baisse = 70€ - (20% x 70€) = 56€

Le pourcentage de réduction totale correspond donc à la différence entre le prix initial et le prix final, exprimé en pourcentage du prix initial :

Pourcentage de réduction totale = ((100€ - 56€) / 100€) x 100% = 44%

Le vêtement a donc subi une réduction de 44% au total après les deux baisses successives de prix.

Based on the table provided, the statement that is true is of all the cats adopted, about 66% were adopted in a month or less.

To calculate a percentage, you need to divide a part by the whole and then multiply the result by 100. The formula for calculating a percentage is:

Percentage = (Part / Whole) x 100

Alternatively, if you have the percentage and the whole, you can use the following formula to find the part:

Part = (Percentage / 100) x Whole

Based on this, let's prove the true statement:

Total of animals: 58 cats

Total of cats adopted in a month or less: 38

Percentage = 38/ 58 x 100 = 65.5% which can be rounded as 66%

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Step-by-step explanation:

12 lemons   / 4 people      *     30 people =   90 lemons

1 liter/ 12 lemons   *  90 lemons =  7.5 liters of water

Answer:

Step-by-step explanation:

Answer:

yes

Step-by-step explanation:

To solve this system of equations using the elimination method, we need to eliminate one of the variables. One way to do this is to add the two equations together in a way that will eliminate one of the variables.

First, we can multiply the second equation by -1 to change the sign of all its terms:

-6x + 5y = 1

-6x - 4y = 10

Now we can add the two equations together to eliminate x:

( -6x + 5y ) + ( -6x - 4y ) = 1 + 10

Simplifying the left side and the right side:

-12x + y = 11

Now we have one equation with one variable, y. We can solve for y by isolating it on one side of the equation:

-12x + y = 11

y = 12x + 11

We can substitute this expression for y into either of the original equations to solve for x. Let's use the first equation:

-6x + 5y = 1

-6x + 5(12x + 11) = 1

Simplifying the left side:

54x + 55 = 1

Subtracting 55 from both sides:

54x = -54

Dividing both sides by 54:

x = -1

So the solution to the system of equations is x = -1 and y = 1. We can check this solution by substituting these values into both original equations and verifying that they are true.

As a result, the difference is 837 grammes between the cones' respective masses.

Describe density.

Mass per unit volume is measured using density. It is described as a substance's mass per unit volume. The following is the density formula:

Mass / Volume equals density.

So, The formula m = V, where m is the mass, is the density, and V is the volume, can be used to determine each cone's mass.

The formula below can be used to determine each cone's mass:

- The mass of a steel cone weighs 1298.21 grams or 7.75 grams per cubic centimeter.

- The granite cone weighs 460.76 grams, or 2.75 grams per cubic centimeter.

- Mass of steel cone = 7.75 g/cm³ × 167.55 cm³ = 1298.21 grams

- Mass of granite cone = 2.75 g/cm³ × 167.55 cm³ = 460.76 grams

for difference of masses = 1298.2 grams - 460.76  grams = 837 grams

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[tex](\stackrel{x_1}{7}~,~\stackrel{y_1}{p})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{p}}}{\underset{\textit{\large run}} {\underset{x_2}{8}-\underset{x_1}{7}}} ~~ = ~~\stackrel{\stackrel{\textit{\small slope}}{\downarrow }}{ -5 }\implies \cfrac{4-p}{1}=-5 \\\\\\ 4-p=-5\implies 4=-5+p\implies 9=p[/tex]

Answer:

8.

Step-by-step explanation:

That would be by a factor of 2^3 = 8.

The slope and y-intercept of the following equation are:

Slope = 4.

y-intercept = 1.

In Mathematics, the slope-intercept form of the equation of a straight line is represented by this mathematical expression;

y = mx + c

Where:

Based on the information provided above, an equation that models the line is represented by this mathematical equation;

y = mx + c

y = 11x + 41

By comparison, we have the following:

mx = 4x

Slope, m = 4.

Initial number or y-intercept, c = 1.

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Answer:22.1%

Step-by-step explanation:

These are the polynomial equations:

A = x^ (1/4) - ∛x + 4√x - 8x + 16

B = 3x² - 5x⁴ + 2x - 12

C = x³ - 7x² + 9x - 5x⁴ - 20

D = x⁵ - 5x⁴ + 4x³ - 3x² + 2x - 1

These are the non-polynomial equations:

E = 4/x⁴ + 3/x³ - 2/x² - 1

F = x⁻⁵ - 5x⁻⁴ + 4x⁻³ - 3x⁻² + 2x⁻¹ - 1

Polynomials are mathematical expressions that only use addition, subtraction, multiplication, and non-negative exponentiation of the variables, along with coefficients (constants that multiply with the variables), coefficients, and constants.

Some of the elements of an equation are coefficients, variables, operators, constants, terms, expressions, and the equal to sign. An equation must always begin with the "=" sign and have terms on both sides.

Let the polynomial equations be represented by the following letters: A, B, C, D, E, and F.

In the equation, we can solve for other values to obtain:

Moreover, a polynomial equation is not an algebraic equation that has a negative exponent or an exponent that is fractional. Thus, negative exponent expressions are not polynomials.

A = x^ (1/4) - ∛x + 4√x - 8x + 16

This polynomial exists.

B = 3x² - 5x⁴ + 2x - 12

This polynomial exists.

C = x³ - 7x² + 9x - 5x⁴ - 20

This polynomial exists.

D = x⁵ - 5x⁴ + 4x³ - 3x² + 2x - 1

It is a polynomial.

E = 4/x⁴ + 3/x³ - 2/x² - 1

It is not a polynomial.

F = x⁻⁵ - 5x⁻⁴ + 4x⁻³ - 3x⁻² + 2x⁻¹ - 1

A polynomial is not what it is.

The polynomials are thus resolved.

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